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{{task|Mathematics}} [[File:taxi1729.png|600px||right]]
A [[wp:Hardy–Ramanujan number|taxicab number]] (the definition that is being used here) is a positive integer that can be expressed as the sum of two positive cubes in more than one way.
The first taxicab number is '''1729''', which is: ::: 13 + 123 and ::: 93 + 103.
Taxicab numbers are also known as: ::* taxi numbers ::* taxi-cab numbers ::* taxi cab numbers ::* Hardy-Ramanujan numbers
;Task:
- Compute and display the lowest 25 taxicab numbers (in numeric order, and in a human-readable format).
- For each of the taxicab numbers, show the number as well as it's constituent cubes.
;Extra credit
- Show the 2,000th taxicab number, and a half dozen more
;See also:
- [http://oeis.org/A001235 A001235 taxicab numbers] on The On-Line Encyclopedia of Integer Sequences.
- [http://mathworld.wolfram.com/Hardy-RamanujanNumber.html Hardy-Ramanujan Number] on MathWorld.
- [http://mathworld.wolfram.com/TaxicabNumber.html taxicab number] on MathWorld.
- [https://en.wikipedia.org/wiki/Taxicab_number taxicab number] on Wikipedia.
Befunge
This is quite slow in most interpreters, although a decent compiler should allow it to complete in a matter of seconds. Regardless of the speed, though, the range in a standard Befunge-93 implementation is limited to the first 64 numbers in the series, after which the 8-bit memory cells will overflow. That range could be extended in Befunge-98, but realistically you're not likely to wait that long for the results.
v+1$$<_v#!`**::+1g42$$_v#<!`**::+1g43\g43::<<v,,.g42,<
>004p:0>1+24p:24g\:24g>>1+:34p::**24g::**+-|p>9,,,14v,
,,,"^3 + ^3= ^3 + ^3".\,,,9"= ".:\_v#g40g43<^v,,,,.g<^
5+,$$$\1+:38*`#@_\::"~"1+:24p34p0\0>14p24g04^>,04g.,,5
{{out}}
1729 = 10 ^3 + 9 ^3 = 12 ^3 + 1 ^3
4104 = 15 ^3 + 9 ^3 = 16 ^3 + 2 ^3
13832 = 20 ^3 + 18 ^3 = 24 ^3 + 2 ^3
20683 = 24 ^3 + 19 ^3 = 27 ^3 + 10 ^3
32832 = 30 ^3 + 18 ^3 = 32 ^3 + 4 ^3
39312 = 33 ^3 + 15 ^3 = 34 ^3 + 2 ^3
40033 = 33 ^3 + 16 ^3 = 34 ^3 + 9 ^3
46683 = 30 ^3 + 27 ^3 = 36 ^3 + 3 ^3
64232 = 36 ^3 + 26 ^3 = 39 ^3 + 17 ^3
65728 = 33 ^3 + 31 ^3 = 40 ^3 + 12 ^3
110656 = 40 ^3 + 36 ^3 = 48 ^3 + 4 ^3
110808 = 45 ^3 + 27 ^3 = 48 ^3 + 6 ^3
134379 = 43 ^3 + 38 ^3 = 51 ^3 + 12 ^3
149389 = 50 ^3 + 29 ^3 = 53 ^3 + 8 ^3
165464 = 48 ^3 + 38 ^3 = 54 ^3 + 20 ^3
171288 = 54 ^3 + 24 ^3 = 55 ^3 + 17 ^3
195841 = 57 ^3 + 22 ^3 = 58 ^3 + 9 ^3
216027 = 59 ^3 + 22 ^3 = 60 ^3 + 3 ^3
216125 = 50 ^3 + 45 ^3 = 60 ^3 + 5 ^3
262656 = 60 ^3 + 36 ^3 = 64 ^3 + 8 ^3
314496 = 66 ^3 + 30 ^3 = 68 ^3 + 4 ^3
320264 = 66 ^3 + 32 ^3 = 68 ^3 + 18 ^3
327763 = 58 ^3 + 51 ^3 = 67 ^3 + 30 ^3
373464 = 60 ^3 + 54 ^3 = 72 ^3 + 6 ^3
402597 = 61 ^3 + 56 ^3 = 69 ^3 + 42 ^3
C
Using a priority queue to emit sum of two cubs in order. It's reasonably fast and doesn't use excessive amount of memory (the heap is only at 245 length upon the 2006th taxi).
#include <stdio.h> #include <stdlib.h> typedef unsigned long long xint; typedef unsigned uint; typedef struct { uint x, y; // x > y always xint value; } sum_t; xint *cube; uint n_cubes; sum_t *pq; uint pq_len, pq_cap; void add_cube(void) { uint x = n_cubes++; cube = realloc(cube, sizeof(xint) * (n_cubes + 1)); cube[n_cubes] = (xint) n_cubes*n_cubes*n_cubes; if (x < 2) return; // x = 0 or 1 is useless if (++pq_len >= pq_cap) { if (!(pq_cap *= 2)) pq_cap = 2; pq = realloc(pq, sizeof(*pq) * pq_cap); } sum_t tmp = (sum_t) { x, 1, cube[x] + 1 }; // upheap uint i, j; for (i = pq_len; i >= 1 && pq[j = i>>1].value > tmp.value; i = j) pq[i] = pq[j]; pq[i] = tmp; } void next_sum(void) { redo: while (!pq_len || pq[1].value >= cube[n_cubes]) add_cube(); sum_t tmp = pq[0] = pq[1]; // pq[0] always stores last seen value if (++tmp.y >= tmp.x) { // done with this x; throw it away tmp = pq[pq_len--]; if (!pq_len) goto redo; // refill empty heap } else tmp.value += cube[tmp.y] - cube[tmp.y-1]; uint i, j; // downheap for (i = 1; (j = i<<1) <= pq_len; pq[i] = pq[j], i = j) { if (j < pq_len && pq[j+1].value < pq[j].value) ++j; if (pq[j].value >= tmp.value) break; } pq[i] = tmp; } uint next_taxi(sum_t *hist) { do next_sum(); while (pq[0].value != pq[1].value); uint len = 1; hist[0] = pq[0]; do { hist[len++] = pq[1]; next_sum(); } while (pq[0].value == pq[1].value); return len; } int main(void) { uint i, l; sum_t x[10]; for (i = 1; i <= 2006; i++) { l = next_taxi(x); if (25 < i && i < 2000) continue; printf("%4u:%10llu", i, x[0].value); while (l--) printf(" = %4u^3 + %4u^3", x[l].x, x[l].y); putchar('\n'); } return 0; }
{{out}}
1: 1729 = 12^3 + 1^3 = 10^3 + 9^3
2: 4104 = 15^3 + 9^3 = 16^3 + 2^3
3: 13832 = 20^3 + 18^3 = 24^3 + 2^3
4: 20683 = 27^3 + 10^3 = 24^3 + 19^3
5: 32832 = 30^3 + 18^3 = 32^3 + 4^3
6: 39312 = 33^3 + 15^3 = 34^3 + 2^3
7: 40033 = 33^3 + 16^3 = 34^3 + 9^3
8: 46683 = 30^3 + 27^3 = 36^3 + 3^3
9: 64232 = 36^3 + 26^3 = 39^3 + 17^3
10: 65728 = 33^3 + 31^3 = 40^3 + 12^3
11: 110656 = 40^3 + 36^3 = 48^3 + 4^3
12: 110808 = 45^3 + 27^3 = 48^3 + 6^3
13: 134379 = 43^3 + 38^3 = 51^3 + 12^3
14: 149389 = 50^3 + 29^3 = 53^3 + 8^3
15: 165464 = 48^3 + 38^3 = 54^3 + 20^3
16: 171288 = 54^3 + 24^3 = 55^3 + 17^3
17: 195841 = 57^3 + 22^3 = 58^3 + 9^3
18: 216027 = 59^3 + 22^3 = 60^3 + 3^3
19: 216125 = 50^3 + 45^3 = 60^3 + 5^3
20: 262656 = 60^3 + 36^3 = 64^3 + 8^3
21: 314496 = 66^3 + 30^3 = 68^3 + 4^3
22: 320264 = 66^3 + 32^3 = 68^3 + 18^3
23: 327763 = 58^3 + 51^3 = 67^3 + 30^3
24: 373464 = 60^3 + 54^3 = 72^3 + 6^3
25: 402597 = 61^3 + 56^3 = 69^3 + 42^3
2000:1671816384 = 1168^3 + 428^3 = 944^3 + 940^3
2001:1672470592 = 1124^3 + 632^3 = 1187^3 + 29^3
2002:1673170856 = 1034^3 + 828^3 = 1164^3 + 458^3
2003:1675045225 = 1153^3 + 522^3 = 1081^3 + 744^3
2004:1675958167 = 1096^3 + 711^3 = 1159^3 + 492^3
2005:1676926719 = 1188^3 + 63^3 = 1095^3 + 714^3
2006:1677646971 = 990^3 + 891^3 = 1188^3 + 99^3
C#
using System; using System.Collections.Generic; using System.Linq; using System.Text; namespace TaxicabNumber { class Program { static void Main(string[] args) { IDictionary<long, IList<Tuple<int, int>>> taxicabNumbers = GetTaxicabNumbers(2006); PrintTaxicabNumbers(taxicabNumbers); Console.ReadKey(); } private static IDictionary<long, IList<Tuple<int, int>>> GetTaxicabNumbers(int length) { SortedList<long, IList<Tuple<int, int>>> sumsOfTwoCubes = new SortedList<long, IList<Tuple<int, int>>>(); for (int i = 1; i < int.MaxValue; i++) { for (int j = 1; j < int.MaxValue; j++) { long sum = (long)(Math.Pow((double)i, 3) + Math.Pow((double)j, 3)); if (!sumsOfTwoCubes.ContainsKey(sum)) { sumsOfTwoCubes.Add(sum, new List<Tuple<int, int>>()); } sumsOfTwoCubes[sum].Add(new Tuple<int, int>(i, j)); if (j >= i) { break; } } // Found that you need to keep going for a while after the length, because higher i values fill in gaps if (sumsOfTwoCubes.Count(t => t.Value.Count >= 2) >= length * 1.1) { break; } } IDictionary<long, IList<Tuple<int, int>>> values = (from t in sumsOfTwoCubes where t.Value.Count >= 2 select t) .Take(2006) .ToDictionary(u => u.Key, u => u.Value); return values; } private static void PrintTaxicabNumbers(IDictionary<long, IList<Tuple<int, int>>> values) { int i = 1; foreach (long taxicabNumber in values.Keys) { StringBuilder output = new StringBuilder().AppendFormat("{0,10}\t{1,4}", i, taxicabNumber); foreach (Tuple<int, int> numbers in values[taxicabNumber]) { output.AppendFormat("\t= {0}^3 + {1}^3", numbers.Item1, numbers.Item2); } if (i <= 25 || (i >= 2000 && i <= 2006)) { Console.WriteLine(output.ToString()); } i++; } } } }
Clojure
(ns test-project-intellij.core (:gen-class)) (defn cube [x] "Cube a number through triple multiplication" (* x x x)) (defn sum3 [[i j]] " [i j] -> i^3 + j^3" (+ (cube i) (cube j))) (defn next-pair [[i j]] " Generate next [i j] pair of sequence (producing lower triangle pairs) " (if (< j i) [i (inc j)] [(inc i) 1])) ;; Pair sequence generator [1 1] [2 1] [2 2] [3 1] [3 2] [3 3] ... (def pairs-seq (iterate next-pair [1 1])) (defn dict-inc [m pair] " Add pair to pair map m, with the key of the map based upon the cubic sum (sum3) and the value appends the pair " (update-in m [(sum3 pair)] (fnil #(conj % pair) []))) (defn enough? [m n-to-generate] " Checks if we have enough taxi numbers (i.e. if number in map >= count-needed " (->> m ; hash-map of sum of cube of numbers [key] and their pairs as value (filter #(if (> (count (second %)) 1) true false)) ; filter out ones which don't have more than 1 entry (count) ; count the item remaining (<= n-to-generate))) ; true iff count-needed is less or equal to the nubmer filtered (defn find-taxi-numbers [n-to-generate] " Generates 1st n-to-generate taxi numbers" (loop [m {} ; Hash-map containing cube of pairs (key) and set of pairs that produce sum (value) p pairs-seq ; select pairs from our pair sequence generator (i.e. [1 1] [2 1] [2 2] ...) num-tried 0 ; Since its expensve to count how many taxi numbers we have found check-after 1] ; we only check if we have enough numbers every time (num-tried equals check-after) ; num-tried increments by 1 each time we try the next pair and ; check-after doubles if we don't have enough taxi numbers (if (and (= num-tried check-after) (enough? m n-to-generate)) ; check if we found enough taxi numbers (sort-by first (into [] (filter #(> (count (second %)) 1) m))) ; sort the taxi numbers and this is the result (if (= num-tried check-after) ; Check if we need to increase our count between checking (recur (dict-inc m (first p)) (rest p) (inc num-tried) (* 2 check-after)) ; increased count between checking (recur (dict-inc m (first p)) (rest p) (inc num-tried) check-after))))) ; didn't increase the count ; Generate 1st 2006 taxi numbers (def result (find-taxi-numbers 2006)) ;; Show First 25 (defn show-result [n sample] " Prints one line of result " (print (format "%4d:%10d" n (first sample))) (doseq [q (second sample) :let [[i j] q]] (print (format " = %4d^3 + %4d^3" i j))) (println)) ; 1st 25 taxi numbers (doseq [n (range 1 26) :let [sample (nth result (dec n))]] (show-result n sample)) ; taxi numbers from 2000th to 2006th (doseq [n (range 2000 2007) :let [sample (nth result (dec n))]] (show-result n sample)) }
{{out}}
1: 1729 = 10^3 + 9^3 = 12^3 + 1^3
2: 4104 = 15^3 + 9^3 = 16^3 + 2^3
3: 13832 = 20^3 + 18^3 = 24^3 + 2^3
4: 20683 = 24^3 + 19^3 = 27^3 + 10^3
5: 32832 = 30^3 + 18^3 = 32^3 + 4^3
6: 39312 = 33^3 + 15^3 = 34^3 + 2^3
7: 40033 = 33^3 + 16^3 = 34^3 + 9^3
8: 46683 = 30^3 + 27^3 = 36^3 + 3^3
9: 64232 = 36^3 + 26^3 = 39^3 + 17^3
10: 65728 = 33^3 + 31^3 = 40^3 + 12^3
11: 110656 = 40^3 + 36^3 = 48^3 + 4^3
12: 110808 = 45^3 + 27^3 = 48^3 + 6^3
13: 134379 = 43^3 + 38^3 = 51^3 + 12^3
14: 149389 = 50^3 + 29^3 = 53^3 + 8^3
15: 165464 = 48^3 + 38^3 = 54^3 + 20^3
16: 171288 = 54^3 + 24^3 = 55^3 + 17^3
17: 195841 = 57^3 + 22^3 = 58^3 + 9^3
18: 216027 = 59^3 + 22^3 = 60^3 + 3^3
19: 216125 = 50^3 + 45^3 = 60^3 + 5^3
20: 262656 = 60^3 + 36^3 = 64^3 + 8^3
21: 314496 = 66^3 + 30^3 = 68^3 + 4^3
22: 320264 = 66^3 + 32^3 = 68^3 + 18^3
23: 327763 = 58^3 + 51^3 = 67^3 + 30^3
24: 373464 = 60^3 + 54^3 = 72^3 + 6^3
25: 402597 = 61^3 + 56^3 = 69^3 + 42^3
2000:1671816384 = 944^3 + 940^3 = 1168^3 + 428^3
2001:1672470592 = 1124^3 + 632^3 = 1187^3 + 29^3
2002:1673170856 = 1034^3 + 828^3 = 1164^3 + 458^3
2003:1675045225 = 1081^3 + 744^3 = 1153^3 + 522^3
2004:1675958167 = 1096^3 + 711^3 = 1159^3 + 492^3
2005:1676926719 = 1095^3 + 714^3 = 1188^3 + 63^3
2006:1677646971 = 990^3 + 891^3 = 1188^3 + 99^3
D
High Level Version
{{trans|Python}}
void main() /*@safe*/ { import std.stdio, std.range, std.algorithm, std.typecons, std.string; auto iCubes = iota(1u, 1201u).map!(x => tuple(x, x ^^ 3)); bool[Tuple!(uint, uint)][uint] sum2cubes; foreach (i, immutable i3; iCubes) foreach (j, immutable j3; iCubes[i .. $]) sum2cubes[i3 + j3][tuple(i, j)] = true; const taxis = sum2cubes.byKeyValue.filter!(p => p.value.length > 1) .array.schwartzSort!(p => p.key).release; foreach (/*immutable*/ const r; [[0, 25], [2000 - 1, 2000 + 6]]) { foreach (immutable i, const t; taxis[r[0] .. r[1]]) writefln("%4d: %10d =%-(%s =%)", i + r[0] + 1, t.key, t.value.keys.sort().map!q{"%4d^3 + %4d^3".format(a[])}); writeln; } }
{{out}}
1: 1729 = 1^3 + 12^3 = 9^3 + 10^3
2: 4104 = 2^3 + 16^3 = 9^3 + 15^3
3: 13832 = 2^3 + 24^3 = 18^3 + 20^3
4: 20683 = 10^3 + 27^3 = 19^3 + 24^3
5: 32832 = 4^3 + 32^3 = 18^3 + 30^3
6: 39312 = 2^3 + 34^3 = 15^3 + 33^3
7: 40033 = 9^3 + 34^3 = 16^3 + 33^3
8: 46683 = 3^3 + 36^3 = 27^3 + 30^3
9: 64232 = 17^3 + 39^3 = 26^3 + 36^3
10: 65728 = 12^3 + 40^3 = 31^3 + 33^3
11: 110656 = 4^3 + 48^3 = 36^3 + 40^3
12: 110808 = 6^3 + 48^3 = 27^3 + 45^3
13: 134379 = 12^3 + 51^3 = 38^3 + 43^3
14: 149389 = 8^3 + 53^3 = 29^3 + 50^3
15: 165464 = 20^3 + 54^3 = 38^3 + 48^3
16: 171288 = 17^3 + 55^3 = 24^3 + 54^3
17: 195841 = 9^3 + 58^3 = 22^3 + 57^3
18: 216027 = 3^3 + 60^3 = 22^3 + 59^3
19: 216125 = 5^3 + 60^3 = 45^3 + 50^3
20: 262656 = 8^3 + 64^3 = 36^3 + 60^3
21: 314496 = 4^3 + 68^3 = 30^3 + 66^3
22: 320264 = 18^3 + 68^3 = 32^3 + 66^3
23: 327763 = 30^3 + 67^3 = 51^3 + 58^3
24: 373464 = 6^3 + 72^3 = 54^3 + 60^3
25: 402597 = 42^3 + 69^3 = 56^3 + 61^3
2000: 1671816384 = 428^3 + 1168^3 = 940^3 + 944^3
2001: 1672470592 = 29^3 + 1187^3 = 632^3 + 1124^3
2002: 1673170856 = 458^3 + 1164^3 = 828^3 + 1034^3
2003: 1675045225 = 522^3 + 1153^3 = 744^3 + 1081^3
2004: 1675958167 = 492^3 + 1159^3 = 711^3 + 1096^3
2005: 1676926719 = 63^3 + 1188^3 = 714^3 + 1095^3
2006: 1677646971 = 99^3 + 1188^3 = 891^3 + 990^3
Run-time: about 2.9 seconds with dmd compiler.
===Heap-Based Version=== {{trans|Java}}
import std.stdio, std.string, std.container; struct CubeSum { ulong x, y, value; this(in ulong x_, in ulong y_) pure nothrow @safe @nogc { this.x = x_; this.y = y_; this.value = x_ ^^ 3 + y_ ^^ 3; } } final class Taxi { BinaryHeap!(Array!CubeSum, "a.value > b.value") pq; CubeSum last; ulong n = 0; this() { last = nextSum(); } CubeSum nextSum() { while (pq.empty || pq.front.value >= n ^^ 3) pq.insert(CubeSum(++n, 1)); auto s = pq.front; pq.removeFront; if (s.x > s.y + 1) pq.insert(CubeSum(s.x, s.y + 1)); return s; } CubeSum[] nextTaxi() { CubeSum s; typeof(return) train; while ((s = nextSum).value != last.value) last = s; train ~= last; do { train ~= s; } while ((s = nextSum).value == last.value); last = s; return train; } } void main() { auto taxi = new Taxi; foreach (immutable i; 1 .. 2007) { const t = taxi.nextTaxi; if (i > 25 && i < 2000) continue; writef("%4d: %10d", i, t[0].value); foreach (const s; t) writef(" = %4d^3 + %4d^3", s.x, s.y); writeln; } }
{{out}}
1: 1729 = 10^3 + 9^3 = 12^3 + 1^3
2: 4104 = 15^3 + 9^3 = 16^3 + 2^3
3: 13832 = 20^3 + 18^3 = 24^3 + 2^3
4: 20683 = 24^3 + 19^3 = 27^3 + 10^3
5: 32832 = 30^3 + 18^3 = 32^3 + 4^3
6: 39312 = 33^3 + 15^3 = 34^3 + 2^3
7: 40033 = 33^3 + 16^3 = 34^3 + 9^3
8: 46683 = 30^3 + 27^3 = 36^3 + 3^3
9: 64232 = 39^3 + 17^3 = 36^3 + 26^3
10: 65728 = 40^3 + 12^3 = 33^3 + 31^3
11: 110656 = 40^3 + 36^3 = 48^3 + 4^3
12: 110808 = 45^3 + 27^3 = 48^3 + 6^3
13: 134379 = 51^3 + 12^3 = 43^3 + 38^3
14: 149389 = 50^3 + 29^3 = 53^3 + 8^3
15: 165464 = 48^3 + 38^3 = 54^3 + 20^3
16: 171288 = 54^3 + 24^3 = 55^3 + 17^3
17: 195841 = 57^3 + 22^3 = 58^3 + 9^3
18: 216027 = 59^3 + 22^3 = 60^3 + 3^3
19: 216125 = 50^3 + 45^3 = 60^3 + 5^3
20: 262656 = 60^3 + 36^3 = 64^3 + 8^3
21: 314496 = 66^3 + 30^3 = 68^3 + 4^3
22: 320264 = 68^3 + 18^3 = 66^3 + 32^3
23: 327763 = 67^3 + 30^3 = 58^3 + 51^3
24: 373464 = 60^3 + 54^3 = 72^3 + 6^3
25: 402597 = 69^3 + 42^3 = 61^3 + 56^3
2000: 1671816384 = 1168^3 + 428^3 = 944^3 + 940^3
2001: 1672470592 = 1124^3 + 632^3 = 1187^3 + 29^3
2002: 1673170856 = 1164^3 + 458^3 = 1034^3 + 828^3
2003: 1675045225 = 1153^3 + 522^3 = 1081^3 + 744^3
2004: 1675958167 = 1159^3 + 492^3 = 1096^3 + 711^3
2005: 1676926719 = 1095^3 + 714^3 = 1188^3 + 63^3
2006: 1677646971 = 990^3 + 891^3 = 1188^3 + 99^3
Run-time: about 0.31 seconds with ldc2 compiler. It's faster than the Java solution.
===Low Level Heap-Based Version=== {{trans|C}}
struct Taxicabs { alias CubesSumT = uint; // Or ulong. static struct Sum { CubesSumT value; uint x, y; } // The cubes can be pre-computed if CubesSumT is a BigInt. private uint nCubes; private Sum[] pq; private uint pq_len; private void addCube() pure nothrow @safe { nCubes = nCubes ? nCubes + 1 : 2; if (nCubes < 2) return; // 0 or 1 is useless. pq_len++; if (pq_len >= pq.length) pq.length = (pq.length == 0) ? 2 : (pq.length * 2); immutable tmp = Sum(CubesSumT(nCubes - 2) ^^ 3 + 1, nCubes - 2, 1); // Upheap. uint i = pq_len; for (; i >= 1 && pq[i >> 1].value > tmp.value; i >>= 1) pq[i] = pq[i >> 1]; pq[i] = tmp; } private void nextSum() pure nothrow @safe { while (!pq_len || pq[1].value >= (nCubes - 1) ^^ 3) addCube(); Sum tmp = pq[0] = pq[1]; //pq[0] always stores last seen value. tmp.y++; if (tmp.y >= tmp.x) { // Done with this x; throw it away. tmp = pq[pq_len]; pq_len--; if (!pq_len) return nextSum(); // Refill empty heap. } else tmp.value += tmp.y ^^ 3 - (tmp.y - 1) ^^ 3; // Downheap. uint i = 1; while (true) { uint j = i << 1; if (j > pq_len) break; if (j < pq_len && pq[j + 1].value < pq[j].value) j++; if (pq[j].value >= tmp.value) break; pq[i] = pq[j]; i = j; } pq[i] = tmp; } Sum[] nextTaxi(size_t N)(ref Sum[N] hist) pure nothrow @safe { do { nextSum(); } while (pq[0].value != pq[1].value); uint len = 1; hist[0] = pq[0]; do { hist[len] = pq[1]; len++; nextSum(); } while (pq[0].value == pq[1].value); return hist[0 .. len]; } } void main() nothrow { import core.stdc.stdio; Taxicabs t; Taxicabs.Sum[3] x; foreach (immutable uint i; 1 .. 2007) { const triples = t.nextTaxi(x); if (i > 25 && i < 2000) continue; printf("%4u: %10lu", i, triples[0].value); foreach_reverse (const s; triples) printf(" = %4u^3 + %4u^3", s.x, s.y); '\n'.putchar; } }
{{out}}
1: 1729 = 12^3 + 1^3 = 10^3 + 9^3
2: 4104 = 15^3 + 9^3 = 16^3 + 2^3
3: 13832 = 20^3 + 18^3 = 24^3 + 2^3
4: 20683 = 27^3 + 10^3 = 24^3 + 19^3
5: 32832 = 30^3 + 18^3 = 32^3 + 4^3
6: 39312 = 33^3 + 15^3 = 34^3 + 2^3
7: 40033 = 33^3 + 16^3 = 34^3 + 9^3
8: 46683 = 30^3 + 27^3 = 36^3 + 3^3
9: 64232 = 36^3 + 26^3 = 39^3 + 17^3
10: 65728 = 33^3 + 31^3 = 40^3 + 12^3
11: 110656 = 40^3 + 36^3 = 48^3 + 4^3
12: 110808 = 45^3 + 27^3 = 48^3 + 6^3
13: 134379 = 43^3 + 38^3 = 51^3 + 12^3
14: 149389 = 50^3 + 29^3 = 53^3 + 8^3
15: 165464 = 48^3 + 38^3 = 54^3 + 20^3
16: 171288 = 54^3 + 24^3 = 55^3 + 17^3
17: 195841 = 57^3 + 22^3 = 58^3 + 9^3
18: 216027 = 59^3 + 22^3 = 60^3 + 3^3
19: 216125 = 50^3 + 45^3 = 60^3 + 5^3
20: 262656 = 60^3 + 36^3 = 64^3 + 8^3
21: 314496 = 66^3 + 30^3 = 68^3 + 4^3
22: 320264 = 66^3 + 32^3 = 68^3 + 18^3
23: 327763 = 58^3 + 51^3 = 67^3 + 30^3
24: 373464 = 60^3 + 54^3 = 72^3 + 6^3
25: 402597 = 61^3 + 56^3 = 69^3 + 42^3
2000: 1671816384 = 1168^3 + 428^3 = 944^3 + 940^3
2001: 1672470592 = 1124^3 + 632^3 = 1187^3 + 29^3
2002: 1673170856 = 1034^3 + 828^3 = 1164^3 + 458^3
2003: 1675045225 = 1153^3 + 522^3 = 1081^3 + 744^3
2004: 1675958167 = 1096^3 + 711^3 = 1159^3 + 492^3
2005: 1676926719 = 1188^3 + 63^3 = 1095^3 + 714^3
2006: 1677646971 = 990^3 + 891^3 = 1188^3 + 99^3
Run-time: about 0.08 seconds with ldc2 compiler.
DCL
We invoke external utility SORT which I suppose technically speaking is not a formal part of the language but is darn handy at times;
$ close /nolog sums_of_cubes
$ on control_y then $ goto clean
$ open /write sums_of_cubes sums_of_cubes.txt
$ i = 1
$ loop1:
$ write sys$output i
$ j = 1
$ loop2:
$ sum = i * i * i + j * j * j
$ if sum .lt. 0
$ then
$ write sys$output "overflow at ", j
$ goto next_i
$ endif
$ write sums_of_cubes f$fao( "!10SL,!10SL,!10SL", sum, i, j )
$ j = j + 1
$ if j .le. i then $ goto loop2
$ next_i:
$ i = i + 1
$ if i .le. 1289 then $ goto loop1 ! cube_root of 2^31-1
$ close sums_of_cubes
$ sort sums_of_cubes.txt sorted_sums_of_cubes.txt
$ close /nolog sorted_sums_of_cubes
$ open sorted_sums_of_cubes sorted_sums_of_cubes.txt
$ count = 0
$ read sorted_sums_of_cubes prev_prev_line ! need to detect when there are more than just 2 different sums, e.g. 456
$ prev_prev_sum = f$element( 0, ",", f$edit( prev_prev_line, "collapse" ))
$ read sorted_sums_of_cubes prev_line
$ prev_sum = f$element( 0,",", f$edit( prev_line, "collapse" ))
$ loop3:
$ read /end_of_file = done sorted_sums_of_cubes line
$ sum = f$element( 0, ",", f$edit( line, "collapse" ))
$ if sum .eqs. prev_sum
$ then
$ if sum .nes. prev_prev_sum then $ count = count + 1
$ int_sum = f$integer( sum )
$ i1 = f$integer( f$element( 1, ",", prev_line ))
$ j1 = f$integer( f$element( 2, ",", prev_line ))
$ i2 = f$integer( f$element( 1, ",", line ))
$ j2 = f$integer( f$element( 2, ",", line ))
$ if count .le. 25 .or. ( count .ge. 2000 .and. count .le. 2006 ) then -
$ write sys$output f$fao( "!4SL:!11SL =!5SL^3 +!5SL^3 =!5SL^3 +!5SL^3", count, int_sum, i1, j1, i2, j2 )
$ endif
$ prev_prev_line = prev_line
$ prev_prev_sum = prev_sum
$ prev_line = line
$ prev_sum = sum
$ goto loop3
$ done:
$ close sorted_sums_of_cubes
$ exit
$
$ clean:
$ close /nolog sorted_sums_of_cubes
$ close /nolog sums_of_cubes
{{out}}
$ @taxicab_numbers
1: 1729 = 10^3 + 9^3 = 12^3 + 1^3
2: 4104 = 15^3 + 9^3 = 16^3 + 2^3
3: 13832 = 20^3 + 18^3 = 24^3 + 2^3
4: 20683 = 24^3 + 19^3 = 27^3 + 10^3
5: 32832 = 30^3 + 18^3 = 32^3 + 4^3
6: 39312 = 33^3 + 15^3 = 34^3 + 2^3
7: 40033 = 33^3 + 16^3 = 34^3 + 9^3
8: 46683 = 30^3 + 27^3 = 36^3 + 3^3
9: 64232 = 36^3 + 26^3 = 39^3 + 17^3
10: 65728 = 33^3 + 31^3 = 40^3 + 12^3
11: 110656 = 40^3 + 36^3 = 48^3 + 4^3
12: 110808 = 45^3 + 27^3 = 48^3 + 6^3
13: 134379 = 43^3 + 38^3 = 51^3 + 12^3
14: 149389 = 50^3 + 29^3 = 53^3 + 8^3
15: 165464 = 48^3 + 38^3 = 54^3 + 20^3
16: 171288 = 54^3 + 24^3 = 55^3 + 17^3
17: 195841 = 57^3 + 22^3 = 58^3 + 9^3
18: 216027 = 59^3 + 22^3 = 60^3 + 3^3
19: 216125 = 50^3 + 45^3 = 60^3 + 5^3
20: 262656 = 60^3 + 36^3 = 64^3 + 8^3
21: 314496 = 66^3 + 30^3 = 68^3 + 4^3
22: 320264 = 66^3 + 32^3 = 68^3 + 18^3
23: 327763 = 58^3 + 51^3 = 67^3 + 30^3
24: 373464 = 60^3 + 54^3 = 72^3 + 6^3
25: 402597 = 61^3 + 56^3 = 69^3 + 42^3
2000: 1671816384 = 944^3 + 940^3 = 1168^3 + 428^3
2001: 1672470592 = 1124^3 + 632^3 = 1187^3 + 29^3
2002: 1673170856 = 1034^3 + 828^3 = 1164^3 + 458^3
2003: 1675045225 = 1081^3 + 744^3 = 1153^3 + 522^3
2004: 1675958167 = 1096^3 + 711^3 = 1159^3 + 492^3
2005: 1676926719 = 1095^3 + 714^3 = 1188^3 + 63^3
2006: 1677646971 = 990^3 + 891^3 = 1188^3 + 99^3
EchoLisp
Using the '''heap''' library, and a heap to store the taxicab numbers. For taxi tuples - decomposition in more than two sums - we use the '''group''' function which transforms a list ( 3 5 5 6 8 ...) into ((3) (5 5) (6) ...).
(require '(heap compile))
(define (scube a b) (+ (* a a a) (* b b b)))
(compile 'scube "-f") ; "-f" means : no bigint, no rational used
;; is n - a^3 a cube b^3?
;; if yes return b, else #f
(define (taxi? n a (b 0))
(set! b (cbrt (- n (* a a a)))) ;; cbrt is ∛
(when (and (< b a) (integer? b)) b))
(compile 'taxi? "-f")
#|-------------------
looking for taxis
--------------------|#
;; remove from heap until heap-top >= a
;; when twins are removed, it is a taxicab number : push it
;; at any time (top stack) = last removed
(define (clean-taxi H limit: a min-of-heap: htop)
(when (and htop (> a htop))
(when (!= (stack-top S) htop) (pop S))
(push S htop)
(heap-pop H)
(clean-taxi H a (heap-top H))))
(compile 'clean-taxi "-f")
;; loop on a and b, b <=a , until n taxicabs found
(define (taxicab (n 2100))
(for ((a (in-naturals)))
(clean-taxi H (* a a a) (heap-top H))
#:break (> (stack-length S) n)
(for ((b a))
(heap-push H (scube a b)))))
#|------------------
printing taxis
---------------------|#
;; string of all decompositions
(define (taxi->string i n)
(string-append (format "%d. %d " (1+ i) n)
(for/string ((a (cbrt n)))
#:when (taxi? n a)
(format " = %4d^3 + %4d^3" a (taxi? n a)))))
(define (taxi-print taxis (nfrom 0) (nto 26))
(for ((i (in-naturals nfrom)) (taxi (sublist taxis nfrom nto)))
(writeln (taxi->string i (first taxi)))))
{{out}}
(define S (stack 'S)) ;; to push taxis
(define H (make-heap < )) ;; make min heap of all scubes
(taxicab 2100)
(define taxis (group (stack->list S)))
(taxi-print taxis )
1. 1729 = 10^3 + 9^3 = 12^3 + 1^3
2. 4104 = 15^3 + 9^3 = 16^3 + 2^3
3. 13832 = 20^3 + 18^3 = 24^3 + 2^3
4. 20683 = 24^3 + 19^3 = 27^3 + 10^3
#| ... |#
24. 373464 = 60^3 + 54^3 = 72^3 + 6^3
25. 402597 = 61^3 + 56^3 = 69^3 + 42^3
26. 439101 = 69^3 + 48^3 = 76^3 + 5^3
(taxi-print taxis 1999 2006)
2000. 1671816384 = 944^3 + 940^3 = 1168^3 + 428^3
2001. 1672470592 = 1124^3 + 632^3 = 1187^3 + 29^3
2002. 1673170856 = 1034^3 + 828^3 = 1164^3 + 458^3
2003. 1675045225 = 1081^3 + 744^3 = 1153^3 + 522^3
2004. 1675958167 = 1096^3 + 711^3 = 1159^3 + 492^3
2005. 1676926719 = 1095^3 + 714^3 = 1188^3 + 63^3
2006. 1677646971 = 990^3 + 891^3 = 1188^3 + 99^3
;; extra bonus : print all taxis which are triplets
(define (taxi-tuples taxis (nfrom 0) (nto 2000))
(for ((i (in-naturals nfrom)) (taxi (sublist taxis nfrom nto)))
#:when (> (length taxi) 1) ;; filter for tuples is here
(writeln (taxi->string i (first taxi)))))
(taxi-tuples taxis)
455. 87539319 = 414^3 + 255^3 = 423^3 + 228^3 = 436^3 + 167^3
535. 119824488 = 428^3 + 346^3 = 492^3 + 90^3 = 493^3 + 11^3
588. 143604279 = 423^3 + 408^3 = 460^3 + 359^3 = 522^3 + 111^3
655. 175959000 = 525^3 + 315^3 = 552^3 + 198^3 = 560^3 + 70^3
888. 327763000 = 580^3 + 510^3 = 661^3 + 339^3 = 670^3 + 300^3
1299. 700314552 = 828^3 + 510^3 = 846^3 + 456^3 = 872^3 + 334^3
1398. 804360375 = 920^3 + 295^3 = 927^3 + 198^3 = 930^3 + 15^3
1515. 958595904 = 856^3 + 692^3 = 984^3 + 180^3 = 986^3 + 22^3
1660. 1148834232 = 846^3 + 816^3 = 920^3 + 718^3 = 1044^3 + 222^3
1837. 1407672000 = 1050^3 + 630^3 = 1104^3 + 396^3 = 1120^3 + 140^3
Elixir
defmodule Taxicab do def numbers(n \\ 1200) do (for i <- 1..n, j <- i..n, do: {i,j}) |> Enum.group_by(fn {i,j} -> i*i*i + j*j*j end) |> Enum.filter(fn {_,v} -> length(v)>1 end) |> Enum.sort end end nums = Taxicab.numbers |> Enum.with_index Enum.each(nums, fn {x,i} -> if i in 0..24 or i in 1999..2005 do IO.puts "#{i+1} : #{inspect x}" end end)
{{out}}
1 : {1729, [{9, 10}, {1, 12}]}
2 : {4104, [{9, 15}, {2, 16}]}
3 : {13832, [{18, 20}, {2, 24}]}
4 : {20683, [{19, 24}, {10, 27}]}
5 : {32832, [{18, 30}, {4, 32}]}
6 : {39312, [{15, 33}, {2, 34}]}
7 : {40033, [{16, 33}, {9, 34}]}
8 : {46683, [{27, 30}, {3, 36}]}
9 : {64232, [{26, 36}, {17, 39}]}
10 : {65728, [{31, 33}, {12, 40}]}
11 : {110656, [{36, 40}, {4, 48}]}
12 : {110808, [{27, 45}, {6, 48}]}
13 : {134379, [{38, 43}, {12, 51}]}
14 : {149389, [{29, 50}, {8, 53}]}
15 : {165464, [{38, 48}, {20, 54}]}
16 : {171288, [{24, 54}, {17, 55}]}
17 : {195841, [{22, 57}, {9, 58}]}
18 : {216027, [{22, 59}, {3, 60}]}
19 : {216125, [{45, 50}, {5, 60}]}
20 : {262656, [{36, 60}, {8, 64}]}
21 : {314496, [{30, 66}, {4, 68}]}
22 : {320264, [{32, 66}, {18, 68}]}
23 : {327763, [{51, 58}, {30, 67}]}
24 : {373464, [{54, 60}, {6, 72}]}
25 : {402597, [{56, 61}, {42, 69}]}
2000 : {1671816384, [{940, 944}, {428, 1168}]}
2001 : {1672470592, [{632, 1124}, {29, 1187}]}
2002 : {1673170856, [{828, 1034}, {458, 1164}]}
2003 : {1675045225, [{744, 1081}, {522, 1153}]}
2004 : {1675958167, [{711, 1096}, {492, 1159}]}
2005 : {1676926719, [{714, 1095}, {63, 1188}]}
2006 : {1677646971, [{891, 990}, {99, 1188}]}
FreeBASIC
' version 11-10-2016
' compile with: fbc -s console
' Brute force
' adopted from "Sorting algorithms/Shell" sort Task
Sub shellsort(s() As String)
' sort from lower bound to the highter bound
Dim As UInteger lb = LBound(s)
Dim As UInteger ub = UBound(s)
Dim As Integer done, i, inc = ub - lb
Do
inc = inc / 2.2
If inc < 1 Then inc = 1
Do
done = 0
For i = lb To ub - inc
If s(i) > s(i + inc) Then
Swap s(i), s(i + inc)
done = 1
End If
Next
Loop Until done = 0
Loop Until inc = 1
End Sub
' ------=< MAIN >=------
Dim As UInteger x, y, count, c, sum
Dim As UInteger cube(1290)
Dim As String result(), str1, str2, str3
Dim As String buf11 = Space(11), buf5 = Space(5)
ReDim result(900000) ' ~1291*1291\2
' set up the cubes
Print : Print " Calculate cubes"
For x = 1 To 1290
cube(x) = x*x*x
Next
' combine and store
Print : Print " Combine cubes"
For x = 1 To 1290
For y = x To 1290
sum = cube(x)+cube(y)
RSet buf11, Str(sum) : str1 = buf11
RSet buf5, Str(x) : str2 = buf5
RSet buf5, Str(y) : Str3 = buf5
result(count)=buf11 + " = " + str2 + " ^ 3 + " + str3 + " ^ 3"
count = count +1
Next
Next
count= count -1
ReDim Preserve result(count) ' trim the array
Print : Print " Sort (takes some time)"
shellsort(result()) ' sort
Print : Print " Find the Taxicab numbers"
c = 1 ' start at index 1
For x = 0 To count -1
' find sums that match
If Left(result(x), 11) = Left(result(x + 1), 11) Then
result(c) = result(x)
y = x +1
Do ' merge the other solution(s)
result(c) = result(c) + Mid(result(y), 12)
y = y +1
Loop Until Left(result(x), 11) <> Left(result(y), 11)
x = y -1 ' let x point to last match result
c = c +1
End If
Next
c = c -1
Print : Print " "; c; " Taxicab numbers found"
ReDim Preserve result(c) ' trim the array again
cls
Print : Print " Print first 25 numbers" : Print
For x = 1 To 25
Print result(x)
Next
Print : Print " The 2000th to the 2006th" : Print
For x = 2000 To 2006
Print result(x)
Next
' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
{{out}}
Print first 25 numbers
1729 = 1 ^ 3 + 12 ^ 3 = 9 ^ 3 + 10 ^ 3
4104 = 2 ^ 3 + 16 ^ 3 = 9 ^ 3 + 15 ^ 3
13832 = 2 ^ 3 + 24 ^ 3 = 18 ^ 3 + 20 ^ 3
20683 = 10 ^ 3 + 27 ^ 3 = 19 ^ 3 + 24 ^ 3
32832 = 4 ^ 3 + 32 ^ 3 = 18 ^ 3 + 30 ^ 3
39312 = 2 ^ 3 + 34 ^ 3 = 15 ^ 3 + 33 ^ 3
40033 = 9 ^ 3 + 34 ^ 3 = 16 ^ 3 + 33 ^ 3
46683 = 3 ^ 3 + 36 ^ 3 = 27 ^ 3 + 30 ^ 3
64232 = 17 ^ 3 + 39 ^ 3 = 26 ^ 3 + 36 ^ 3
65728 = 12 ^ 3 + 40 ^ 3 = 31 ^ 3 + 33 ^ 3
110656 = 4 ^ 3 + 48 ^ 3 = 36 ^ 3 + 40 ^ 3
110808 = 6 ^ 3 + 48 ^ 3 = 27 ^ 3 + 45 ^ 3
134379 = 12 ^ 3 + 51 ^ 3 = 38 ^ 3 + 43 ^ 3
149389 = 8 ^ 3 + 53 ^ 3 = 29 ^ 3 + 50 ^ 3
165464 = 20 ^ 3 + 54 ^ 3 = 38 ^ 3 + 48 ^ 3
171288 = 17 ^ 3 + 55 ^ 3 = 24 ^ 3 + 54 ^ 3
195841 = 9 ^ 3 + 58 ^ 3 = 22 ^ 3 + 57 ^ 3
216027 = 3 ^ 3 + 60 ^ 3 = 22 ^ 3 + 59 ^ 3
216125 = 5 ^ 3 + 60 ^ 3 = 45 ^ 3 + 50 ^ 3
262656 = 8 ^ 3 + 64 ^ 3 = 36 ^ 3 + 60 ^ 3
314496 = 4 ^ 3 + 68 ^ 3 = 30 ^ 3 + 66 ^ 3
320264 = 18 ^ 3 + 68 ^ 3 = 32 ^ 3 + 66 ^ 3
327763 = 30 ^ 3 + 67 ^ 3 = 51 ^ 3 + 58 ^ 3
373464 = 6 ^ 3 + 72 ^ 3 = 54 ^ 3 + 60 ^ 3
402597 = 42 ^ 3 + 69 ^ 3 = 56 ^ 3 + 61 ^ 3
The 2000th to the 2006th
1671816384 = 428 ^ 3 + 1168 ^ 3 = 940 ^ 3 + 944 ^ 3
1672470592 = 29 ^ 3 + 1187 ^ 3 = 632 ^ 3 + 1124 ^ 3
1673170856 = 458 ^ 3 + 1164 ^ 3 = 828 ^ 3 + 1034 ^ 3
1675045225 = 522 ^ 3 + 1153 ^ 3 = 744 ^ 3 + 1081 ^ 3
1675958167 = 492 ^ 3 + 1159 ^ 3 = 711 ^ 3 + 1096 ^ 3
1676926719 = 63 ^ 3 + 1188 ^ 3 = 714 ^ 3 + 1095 ^ 3
1677646971 = 99 ^ 3 + 1188 ^ 3 = 891 ^ 3 + 990 ^ 3
Go
package main import ( "container/heap" "fmt" "strings" ) type CubeSum struct { x, y uint16 value uint64 } func (c *CubeSum) fixvalue() { c.value = cubes[c.x] + cubes[c.y] } type CubeSumHeap []*CubeSum func (h CubeSumHeap) Len() int { return len(h) } func (h CubeSumHeap) Less(i, j int) bool { return h[i].value < h[j].value } func (h CubeSumHeap) Swap(i, j int) { h[i], h[j] = h[j], h[i] } func (h *CubeSumHeap) Push(x interface{}) { (*h) = append(*h, x.(*CubeSum)) } func (h *CubeSumHeap) Pop() interface{} { x := (*h)[len(*h)-1] *h = (*h)[:len(*h)-1] return x } type TaxicabGen struct { n int h CubeSumHeap } var cubes []uint64 // cubes[i] == i*i*i func cubesExtend(i int) { for n := uint64(len(cubes)); n <= uint64(i); n++ { cubes = append(cubes, n*n*n) } } func (g *TaxicabGen) min() CubeSum { for len(g.h) == 0 || g.h[0].value > cubes[g.n] { g.n++ cubesExtend(g.n) heap.Push(&g.h, &CubeSum{uint16(g.n), 1, cubes[g.n] + 1}) } // Note, we use g.h[0] to "peek" at the min heap entry. c := *(g.h[0]) if c.y+1 <= c.x { // Instead of Pop and Push we modify in place and fix. g.h[0].y++ g.h[0].fixvalue() heap.Fix(&g.h, 0) } else { heap.Pop(&g.h) } return c } // Originally this was just: type Taxicab [2]CubeSum // and we always returned two sums. Now we return all the sums. type Taxicab []CubeSum func (t Taxicab) String() string { var b strings.Builder fmt.Fprintf(&b, "%12d", t[0].value) for _, p := range t { fmt.Fprintf(&b, " =%5d³ +%5d³", p.x, p.y) } return b.String() } func (g *TaxicabGen) Next() Taxicab { a, b := g.min(), g.min() for a.value != b.value { a, b = b, g.min() } //return Taxicab{a,b} // Originally this just returned Taxicab{a,b} and we didn't look // further into the heap. Since we start by looking at the next // pair, that is okay until the first Taxicab number with four // ways of expressing the cube, which doesn't happen until the // 97,235th Taxicab: // 6963472309248 = 16630³ + 13322³ = 18072³ + 10200³ // = 18948³ + 5436³ = 19083³ + 2421³ // Now we return all ways so we need to peek into the heap. t := Taxicab{a, b} for g.h[0].value == b.value { t = append(t, g.min()) } return t } func main() { const ( low = 25 mid = 2e3 high = 4e4 ) var tg TaxicabGen firstn := 3 // To show the first triple, quadruple, etc for i := 1; i <= high+6; i++ { t := tg.Next() switch { case len(t) >= firstn: firstn++ fallthrough case i <= low || (mid <= i && i <= mid+6) || i >= high: //fmt.Printf("h:%-4d ", len(tg.h)) fmt.Printf("%5d: %v\n", i, t) } } }
{{out}}
1: 1729 = 12³ + 1³ = 10³ + 9³
2: 4104 = 16³ + 2³ = 15³ + 9³
3: 13832 = 24³ + 2³ = 20³ + 18³
4: 20683 = 27³ + 10³ = 24³ + 19³
5: 32832 = 32³ + 4³ = 30³ + 18³
6: 39312 = 34³ + 2³ = 33³ + 15³
7: 40033 = 34³ + 9³ = 33³ + 16³
8: 46683 = 36³ + 3³ = 30³ + 27³
9: 64232 = 36³ + 26³ = 39³ + 17³
10: 65728 = 40³ + 12³ = 33³ + 31³
11: 110656 = 48³ + 4³ = 40³ + 36³
12: 110808 = 48³ + 6³ = 45³ + 27³
13: 134379 = 51³ + 12³ = 43³ + 38³
14: 149389 = 53³ + 8³ = 50³ + 29³
15: 165464 = 54³ + 20³ = 48³ + 38³
16: 171288 = 55³ + 17³ = 54³ + 24³
17: 195841 = 58³ + 9³ = 57³ + 22³
18: 216027 = 60³ + 3³ = 59³ + 22³
19: 216125 = 60³ + 5³ = 50³ + 45³
20: 262656 = 64³ + 8³ = 60³ + 36³
21: 314496 = 68³ + 4³ = 66³ + 30³
22: 320264 = 66³ + 32³ = 68³ + 18³
23: 327763 = 58³ + 51³ = 67³ + 30³
24: 373464 = 72³ + 6³ = 60³ + 54³
25: 402597 = 69³ + 42³ = 61³ + 56³
455: 87539319 = 436³ + 167³ = 423³ + 228³ = 414³ + 255³
2000: 1671816384 = 1168³ + 428³ = 944³ + 940³
2001: 1672470592 = 1187³ + 29³ = 1124³ + 632³
2002: 1673170856 = 1164³ + 458³ = 1034³ + 828³
2003: 1675045225 = 1081³ + 744³ = 1153³ + 522³
2004: 1675958167 = 1096³ + 711³ = 1159³ + 492³
2005: 1676926719 = 1188³ + 63³ = 1095³ + 714³
2006: 1677646971 = 990³ + 891³ = 1188³ + 99³
40000: 976889700163 = 8659³ + 6894³ = 9891³ + 2098³
40001: 976942087381 = 7890³ + 7861³ = 8680³ + 6861³
40002: 976946344920 = 9476³ + 5014³ = 9798³ + 3312³
40003: 976962998375 = 9912³ + 1463³ = 8415³ + 7250³
40004: 976974757064 = 9365³ + 5379³ = 9131³ + 5997³
40005: 977025552984 = 9894³ + 2040³ = 9792³ + 3366³
40006: 977104161000 = 9465³ + 5055³ = 9920³ + 970³
Haskell
import Data.List (sortBy, groupBy, tails, transpose) import Data.Ord (comparing) -- TAXICAB NUMBERS ---------------------------------------------------- taxis :: Int -> [[(Int, ((Int, Int), (Int, Int)))]] taxis nCubes = filter ((> 1) . length) $ groupBy ((. fst) . (==) . fst) $ sortBy (comparing fst) [ (fst x + fst y, (x, y)) | (x:t) <- tails $ ((^ 3) >>= (,)) <$> [1 .. nCubes] , y <- t ] -- Taxicab numbers composed from first 1200 cubes xs :: [(Int, [(Int, ((Int, Int), (Int, Int)))])] xs = zip [1 ..] (taxis 1200) -- PRETTY PRINTING ---------------------------------------------------- taxiRow :: (Int, [(Int, ((Int, Int), (Int, Int)))]) -> [String] taxiRow (n, [(a, ((axc, axr), (ayc, ayr))), (b, ((bxc, bxr), (byc, byr)))]) = concat [ [show n, ". ", show a, " = "] , term axr axc " + " , term ayr ayc " or " , term bxr bxc " + " , term byr byc [] ] where term r c l = ["(", show r, "^3=", show c, ")", l] -- OUTPUT ------------------------------------------------------------- main :: IO () main = mapM_ putStrLn $ concat <$> transpose (((<$>) =<< flip justifyRight ' ' . maximum . (length <$>)) <$> transpose (taxiRow <$> (take 25 xs ++ take 7 (drop 1999 xs)))) where justifyRight n c s = drop (length s) (replicate n c ++ s)
{{Out}}
1. 1729 = ( 1^3= 1) + ( 12^3= 1728) or ( 9^3= 729) + ( 10^3= 1000)
2. 4104 = ( 2^3= 8) + ( 16^3= 4096) or ( 9^3= 729) + ( 15^3= 3375)
3. 13832 = ( 2^3= 8) + ( 24^3= 13824) or ( 18^3= 5832) + ( 20^3= 8000)
4. 20683 = ( 10^3= 1000) + ( 27^3= 19683) or ( 19^3= 6859) + ( 24^3= 13824)
5. 32832 = ( 4^3= 64) + ( 32^3= 32768) or ( 18^3= 5832) + ( 30^3= 27000)
6. 39312 = ( 2^3= 8) + ( 34^3= 39304) or ( 15^3= 3375) + ( 33^3= 35937)
7. 40033 = ( 9^3= 729) + ( 34^3= 39304) or ( 16^3= 4096) + ( 33^3= 35937)
8. 46683 = ( 3^3= 27) + ( 36^3= 46656) or ( 27^3= 19683) + ( 30^3= 27000)
9. 64232 = ( 17^3= 4913) + ( 39^3= 59319) or ( 26^3= 17576) + ( 36^3= 46656)
10. 65728 = ( 12^3= 1728) + ( 40^3= 64000) or ( 31^3= 29791) + ( 33^3= 35937)
11. 110656 = ( 4^3= 64) + ( 48^3= 110592) or ( 36^3= 46656) + ( 40^3= 64000)
12. 110808 = ( 6^3= 216) + ( 48^3= 110592) or ( 27^3= 19683) + ( 45^3= 91125)
13. 134379 = ( 12^3= 1728) + ( 51^3= 132651) or ( 38^3= 54872) + ( 43^3= 79507)
14. 149389 = ( 8^3= 512) + ( 53^3= 148877) or ( 29^3= 24389) + ( 50^3= 125000)
15. 165464 = ( 20^3= 8000) + ( 54^3= 157464) or ( 38^3= 54872) + ( 48^3= 110592)
16. 171288 = ( 17^3= 4913) + ( 55^3= 166375) or ( 24^3= 13824) + ( 54^3= 157464)
17. 195841 = ( 9^3= 729) + ( 58^3= 195112) or ( 22^3= 10648) + ( 57^3= 185193)
18. 216027 = ( 3^3= 27) + ( 60^3= 216000) or ( 22^3= 10648) + ( 59^3= 205379)
19. 216125 = ( 5^3= 125) + ( 60^3= 216000) or ( 45^3= 91125) + ( 50^3= 125000)
20. 262656 = ( 8^3= 512) + ( 64^3= 262144) or ( 36^3= 46656) + ( 60^3= 216000)
21. 314496 = ( 4^3= 64) + ( 68^3= 314432) or ( 30^3= 27000) + ( 66^3= 287496)
22. 320264 = ( 18^3= 5832) + ( 68^3= 314432) or ( 32^3= 32768) + ( 66^3= 287496)
23. 327763 = ( 30^3= 27000) + ( 67^3= 300763) or ( 51^3= 132651) + ( 58^3= 195112)
24. 373464 = ( 6^3= 216) + ( 72^3= 373248) or ( 54^3= 157464) + ( 60^3= 216000)
25. 402597 = ( 42^3= 74088) + ( 69^3= 328509) or ( 56^3= 175616) + ( 61^3= 226981)
2000. 1671816384 = (428^3= 78402752) + (1168^3=1593413632) or (940^3=830584000) + ( 944^3= 841232384)
2001. 1672470592 = ( 29^3= 24389) + (1187^3=1672446203) or (632^3=252435968) + (1124^3=1420034624)
2002. 1673170856 = (458^3= 96071912) + (1164^3=1577098944) or (828^3=567663552) + (1034^3=1105507304)
2003. 1675045225 = (522^3=142236648) + (1153^3=1532808577) or (744^3=411830784) + (1081^3=1263214441)
2004. 1675958167 = (492^3=119095488) + (1159^3=1556862679) or (711^3=359425431) + (1096^3=1316532736)
2005. 1676926719 = ( 63^3= 250047) + (1188^3=1676676672) or (714^3=363994344) + (1095^3=1312932375)
2006. 1677646971 = ( 99^3= 970299) + (1188^3=1676676672) or (891^3=707347971) + ( 990^3= 970299000)
J
cubes=: 3^~1+i.100 NB. first 100 cubes
triples=: /:~ ~. ,/ (+ , /:~@,)"0/~cubes NB. ordered pairs of cubes (each with their sum)
candidates=: ;({."#. <@(0&#`({.@{.(;,)<@}."1)@.(1<#))/. ])triples
NB. we just want the first 25 taxicab numbers
25{.(,.~ <@>:@i.@#) candidates
┌──┬──────┬────────────┬─────────────┐
│1 │1729 │1 1728 │729 1000 │
├──┼──────┼────────────┼─────────────┤
│2 │4104 │8 4096 │729 3375 │
├──┼──────┼────────────┼─────────────┤
│3 │13832 │8 13824 │5832 8000 │
├──┼──────┼────────────┼─────────────┤
│4 │20683 │1000 19683 │6859 13824 │
├──┼──────┼────────────┼─────────────┤
│5 │32832 │64 32768 │5832 27000 │
├──┼──────┼────────────┼─────────────┤
│6 │39312 │8 39304 │3375 35937 │
├──┼──────┼────────────┼─────────────┤
│7 │40033 │729 39304 │4096 35937 │
├──┼──────┼────────────┼─────────────┤
│8 │46683 │27 46656 │19683 27000 │
├──┼──────┼────────────┼─────────────┤
│9 │64232 │4913 59319 │17576 46656 │
├──┼──────┼────────────┼─────────────┤
│10│65728 │1728 64000 │29791 35937 │
├──┼──────┼────────────┼─────────────┤
│11│110656│64 110592 │46656 64000 │
├──┼──────┼────────────┼─────────────┤
│12│110808│216 110592 │19683 91125 │
├──┼──────┼────────────┼─────────────┤
│13│134379│1728 132651 │54872 79507 │
├──┼──────┼────────────┼─────────────┤
│14│149389│512 148877 │24389 125000 │
├──┼──────┼────────────┼─────────────┤
│15│165464│8000 157464 │54872 110592 │
├──┼──────┼────────────┼─────────────┤
│16│171288│4913 166375 │13824 157464 │
├──┼──────┼────────────┼─────────────┤
│17│195841│729 195112 │10648 185193 │
├──┼──────┼────────────┼─────────────┤
│18│216027│27 216000 │10648 205379 │
├──┼──────┼────────────┼─────────────┤
│19│216125│125 216000 │91125 125000 │
├──┼──────┼────────────┼─────────────┤
│20│262656│512 262144 │46656 216000 │
├──┼──────┼────────────┼─────────────┤
│21│314496│64 314432 │27000 287496 │
├──┼──────┼────────────┼─────────────┤
│22│320264│5832 314432 │32768 287496 │
├──┼──────┼────────────┼─────────────┤
│23│327763│27000 300763│132651 195112│
├──┼──────┼────────────┼─────────────┤
│24│373464│216 373248 │157464 216000│
├──┼──────┼────────────┼─────────────┤
│25│402597│74088 328509│175616 226981│
└──┴──────┴────────────┴─────────────┘
Explanation:
First, generate 100 cubes.
Then, form a 3 column table of unique rows: sum, small cube, large cube
Then, gather rows where the first entry is the same. Keep the ones with at least two such entries (sorted by ascending order of sum).
Then, place an counting index (starting from 1) in front of each row, so the columns are now: '''counting index''', '''sum''', '''small cube''', '''large cube'''.
Note that the cube root of the 25th entry is slightly smaller than 74, so testing against the first 100 cubes is more than sufficient.
Note that here we have elected to show the constituent cubes as themselves rather than as expressions involving their cube roots.
Extra credit:
x:each 7 {. 1999 }. (,.~ <@>:@i.@#) ;({."#. <@(0&#`({.@{.(;,)<@}."1)@.(1<#))/. ])/:~~.,/(+,/:~@,)"0/~3^~1+i.10000
┌────┬──────────┬────────────────────┬────────────────────┬┐
│2000│1671816384│78402752 1593413632 │830584000 841232384 ││
├────┼──────────┼────────────────────┼────────────────────┼┤
│2001│1672470592│24389 1672446203 │252435968 1420034624││
├────┼──────────┼────────────────────┼────────────────────┼┤
│2002│1673170856│96071912 1577098944 │567663552 1105507304││
├────┼──────────┼────────────────────┼────────────────────┼┤
│2003│1675045225│142236648 1532808577│411830784 1263214441││
├────┼──────────┼────────────────────┼────────────────────┼┤
│2004│1675958167│119095488 1556862679│359425431 1316532736││
├────┼──────────┼────────────────────┼────────────────────┼┤
│2005│1676926719│250047 1676676672 │363994344 1312932375││
├────┼──────────┼────────────────────┼────────────────────┼┤
│2006│1677646971│970299 1676676672 │707347971 970299000 ││
└────┴──────────┴────────────────────┴────────────────────┴┘
The extra blank box at the end is because when tackling this large of a data set, some sums can be achieved by three different pairs of cubes.
Java
import java.util.PriorityQueue; import java.util.ArrayList; import java.util.List; import java.util.Iterator; class CubeSum implements Comparable<CubeSum> { public long x, y, value; public CubeSum(long x, long y) { this.x = x; this.y = y; this.value = x*x*x + y*y*y; } public String toString() { return String.format("%4d^3 + %4d^3", x, y); } public int compareTo(CubeSum that) { return value < that.value ? -1 : value > that.value ? 1 : 0; } } class SumIterator implements Iterator<CubeSum> { PriorityQueue<CubeSum> pq = new PriorityQueue<CubeSum>(); long n = 0; public boolean hasNext() { return true; } public CubeSum next() { while (pq.size() == 0 || pq.peek().value >= n*n*n) pq.add(new CubeSum(++n, 1)); CubeSum s = pq.remove(); if (s.x > s.y + 1) pq.add(new CubeSum(s.x, s.y+1)); return s; } } class TaxiIterator implements Iterator<List<CubeSum>> { Iterator<CubeSum> sumIterator = new SumIterator(); CubeSum last = sumIterator.next(); public boolean hasNext() { return true; } public List<CubeSum> next() { CubeSum s; List<CubeSum> train = new ArrayList<CubeSum>(); while ((s = sumIterator.next()).value != last.value) last = s; train.add(last); do { train.add(s); } while ((s = sumIterator.next()).value == last.value); last = s; return train; } } public class Taxi { public static final void main(String[] args) { Iterator<List<CubeSum>> taxi = new TaxiIterator(); for (int i = 1; i <= 2006; i++) { List<CubeSum> t = taxi.next(); if (i > 25 && i < 2000) continue; System.out.printf("%4d: %10d", i, t.get(0).value); for (CubeSum s: t) System.out.print(" = " + s); System.out.println(); } } }
{{out}}
1: 1729 = 10^3 + 9^3 = 12^3 + 1^3
2: 4104 = 15^3 + 9^3 = 16^3 + 2^3
3: 13832 = 20^3 + 18^3 = 24^3 + 2^3
4: 20683 = 24^3 + 19^3 = 27^3 + 10^3
5: 32832 = 30^3 + 18^3 = 32^3 + 4^3
6: 39312 = 33^3 + 15^3 = 34^3 + 2^3
7: 40033 = 34^3 + 9^3 = 33^3 + 16^3
8: 46683 = 30^3 + 27^3 = 36^3 + 3^3
9: 64232 = 36^3 + 26^3 = 39^3 + 17^3
10: 65728 = 33^3 + 31^3 = 40^3 + 12^3
11: 110656 = 40^3 + 36^3 = 48^3 + 4^3
12: 110808 = 45^3 + 27^3 = 48^3 + 6^3
13: 134379 = 43^3 + 38^3 = 51^3 + 12^3
14: 149389 = 50^3 + 29^3 = 53^3 + 8^3
15: 165464 = 48^3 + 38^3 = 54^3 + 20^3
16: 171288 = 54^3 + 24^3 = 55^3 + 17^3
17: 195841 = 57^3 + 22^3 = 58^3 + 9^3
18: 216027 = 59^3 + 22^3 = 60^3 + 3^3
19: 216125 = 50^3 + 45^3 = 60^3 + 5^3
20: 262656 = 60^3 + 36^3 = 64^3 + 8^3
21: 314496 = 66^3 + 30^3 = 68^3 + 4^3
22: 320264 = 66^3 + 32^3 = 68^3 + 18^3
23: 327763 = 58^3 + 51^3 = 67^3 + 30^3
24: 373464 = 60^3 + 54^3 = 72^3 + 6^3
25: 402597 = 61^3 + 56^3 = 69^3 + 42^3
2000: 1671816384 = 1168^3 + 428^3 = 944^3 + 940^3
2001: 1672470592 = 1124^3 + 632^3 = 1187^3 + 29^3
2002: 1673170856 = 1164^3 + 458^3 = 1034^3 + 828^3
2003: 1675045225 = 1153^3 + 522^3 = 1081^3 + 744^3
2004: 1675958167 = 1159^3 + 492^3 = 1096^3 + 711^3
2005: 1676926719 = 1095^3 + 714^3 = 1188^3 + 63^3
2006: 1677646971 = 990^3 + 891^3 = 1188^3 + 99^3
JavaScript
var n3s = [], s3s = {} for (var n = 1, e = 1200; n < e; n += 1) n3s[n] = n * n * n for (var a = 1; a < e - 1; a += 1) { var a3 = n3s[a] for (var b = a; b < e; b += 1) { var b3 = n3s[b] var s3 = a3 + b3, abs = s3s[s3] if (!abs) s3s[s3] = abs = [] abs.push([a, b]) } } var i = 0 for (var s3 in s3s) { var abs = s3s[s3] if (abs.length < 2) continue i += 1 if (abs.length == 2 && i > 25 && i < 2000) continue if (i > 2006) break document.write(i, ': ', s3) for (var ab of abs) { document.write(' = ', ab[0], '<sup>3</sup>+', ab[1], '<sup>3</sup>') } document.write(' ') }
{{out}} 1: 1729 = 13+123 = 93+103 2: 4104 = 23+163 = 93+153 3: 13832 = 23+243 = 183+203 4: 20683 = 103+273 = 193+243 5: 32832 = 43+323 = 183+303 6: 39312 = 23+343 = 153+333 7: 40033 = 93+343 = 163+333 8: 46683 = 33+363 = 273+303 9: 64232 = 173+393 = 263+363 10: 65728 = 123+403 = 313+333 11: 110656 = 43+483 = 363+403 12: 110808 = 63+483 = 273+453 13: 134379 = 123+513 = 383+433 14: 149389 = 83+533 = 293+503 15: 165464 = 203+543 = 383+483 16: 171288 = 173+553 = 243+543 17: 195841 = 93+583 = 223+573 18: 216027 = 33+603 = 223+593 19: 216125 = 53+603 = 453+503 20: 262656 = 83+643 = 363+603 21: 314496 = 43+683 = 303+663 22: 320264 = 183+683 = 323+663 23: 327763 = 303+673 = 513+583 24: 373464 = 63+723 = 543+603 25: 402597 = 423+693 = 563+613 455: 87539319 = 1673+4363 = 2283+4233 = 2553+4143 535: 119824488 = 113+4933 = 903+4923 = 3463+4283 588: 143604279 = 1113+5223 = 3593+4603 = 4083+4233 655: 175959000 = 703+5603 = 1983+5523 = 3153+5253 888: 327763000 = 3003+6703 = 3393+6613 = 5103+5803 1299: 700314552 = 3343+8723 = 4563+8463 = 5103+8283 1398: 804360375 = 153+9303 = 1983+9273 = 2953+9203 1515: 958595904 = 223+9863 = 1803+9843 = 6923+8563 1660: 1148834232 = 2223+10443 = 7183+9203 = 8163+8463 1837: 1407672000 = 1403+11203 = 3963+11043 = 6303+10503 2000: 1671816384 = 4283+11683 = 9403+9443 2001: 1672470592 = 293+11873 = 6323+11243 2002: 1673170856 = 4583+11643 = 8283+10343 2003: 1675045225 = 5223+11533 = 7443+10813 2004: 1675958167 = 4923+11593 = 7113+10963 2005: 1676926719 = 633+11883 = 7143+10953 2006: 1677646971 = 993+11883 = 8913+9903
jq
{{works with|jq|1.4}}
# Output: an array of the form [i^3 + j^3, [i, j]] sorted by the sum.
# Only cubes of 1 to ($in-1) are considered; the listing is therefore truncated
# as it might not capture taxicab numbers greater than $in ^ 3.
def sum_of_two_cubes:
def cubed: .*.*.;
. as $in
| (cubed + 1) as $limit
| [range(1;$in) as $i | range($i;$in) as $j
| [ ($i|cubed) + ($j|cubed), [$i, $j] ] ] | sort
| map( select( .[0] < $limit ) );
# Output a stream of triples [t, d1, d2], in order of t,
# where t is a taxicab number, and d1 and d2 are distinct
# decompositions [i,j] with i^3 + j^3 == t.
# The stream includes each taxicab number once only.
#
def taxicabs0:
sum_of_two_cubes as $sums
| range(1;$sums|length) as $i
| if $sums[$i][0] == $sums[$i-1][0]
and ($i==1 or $sums[$i][0] != $sums[$i-2][0])
then [$sums[$i][0], $sums[$i-1][1], $sums[$i][1]]
else empty
end;
# Output a stream of $n taxicab triples: [t, d1, d2] as described above,
# without repeating t.
def taxicabs:
# If your jq includes until/2 then the following definition
# can be omitted:
def until(cond; next):
def _until: if cond then . else (next|_until) end; _until;
. as $n
| [10, ($n / 10 | floor)] | max as $increment
| [20, ($n / 2 | floor)] | max
| [ ., [taxicabs0] ]
| until( .[1] | length >= $m; (.[0] + $increment) | [., [taxicabs0]] )
| .[1][0:$n] ;
'''The task'''
2006 | taxicabs as $t
| (range(0;25), range(1999;2006)) as $i
| "\($i+1): \($t[$i][0]) ~ \($t[$i][1]) and \($t[$i][2])"
{{out}}
$ jq -n -r -f Taxicab_numbers.jq 1: 1729 ~ [1,12] and [9,10] 2: 4104 ~ [2,16] and [9,15] 3: 13832 ~ [2,24] and [18,20] 4: 20683 ~ [10,27] and [19,24] 5: 32832 ~ [4,32] and [18,30] 6: 39312 ~ [2,34] and [15,33] 7: 40033 ~ [9,34] and [16,33] 8: 46683 ~ [3,36] and [27,30] 9: 64232 ~ [17,39] and [26,36] 10: 65728 ~ [12,40] and [31,33] 11: 110656 ~ [4,48] and [36,40] 12: 110808 ~ [6,48] and [27,45] 13: 134379 ~ [12,51] and [38,43] 14: 149389 ~ [8,53] and [29,50] 15: 165464 ~ [20,54] and [38,48] 16: 171288 ~ [17,55] and [24,54] 17: 195841 ~ [9,58] and [22,57] 18: 216027 ~ [3,60] and [22,59] 19: 216125 ~ [5,60] and [45,50] 20: 262656 ~ [8,64] and [36,60] 21: 314496 ~ [4,68] and [30,66] 22: 320264 ~ [18,68] and [32,66] 23: 327763 ~ [30,67] and [51,58] 24: 373464 ~ [6,72] and [54,60] 25: 402597 ~ [42,69] and [56,61] 2000: 1671816384 ~ [428,1168] and [940,944] 2001: 1672470592 ~ [29,1187] and [632,1124] 2002: 1673170856 ~ [458,1164] and [828,1034] 2003: 1675045225 ~ [522,1153] and [744,1081] 2004: 1675958167 ~ [492,1159] and [711,1096] 2005: 1676926719 ~ [63,1188] and [714,1095] 2006: 1677646971 ~ [99,1188] and [891,990]
Julia
{{works with|Julia|0.6}} {{trans|Python}}
using DataStructures, IterTools function findtaxinumbers(nmax::Integer) cube2n = Dict{Int,Int}(x ^ 3 => x for x in 0:nmax) sum2cubes = DefaultDict{Int,Set{NTuple{2,Int}}}(Set{NTuple{2,Int}}) for ((c1, _), (c2, _)) in product(cube2n, cube2n) if c1 ≥ c2 push!(sum2cubes[c1 + c2], (cube2n[c1], cube2n[c2])) end end taxied = collect((k, v) for (k, v) in sum2cubes if length(v) ≥ 2) return sort!(taxied, by = first) end taxied = findtaxinumbers(1200) for (ith, (cube, set)) in zip(1:25, taxied[1:25]) @printf "%2i: %7i = %s\n" ith cube join(set, ", ") # println(ith, ": ", cube, " = ", join(set, ", ")) end println("...") for (ith, (cube, set)) in zip(2000:2006, taxied[2000:2006]) @printf "%-4i: %i = %s\n" ith cube join(set, ", ") end # version 2 function findtaxinumbers(nmax::Integer) cubes, crev = collect(x ^ 3 for x in 1:nmax), Dict{Int,Int}() for (x, x3) in enumerate(cubes) crev[x3] = x end sums = collect(x + y for x in cubes for y in cubes if y < x) sort!(sums) idx = 0 for i in 2:(endof(sums) - 1) if sums[i-1] != sums[i] && sums[i] == sums[i+1] idx += 1 if 25 < idx < 2000 || idx > 2006 continue end n, p = sums[i], NTuple{2,Int}[] for x in cubes n < 2x && break if haskey(crev, n - x) push!(p, (crev[x], crev[n - x])) end end @printf "%4d: %10d" idx n for x in p @printf(" = %4d ^ 3 + %4d ^ 3", x...) end println() end end end findtaxinumbers(1200)
{{out}}
1: 1729 = (12, 1), (10, 9)
2: 4104 = (16, 2), (15, 9)
3: 13832 = (24, 2), (20, 18)
4: 20683 = (27, 10), (24, 19)
5: 32832 = (32, 4), (30, 18)
6: 39312 = (33, 15), (34, 2)
7: 40033 = (34, 9), (33, 16)
8: 46683 = (30, 27), (36, 3)
9: 64232 = (36, 26), (39, 17)
10: 65728 = (33, 31), (40, 12)
11: 110656 = (48, 4), (40, 36)
12: 110808 = (48, 6), (45, 27)
13: 134379 = (43, 38), (51, 12)
14: 149389 = (50, 29), (53, 8)
15: 165464 = (54, 20), (48, 38)
16: 171288 = (54, 24), (55, 17)
17: 195841 = (57, 22), (58, 9)
18: 216027 = (59, 22), (60, 3)
19: 216125 = (60, 5), (50, 45)
20: 262656 = (64, 8), (60, 36)
21: 314496 = (66, 30), (68, 4)
22: 320264 = (66, 32), (68, 18)
23: 327763 = (67, 30), (58, 51)
24: 373464 = (60, 54), (72, 6)
25: 402597 = (69, 42), (61, 56)
...
2000: 1671816384 = (944, 940), (1168, 428)
2001: 1672470592 = (1124, 632), (1187, 29)
2002: 1673170856 = (1034, 828), (1164, 458)
2003: 1675045225 = (1081, 744), (1153, 522)
2004: 1675958167 = (1159, 492), (1096, 711)
2005: 1676926719 = (1188, 63), (1095, 714)
2006: 1677646971 = (1188, 99), (990, 891)
1: 1729 = 1 ^ 3 + 12 ^ 3 = 9 ^ 3 + 10 ^ 3
2: 4104 = 2 ^ 3 + 16 ^ 3 = 9 ^ 3 + 15 ^ 3
3: 13832 = 2 ^ 3 + 24 ^ 3 = 18 ^ 3 + 20 ^ 3
4: 20683 = 10 ^ 3 + 27 ^ 3 = 19 ^ 3 + 24 ^ 3
5: 32832 = 4 ^ 3 + 32 ^ 3 = 18 ^ 3 + 30 ^ 3
6: 39312 = 2 ^ 3 + 34 ^ 3 = 15 ^ 3 + 33 ^ 3
7: 40033 = 9 ^ 3 + 34 ^ 3 = 16 ^ 3 + 33 ^ 3
8: 46683 = 3 ^ 3 + 36 ^ 3 = 27 ^ 3 + 30 ^ 3
9: 64232 = 17 ^ 3 + 39 ^ 3 = 26 ^ 3 + 36 ^ 3
10: 65728 = 12 ^ 3 + 40 ^ 3 = 31 ^ 3 + 33 ^ 3
11: 110656 = 4 ^ 3 + 48 ^ 3 = 36 ^ 3 + 40 ^ 3
12: 110808 = 6 ^ 3 + 48 ^ 3 = 27 ^ 3 + 45 ^ 3
13: 134379 = 12 ^ 3 + 51 ^ 3 = 38 ^ 3 + 43 ^ 3
14: 149389 = 8 ^ 3 + 53 ^ 3 = 29 ^ 3 + 50 ^ 3
15: 165464 = 20 ^ 3 + 54 ^ 3 = 38 ^ 3 + 48 ^ 3
16: 171288 = 17 ^ 3 + 55 ^ 3 = 24 ^ 3 + 54 ^ 3
17: 195841 = 9 ^ 3 + 58 ^ 3 = 22 ^ 3 + 57 ^ 3
18: 216027 = 3 ^ 3 + 60 ^ 3 = 22 ^ 3 + 59 ^ 3
19: 216125 = 5 ^ 3 + 60 ^ 3 = 45 ^ 3 + 50 ^ 3
20: 262656 = 8 ^ 3 + 64 ^ 3 = 36 ^ 3 + 60 ^ 3
21: 314496 = 4 ^ 3 + 68 ^ 3 = 30 ^ 3 + 66 ^ 3
22: 320264 = 18 ^ 3 + 68 ^ 3 = 32 ^ 3 + 66 ^ 3
23: 327763 = 30 ^ 3 + 67 ^ 3 = 51 ^ 3 + 58 ^ 3
24: 373464 = 6 ^ 3 + 72 ^ 3 = 54 ^ 3 + 60 ^ 3
25: 402597 = 42 ^ 3 + 69 ^ 3 = 56 ^ 3 + 61 ^ 3
2000: 1671816384 = 428 ^ 3 + 1168 ^ 3 = 940 ^ 3 + 944 ^ 3
2001: 1672470592 = 29 ^ 3 + 1187 ^ 3 = 632 ^ 3 + 1124 ^ 3
2002: 1673170856 = 458 ^ 3 + 1164 ^ 3 = 828 ^ 3 + 1034 ^ 3
2003: 1675045225 = 522 ^ 3 + 1153 ^ 3 = 744 ^ 3 + 1081 ^ 3
2004: 1675958167 = 492 ^ 3 + 1159 ^ 3 = 711 ^ 3 + 1096 ^ 3
2005: 1676926719 = 63 ^ 3 + 1188 ^ 3 = 714 ^ 3 + 1095 ^ 3
2006: 1677646971 = 99 ^ 3 + 1188 ^ 3 = 891 ^ 3 + 990 ^ 3
Kotlin
{{trans|Java}}
// version 1.0.6 import java.util.PriorityQueue class CubeSum(val x: Long, val y: Long) : Comparable<CubeSum> { val value: Long = x * x * x + y * y * y override fun toString() = String.format("%4d^3 + %3d^3", x, y) override fun compareTo(other: CubeSum) = value.compareTo(other.value) } class SumIterator : Iterator<CubeSum> { private val pq = PriorityQueue<CubeSum>() private var n = 0L override fun hasNext() = true override fun next(): CubeSum { while (pq.size == 0 || pq.peek().value >= n * n * n) pq.add(CubeSum(++n, 1)) val s: CubeSum = pq.remove() if (s.x > s.y + 1) pq.add(CubeSum(s.x, s.y + 1)) return s } } class TaxiIterator : Iterator<MutableList<CubeSum>> { private val sumIterator = SumIterator() private var last: CubeSum = sumIterator.next() override fun hasNext() = true override fun next(): MutableList<CubeSum> { var s: CubeSum = sumIterator.next() val train = mutableListOf<CubeSum>() while (s.value != last.value) { last = s s = sumIterator.next() } train.add(last) do { train.add(s) s = sumIterator.next() } while (s.value == last.value) last = s return train } } fun main(args: Array<String>) { val taxi = TaxiIterator() for (i in 1..2006) { val t = taxi.next() if (i in 26 until 2000) continue print(String.format("%4d: %10d", i, t[0].value)) for (s in t) print(" = $s") println() } }
{{out}}
1: 1729 = 10^3 + 9^3 = 12^3 + 1^3
2: 4104 = 15^3 + 9^3 = 16^3 + 2^3
3: 13832 = 20^3 + 18^3 = 24^3 + 2^3
4: 20683 = 24^3 + 19^3 = 27^3 + 10^3
5: 32832 = 30^3 + 18^3 = 32^3 + 4^3
6: 39312 = 33^3 + 15^3 = 34^3 + 2^3
7: 40033 = 33^3 + 16^3 = 34^3 + 9^3
8: 46683 = 30^3 + 27^3 = 36^3 + 3^3
9: 64232 = 39^3 + 17^3 = 36^3 + 26^3
10: 65728 = 40^3 + 12^3 = 33^3 + 31^3
11: 110656 = 40^3 + 36^3 = 48^3 + 4^3
12: 110808 = 45^3 + 27^3 = 48^3 + 6^3
13: 134379 = 51^3 + 12^3 = 43^3 + 38^3
14: 149389 = 50^3 + 29^3 = 53^3 + 8^3
15: 165464 = 48^3 + 38^3 = 54^3 + 20^3
16: 171288 = 54^3 + 24^3 = 55^3 + 17^3
17: 195841 = 57^3 + 22^3 = 58^3 + 9^3
18: 216027 = 59^3 + 22^3 = 60^3 + 3^3
19: 216125 = 50^3 + 45^3 = 60^3 + 5^3
20: 262656 = 60^3 + 36^3 = 64^3 + 8^3
21: 314496 = 66^3 + 30^3 = 68^3 + 4^3
22: 320264 = 68^3 + 18^3 = 66^3 + 32^3
23: 327763 = 67^3 + 30^3 = 58^3 + 51^3
24: 373464 = 60^3 + 54^3 = 72^3 + 6^3
25: 402597 = 69^3 + 42^3 = 61^3 + 56^3
2000: 1671816384 = 1168^3 + 428^3 = 944^3 + 940^3
2001: 1672470592 = 1124^3 + 632^3 = 1187^3 + 29^3
2002: 1673170856 = 1164^3 + 458^3 = 1034^3 + 828^3
2003: 1675045225 = 1153^3 + 522^3 = 1081^3 + 744^3
2004: 1675958167 = 1159^3 + 492^3 = 1096^3 + 711^3
2005: 1676926719 = 1095^3 + 714^3 = 1188^3 + 63^3
2006: 1677646971 = 990^3 + 891^3 = 1188^3 + 99^3
Mathematica
findTaxi[n_] := Sort[Keys[Select[Counts[Flatten[Table[x^3 + y^3, {x, 1, n}, {y, x, n}]]], GreaterThan[1]]]];
Take[findTaxiNumbers[100], 25]
found=findTaxiNumbers[1200][[2000 ;; 2005]]
Map[Reduce[x^3 + y^3 == # && x >= y && x > 0 && y > 0, {x, y}, Integers] &, found]
{{out}}
{1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 262656, 314496, 320264, 327763, 373464, 402597}
{1671816384, 1672470592, 1673170856, 1675045225, 1675958167, 1676926719}
{(x == 944 && y == 940) || (x == 1168 && y == 428),
(x == 1124 && y == 632) || (x == 1187 && y == 29),
(x == 1034 && y == 828) || (x == 1164 && y == 458),
(x == 1081 && y == 744) || (x == 1153 && y == 522),
(x == 1096 && y == 711) || (x == 1159 && y == 492),
(x == 1095 && y == 714) || (x == 1188 && y == 63)}
PARI/GP
taxicab(n)=my(t); for(k=sqrtnint((n-1)\2,3)+1, sqrtnint(n,3), if(ispower(n-k^3, 3), if(t, return(1), t=1))); 0;
cubes(n)=my(t); for(k=sqrtnint((n-1)\2,3)+1, sqrtnint(n,3), if(ispower(n-k^3, 3, &t), print(n" = \t"k"^3\t+ "t"^3")))
select(taxicab, [1..402597])
apply(cubes, %);
{{out}}
%1 = [1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 262656, 314496, 320264, 327763, 373464, 402597]
1729 = 10^3 + 9^3
1729 = 12^3 + 1^3
4104 = 15^3 + 9^3
4104 = 16^3 + 2^3
13832 = 20^3 + 18^3
13832 = 24^3 + 2^3
20683 = 24^3 + 19^3
20683 = 27^3 + 10^3
32832 = 30^3 + 18^3
32832 = 32^3 + 4^3
39312 = 33^3 + 15^3
39312 = 34^3 + 2^3
40033 = 33^3 + 16^3
40033 = 34^3 + 9^3
46683 = 30^3 + 27^3
46683 = 36^3 + 3^3
64232 = 36^3 + 26^3
64232 = 39^3 + 17^3
65728 = 33^3 + 31^3
65728 = 40^3 + 12^3
110656 = 40^3 + 36^3
110656 = 48^3 + 4^3
110808 = 45^3 + 27^3
110808 = 48^3 + 6^3
134379 = 43^3 + 38^3
134379 = 51^3 + 12^3
149389 = 50^3 + 29^3
149389 = 53^3 + 8^3
165464 = 48^3 + 38^3
165464 = 54^3 + 20^3
171288 = 54^3 + 24^3
171288 = 55^3 + 17^3
195841 = 57^3 + 22^3
195841 = 58^3 + 9^3
216027 = 59^3 + 22^3
216027 = 60^3 + 3^3
216125 = 50^3 + 45^3
216125 = 60^3 + 5^3
262656 = 60^3 + 36^3
262656 = 64^3 + 8^3
314496 = 66^3 + 30^3
314496 = 68^3 + 4^3
320264 = 66^3 + 32^3
320264 = 68^3 + 18^3
327763 = 58^3 + 51^3
327763 = 67^3 + 30^3
373464 = 60^3 + 54^3
373464 = 72^3 + 6^3
402597 = 61^3 + 56^3
402597 = 69^3 + 42^3
Pascal
{{works with |Free Pascal}} Brute force: Create all combinations x³+ y³ | y < x one by on and test if there is a combination v < x and v> w > y with the same cube-sum. Combinations to check = n*(n-1)/2.The mean distance of one Combination m is m/2 from m³+1³ to m³+(m-1)³. searchSameSum checks one half of this distance == m/4.So O(n) ~ n³ /8 checks are needed. searchSameSum takes most of the time (>95% ), sorting is neglectable. [[http://rosettacode.org/wiki/Taxicab_numbers#C]]C-Version is ~6 times faster aka 43 vs 247 ms for max = 1290^3. Here limit set to 1190 to just reach the goal of element 2006 ;-) so 200ms are possible. Its impressive, that over all one check takes ~3.5 cpu-cycles on i4330 3.5Ghz
program taxiCabNo; uses sysutils; type tPot3 = Uint32; tPot3Sol = record p3Sum : tPot3; i1,j1, i2,j2 : Word; end; tpPot3 = ^tPot3; tpPot3Sol = ^tPot3Sol; var //1290^3 = 2'146'689'000 < 2^31-1 //1190 is the magic number of the task ;-) pot3 : array[0..1190{1290}] of tPot3;// AllSol : array[0..3000] of tpot3Sol; AllSolHigh : NativeInt; procedure SolOut(const s:tpot3Sol;no: NativeInt); begin with s do writeln(no:5,p3Sum:12,' = ',j1:5,'^3 +',i1:5,'^3 =',j2:5,'^3 +',i2:5,'^3'); end; procedure InsertAllSol; var tmp: tpot3Sol; p :tpPot3Sol; p3Sum: tPot3; i: NativeInt; Begin i := AllSolHigh; IF i > 0 then Begin p := @AllSol[i]; tmp := p^; p3Sum := p^.p3Sum; //search the right place for insertion repeat dec(i); dec(p); IF (p^.p3Sum <= p3Sum) then BREAK; until (i<=0); IF p^.p3Sum = p3Sum then EXIT; //free the right place by moving one place up inc(i); inc(p); IF i<AllSolHigh then Begin move(p^,AllSol[i+1],SizeOf(AllSol[0])*(AllSolHigh-i)); p^ := tmp; end; end; inc(AllSolHigh); end; function searchSameSum(var sol:tpot3Sol):boolean; //try to find a new combination for the same sum //within the limits given by lo and hi var Sum, SumLo: tPot3; hi,lo: NativeInt; Begin with Sol do Begin Sum := p3Sum; lo:= i1; hi:= j1; end; repeat //Move hi down dec(hi); SumLo := Sum-Pot3[hi]; //Move lo up an check until new combination found or implicite lo> hi repeat inc(lo) until (SumLo<=Pot3[lo]); //found? IF SumLo = Pot3[lo] then BREAK; until lo>=hi; IF lo<hi then Begin sol.i2:= lo; sol.j2:= hi; searchSameSum := true; end else searchSameSum := false; end; procedure Search; var i,j: LongInt; Begin AllSolHigh := 0; For j := 2 to High(pot3)-1 do Begin For i := 1 to j-1 do Begin with AllSol[AllSolHigh] do Begin p3Sum:= pot3[i]+pot3[j]; i1:= i; j1:= j; end; IF searchSameSum(AllSol[AllSolHigh]) then BEGIN InsertAllSol; IF AllSolHigh>High(AllSol) then EXIT; end; end; end; end; var i: LongInt; Begin For i := Low(pot3) to High(pot3) do pot3[i] := i*i*i; AllSolHigh := 0; Search; For i := 0 to 24 do SolOut(AllSol[i],i+1); For i := 1999 to 2005 do SolOut(AllSol[i],i+1); writeln('count of solutions ',AllSolHigh); end.
1 1729 = 12^3 + 1^3 = 10^3 + 9^3
2 4104 = 16^3 + 2^3 = 15^3 + 9^3
3 13832 = 24^3 + 2^3 = 20^3 + 18^3
......
24 373464 = 72^3 + 6^3 = 60^3 + 54^3
25 402597 = 69^3 + 42^3 = 61^3 + 56^3
2000 1671816384 = 1168^3 + 428^3 = 944^3 + 940^3
2001 1672470592 = 1187^3 + 29^3 = 1124^3 + 632^3
...
2005 1676926719 = 1188^3 + 63^3 = 1095^3 + 714^3
2006 1677646971 = 1188^3 + 99^3 = 990^3 + 891^3
count of solutions 2050
//checks 196438017
real 0m0.196s
Perl
Uses segmentation so memory use is constrained as high values are searched for. Also has parameter to look for Ta(3) and Ta(4) numbers (which is when segmentation is really needed). By default shows the first 25 numbers; with one argument shows that many; with two arguments shows results in the range.
my($beg, $end) = (@ARGV==0) ? (1,25) : (@ARGV==1) ? (1,shift) : (shift,shift); my $lim = 1e14; # Ought to be dynamic as should segment size my @basis = map { $_*$_*$_ } (1 .. int($lim ** (1.0/3.0) + 1)); my $paira = 2; # We're looking for Ta(2) and larger my ($segsize, $low, $high, $i) = (500_000_000, 0, 0, 0); while ($i < $end) { $low = $high+1; die "lim too low" if $low > $lim; $high = $low + $segsize - 1; $high = $lim if $high > $lim; foreach my $p (_find_pairs_segment(\@basis, $paira, $low, $high, sub { sprintf("%4d^3 + %4d^3", $_[0], $_[1]) }) ) { $i++; next if $i < $beg; last if $i > $end; my $n = shift @$p; printf "%4d: %10d = %s\n", $i, $n, join(" = ", @$p); } } sub _find_pairs_segment { my($p, $len, $start, $end, $formatsub) = @_; my $plen = $#$p; my %allpairs; foreach my $i (0 .. $plen) { my $pi = $p->[$i]; next if ($pi+$p->[$plen]) < $start; last if (2*$pi) > $end; foreach my $j ($i .. $plen) { my $sum = $pi + $p->[$j]; next if $sum < $start; last if $sum > $end; push @{ $allpairs{$sum} }, $i, $j; } # If we wanted to save more memory, we could filter and delete every entry # where $n < 2 * $p->[$i+1]. This can cut memory use in half, but is slow. } my @retlist; foreach my $list (grep { scalar @$_ >= $len*2 } values %allpairs) { my $n = $p->[$list->[0]] + $p->[$list->[1]]; my @pairlist; while (@$list) { push @pairlist, $formatsub->(1 + shift @$list, 1 + shift @$list); } push @retlist, [$n, @pairlist]; } @retlist = sort { $a->[0] <=> $b->[0] } @retlist; return @retlist; }
{{out}}
1: 1729 = 1^3 + 12^3 = 9^3 + 10^3
2: 4104 = 2^3 + 16^3 = 9^3 + 15^3
3: 13832 = 2^3 + 24^3 = 18^3 + 20^3
4: 20683 = 10^3 + 27^3 = 19^3 + 24^3
5: 32832 = 4^3 + 32^3 = 18^3 + 30^3
6: 39312 = 2^3 + 34^3 = 15^3 + 33^3
7: 40033 = 9^3 + 34^3 = 16^3 + 33^3
8: 46683 = 3^3 + 36^3 = 27^3 + 30^3
9: 64232 = 17^3 + 39^3 = 26^3 + 36^3
10: 65728 = 12^3 + 40^3 = 31^3 + 33^3
11: 110656 = 4^3 + 48^3 = 36^3 + 40^3
12: 110808 = 6^3 + 48^3 = 27^3 + 45^3
13: 134379 = 12^3 + 51^3 = 38^3 + 43^3
14: 149389 = 8^3 + 53^3 = 29^3 + 50^3
15: 165464 = 20^3 + 54^3 = 38^3 + 48^3
16: 171288 = 17^3 + 55^3 = 24^3 + 54^3
17: 195841 = 9^3 + 58^3 = 22^3 + 57^3
18: 216027 = 3^3 + 60^3 = 22^3 + 59^3
19: 216125 = 5^3 + 60^3 = 45^3 + 50^3
20: 262656 = 8^3 + 64^3 = 36^3 + 60^3
21: 314496 = 4^3 + 68^3 = 30^3 + 66^3
22: 320264 = 18^3 + 68^3 = 32^3 + 66^3
23: 327763 = 30^3 + 67^3 = 51^3 + 58^3
24: 373464 = 6^3 + 72^3 = 54^3 + 60^3
25: 402597 = 42^3 + 69^3 = 56^3 + 61^3
With arguments 2000 2006:
2000: 1671816384 = 428^3 + 1168^3 = 940^3 + 944^3
2001: 1672470592 = 29^3 + 1187^3 = 632^3 + 1124^3
2002: 1673170856 = 458^3 + 1164^3 = 828^3 + 1034^3
2003: 1675045225 = 522^3 + 1153^3 = 744^3 + 1081^3
2004: 1675958167 = 492^3 + 1159^3 = 711^3 + 1096^3
2005: 1676926719 = 63^3 + 1188^3 = 714^3 + 1095^3
2006: 1677646971 = 99^3 + 1188^3 = 891^3 + 990^3
Perl 6
This uses a pretty simple search algorithm that doesn't necessarily return the Taxicab numbers in order. Assuming we want all the Taxicab numbers within some range S to N, we'll search until we find N values. When we find the Nth value, we continue to search up to the cube root of the largest Taxicab number found up to that point. That ensures we will find all of them inside the desired range without needing to search arbitrarily or use magic numbers. Defaults to returning the Taxicab numbers from 1 to 25. Pass in a different start and end value if you want some other range.
constant @cu = (^Inf).map: { .³ }
sub MAIN ($start = 1, $end = 25) {
my %taxi;
my int $taxis = 0;
my $terminate = 0;
my int $max = 0;
for 1 .. * -> $c1 {
last if ?$terminate && ($terminate < $c1);
for 1 .. $c1 -> $c2 {
my $this = @cu[$c1] + @cu[$c2];
%taxi{$this}.push: [$c2, $c1];
if %taxi{$this}.elems == 2 {
++$taxis;
$max max= $this;
}
$terminate = ceiling $max ** (1/3) if $taxis == $end and !$terminate;
}
}
display( %taxi, $start, $end );
}
sub display (%this_stuff, $start, $end) {
my $i = $start;
printf "%4d %10d =>\t%s\n", $i++, $_.key,
(.value.map({ sprintf "%4d³ + %-s\³", |$_ })).join: ",\t"
for %this_stuff.grep( { $_.value.elems > 1 } ).sort( +*.key )[$start-1..$end-1];
}
{{out}}With no passed parameters (default):
1 1729 => 9³ + 10³, 1³ + 12³
2 4104 => 9³ + 15³, 2³ + 16³
3 13832 => 18³ + 20³, 2³ + 24³
4 20683 => 19³ + 24³, 10³ + 27³
5 32832 => 18³ + 30³, 4³ + 32³
6 39312 => 15³ + 33³, 2³ + 34³
7 40033 => 16³ + 33³, 9³ + 34³
8 46683 => 27³ + 30³, 3³ + 36³
9 64232 => 26³ + 36³, 17³ + 39³
10 65728 => 31³ + 33³, 12³ + 40³
11 110656 => 36³ + 40³, 4³ + 48³
12 110808 => 27³ + 45³, 6³ + 48³
13 134379 => 38³ + 43³, 12³ + 51³
14 149389 => 29³ + 50³, 8³ + 53³
15 165464 => 38³ + 48³, 20³ + 54³
16 171288 => 24³ + 54³, 17³ + 55³
17 195841 => 22³ + 57³, 9³ + 58³
18 216027 => 22³ + 59³, 3³ + 60³
19 216125 => 45³ + 50³, 5³ + 60³
20 262656 => 36³ + 60³, 8³ + 64³
21 314496 => 30³ + 66³, 4³ + 68³
22 320264 => 32³ + 66³, 18³ + 68³
23 327763 => 51³ + 58³, 30³ + 67³
24 373464 => 54³ + 60³, 6³ + 72³
25 402597 => 56³ + 61³, 42³ + 69³
With passed parameters 2000 2006:
2000 1671816384 => 940³ + 944³, 428³ + 1168³
2001 1672470592 => 632³ + 1124³, 29³ + 1187³
2002 1673170856 => 828³ + 1034³, 458³ + 1164³
2003 1675045225 => 744³ + 1081³, 522³ + 1153³
2004 1675958167 => 711³ + 1096³, 492³ + 1159³
2005 1676926719 => 714³ + 1095³, 63³ + 1188³
2006 1677646971 => 891³ + 990³, 99³ + 1188³
Phix
{{trans|Perl_6}} Uses a dictionary to map sum of cubes to either the first/only pair or an integer index into the result set. Turned out to be a fair bit slower (15s) than I first expected.
function get_taxis(integer last)
sequence taxis = {}
integer c1 = 1, maxc1 = 0, c2
atom c3, h3 = 0
while maxc1=0 or c1<maxc1 do
c3 = power(c1,3)
for c2 = 1 to c1 do
atom this = power(c2,3)+c3
integer node = getd_index(this)
if node=NULL then
setd(this,{c2,c1})
else
if this>h3 then h3 = this end if
object data = getd_by_index(node)
if not integer(data) then
taxis = append(taxis,{this,{data}})
data = length(taxis)
setd(this,data)
if data=last then
maxc1 = ceil(power(h3,1/3))
end if
end if
taxis[data][2] &= {{c2,c1}}
end if
end for
c1 += 1
end while
destroy_dict(1,justclear:=true)
taxis = sort(taxis)
return taxis
end function
sequence taxis = get_taxis(2006)
constant sets = {{1,25},{2000,2006}}
for s=1 to length(sets) do
integer {first,last} = sets[s]
for i=first to last do
printf(1,"%d: %d: %s\n",{i,taxis[i][1],sprint(taxis[i][2])})
end for
end for
{{out}}
1: 1729: {{9,10},{1,12}}
2: 4104: {{9,15},{2,16}}
3: 13832: {{18,20},{2,24}}
4: 20683: {{19,24},{10,27}}
5: 32832: {{18,30},{4,32}}
6: 39312: {{15,33},{2,34}}
7: 40033: {{16,33},{9,34}}
8: 46683: {{27,30},{3,36}}
9: 64232: {{26,36},{17,39}}
10: 65728: {{31,33},{12,40}}
11: 110656: {{36,40},{4,48}}
12: 110808: {{27,45},{6,48}}
13: 134379: {{38,43},{12,51}}
14: 149389: {{29,50},{8,53}}
15: 165464: {{38,48},{20,54}}
16: 171288: {{24,54},{17,55}}
17: 195841: {{22,57},{9,58}}
18: 216027: {{22,59},{3,60}}
19: 216125: {{45,50},{5,60}}
20: 262656: {{36,60},{8,64}}
21: 314496: {{30,66},{4,68}}
22: 320264: {{32,66},{18,68}}
23: 327763: {{51,58},{30,67}}
24: 373464: {{54,60},{6,72}}
25: 402597: {{56,61},{42,69}}
2000: 1671816384: {{940,944},{428,1168}}
2001: 1672470592: {{632,1124},{29,1187}}
2002: 1673170856: {{828,1034},{458,1164}}
2003: 1675045225: {{744,1081},{522,1153}}
2004: 1675958167: {{711,1096},{492,1159}}
2005: 1676926719: {{714,1095},{63,1188}}
2006: 1677646971: {{891,990},{99,1188}}
{{trans|C}} Using a [[Priority_queue#Phix|priority queue]], otherwise based on C, quite a bit (18.5x) faster.
Copes with 40000..6, same results as Go, though that increases the runtime from 0.8s to 1min 15s.
sequence cubes = {}
procedure add_cube()
integer n = length(cubes)+1
cubes = append(cubes,n*n*n)
pq_add({{n,1},cubes[n]+1})
end procedure
constant VALUE = PRIORITY
function next_sum()
while length(pq)<=2 or pq[1][VALUE]>=cubes[$] do add_cube() end while
sequence res = pq_pop()
integer {x,y} = res[DATA]
y += 1
if y<x then
pq_add({{x,y},cubes[x]+cubes[y]})
end if
return res
end function
function next_taxi()
sequence top
while 1 do
top = next_sum()
if pq[1][VALUE]=top[VALUE] then exit end if
end while
sequence res = {top}
atom v = top[PRIORITY]
while 1 do
top = next_sum()
res = append(res,top[DATA])
if pq[1][VALUE]!=v then exit end if
end while
return res
end function
for i=1 to 2006 do
sequence x = next_taxi()
if i<=25 or i>=2000 then
atom v = x[1][VALUE]
x[1] = x[1][DATA]
string y = sprintf("%11d+%-10d",sq_power(x[1],3))
for j=2 to length(x) do
y &= sprintf(",%11d+%-10d",sq_power(x[j],3))
end for
printf(1,"%4d: %10d: %-23s [%s]\n",{i,v,sprint(x),y})
end if
end for
{{out}}
1: 1729: {{10,9},{12,1}} [ 1000+729 , 1728+1 ]
2: 4104: {{15,9},{16,2}} [ 3375+729 , 4096+8 ]
3: 13832: {{20,18},{24,2}} [ 8000+5832 , 13824+8 ]
4: 20683: {{24,19},{27,10}} [ 13824+6859 , 19683+1000 ]
5: 32832: {{30,18},{32,4}} [ 27000+5832 , 32768+64 ]
6: 39312: {{33,15},{34,2}} [ 35937+3375 , 39304+8 ]
7: 40033: {{33,16},{34,9}} [ 35937+4096 , 39304+729 ]
8: 46683: {{30,27},{36,3}} [ 27000+19683 , 46656+27 ]
9: 64232: {{39,17},{36,26}} [ 59319+4913 , 46656+17576 ]
10: 65728: {{40,12},{33,31}} [ 64000+1728 , 35937+29791 ]
11: 110656: {{40,36},{48,4}} [ 64000+46656 , 110592+64 ]
12: 110808: {{45,27},{48,6}} [ 91125+19683 , 110592+216 ]
13: 134379: {{51,12},{43,38}} [ 132651+1728 , 79507+54872 ]
14: 149389: {{50,29},{53,8}} [ 125000+24389 , 148877+512 ]
15: 165464: {{48,38},{54,20}} [ 110592+54872 , 157464+8000 ]
16: 171288: {{54,24},{55,17}} [ 157464+13824 , 166375+4913 ]
17: 195841: {{57,22},{58,9}} [ 185193+10648 , 195112+729 ]
18: 216027: {{59,22},{60,3}} [ 205379+10648 , 216000+27 ]
19: 216125: {{50,45},{60,5}} [ 125000+91125 , 216000+125 ]
20: 262656: {{60,36},{64,8}} [ 216000+46656 , 262144+512 ]
21: 314496: {{66,30},{68,4}} [ 287496+27000 , 314432+64 ]
22: 320264: {{68,18},{66,32}} [ 314432+5832 , 287496+32768 ]
23: 327763: {{67,30},{58,51}} [ 300763+27000 , 195112+132651 ]
24: 373464: {{60,54},{72,6}} [ 216000+157464 , 373248+216 ]
25: 402597: {{69,42},{61,56}} [ 328509+74088 , 226981+175616 ]
2000: 1671816384: {{1168,428},{944,940}} [ 1593413632+78402752 , 841232384+830584000 ]
2001: 1672470592: {{1124,632},{1187,29}} [ 1420034624+252435968 , 1672446203+24389 ]
2002: 1673170856: {{1164,458},{1034,828}} [ 1577098944+96071912 , 1105507304+567663552 ]
2003: 1675045225: {{1153,522},{1081,744}} [ 1532808577+142236648 , 1263214441+411830784 ]
2004: 1675958167: {{1159,492},{1096,711}} [ 1556862679+119095488 , 1316532736+359425431 ]
2005: 1676926719: {{1095,714},{1188,63}} [ 1312932375+363994344 , 1676676672+250047 ]
2006: 1677646971: {{990,891},{1188,99}} [ 970299000+707347971 , 1676676672+970299 ]
PicoLisp
(load "@lib/simul.l")
(off 'B)
(for L (subsets 2 (range 1 1200))
(let K (sum '((N) (** N 3)) L)
(ifn (lup B K)
(idx 'B (list K 1 (list L)) T)
(inc (cdr @))
(push (cddr @) L) ) ) )
(setq R
(filter
'((L) (>= (cadr L) 2))
(idx 'B)) )
(for L (head 25 R)
(println (car L) (caddr L)) )
(for L (head 7 (nth R 2000))
(println (car L) (caddr L)) )
{{out}}
1729 ((9 10) (1 12))
4104 ((9 15) (2 16))
13832 ((18 20) (2 24))
20683 ((19 24) (10 27))
32832 ((18 30) (4 32))
39312 ((15 33) (2 34))
40033 ((16 33) (9 34))
46683 ((27 30) (3 36))
64232 ((26 36) (17 39))
65728 ((31 33) (12 40))
110656 ((36 40) (4 48))
110808 ((27 45) (6 48))
134379 ((38 43) (12 51))
149389 ((29 50) (8 53))
165464 ((38 48) (20 54))
171288 ((24 54) (17 55))
195841 ((22 57) (9 58))
216027 ((22 59) (3 60))
216125 ((45 50) (5 60))
262656 ((36 60) (8 64))
314496 ((30 66) (4 68))
320264 ((32 66) (18 68))
327763 ((51 58) (30 67))
373464 ((54 60) (6 72))
402597 ((56 61) (42 69))
1671816384 ((940 944) (428 1168))
1672470592 ((632 1124) (29 1187))
1673170856 ((828 1034) (458 1164))
1675045225 ((744 1081) (522 1153))
1675958167 ((711 1096) (492 1159))
1676926719 ((714 1095) (63 1188))
1677646971 ((891 990) (99 1188))
Python
(Magic number 1201 found by trial and error)
from collections import defaultdict from itertools import product from pprint import pprint as pp cube2n = {x**3:x for x in range(1, 1201)} sum2cubes = defaultdict(set) for c1, c2 in product(cube2n, cube2n): if c1 >= c2: sum2cubes[c1 + c2].add((cube2n[c1], cube2n[c2])) taxied = sorted((k, v) for k,v in sum2cubes.items() if len(v) >= 2) #pp(len(taxied)) # 2068 for t in enumerate(taxied[:25], 1): pp(t) print('...') for t in enumerate(taxied[2000-1:2000+6], 2000): pp(t)
{{out}}
(1, (1729, {(12, 1), (10, 9)}))
(2, (4104, {(16, 2), (15, 9)}))
(3, (13832, {(20, 18), (24, 2)}))
(4, (20683, {(27, 10), (24, 19)}))
(5, (32832, {(30, 18), (32, 4)}))
(6, (39312, {(33, 15), (34, 2)}))
(7, (40033, {(33, 16), (34, 9)}))
(8, (46683, {(30, 27), (36, 3)}))
(9, (64232, {(36, 26), (39, 17)}))
(10, (65728, {(33, 31), (40, 12)}))
(11, (110656, {(48, 4), (40, 36)}))
(12, (110808, {(48, 6), (45, 27)}))
(13, (134379, {(51, 12), (43, 38)}))
(14, (149389, {(50, 29), (53, 8)}))
(15, (165464, {(54, 20), (48, 38)}))
(16, (171288, {(54, 24), (55, 17)}))
(17, (195841, {(57, 22), (58, 9)}))
(18, (216027, {(60, 3), (59, 22)}))
(19, (216125, {(60, 5), (50, 45)}))
(20, (262656, {(64, 8), (60, 36)}))
(21, (314496, {(66, 30), (68, 4)}))
(22, (320264, {(66, 32), (68, 18)}))
(23, (327763, {(58, 51), (67, 30)}))
(24, (373464, {(72, 6), (60, 54)}))
(25, (402597, {(69, 42), (61, 56)}))
...
(2000, (1671816384, {(1168, 428), (944, 940)}))
(2001, (1672470592, {(1187, 29), (1124, 632)}))
(2002, (1673170856, {(1164, 458), (1034, 828)}))
(2003, (1675045225, {(1153, 522), (1081, 744)}))
(2004, (1675958167, {(1159, 492), (1096, 711)}))
(2005, (1676926719, {(1188, 63), (1095, 714)}))
(2006, (1677646971, {(990, 891), (1188, 99)}))
Although, for this task it's simply faster to look up the cubes in the sum when we need to print them, because we can now store and sort only the sums:
cubes, crev = [x**3 for x in range(1,1200)], {} # for cube root lookup for x,x3 in enumerate(cubes): crev[x3] = x + 1 sums = sorted(x+y for x in cubes for y in cubes if y < x) idx = 0 for i in range(1, len(sums)-1): if sums[i-1] != sums[i] and sums[i] == sums[i+1]: idx += 1 if idx > 25 and idx < 2000 or idx > 2006: continue n,p = sums[i],[] for x in cubes: if n-x < x: break if n-x in crev: p.append((crev[x], crev[n-x])) print "%4d: %10d"%(idx,n), for x in p: print " = %4d^3 + %4d^3"%x, print
{{out}}Output trimmed to reduce clutter.
1: 1729 = 1^3 + 12^3 = 9^3 + 10^3
2: 4104 = 2^3 + 16^3 = 9^3 + 15^3
3: 13832 = 2^3 + 24^3 = 18^3 + 20^3
4: 20683 = 10^3 + 27^3 = 19^3 + 24^3
5: 32832 = 4^3 + 32^3 = 18^3 + 30^3
...
2004: 1675958167 = 492^3 + 1159^3 = 711^3 + 1096^3
2005: 1676926719 = 63^3 + 1188^3 = 714^3 + 1095^3
2006: 1677646971 = 99^3 + 1188^3 = 891^3 + 990^3
Using heapq module
A priority queue that holds cube sums. When consecutive sums come out with the same value, they are taxis.
from heapq import heappush, heappop def cubesum(): h,n = [],1 while True: while not h or h[0][0] > n**3: # could also pre-calculate cubes heappush(h, (n**3 + 1, n, 1)) n += 1 (s, x, y) = heappop(h) yield((s, x, y)) y += 1 if y < x: # should be y <= x? heappush(h, (x**3 + y**3, x, y)) def taxis(): out = [(0,0,0)] for s in cubesum(): if s[0] == out[-1][0]: out.append(s) else: if len(out) > 1: yield(out) out = [s] n = 0 for x in taxis(): n += 1 if n >= 2006: break if n <= 25 or n >= 2000: print(n, x)
{{out}}
(1, [(1729, 10, 9), (1729, 12, 1)])
(2, [(4104, 15, 9), (4104, 16, 2)])
(3, [(13832, 20, 18), (13832, 24, 2)])
(4, [(20683, 24, 19), (20683, 27, 10)])
(5, [(32832, 30, 18), (32832, 32, 4)])
(6, [(39312, 33, 15), (39312, 34, 2)])
(7, [(40033, 33, 16), (40033, 34, 9)])
(8, [(46683, 30, 27), (46683, 36, 3)])
(9, [(64232, 36, 26), (64232, 39, 17)])
(10, [(65728, 33, 31), (65728, 40, 12)])
(11, [(110656, 40, 36), (110656, 48, 4)])
(12, [(110808, 45, 27), (110808, 48, 6)])
(13, [(134379, 43, 38), (134379, 51, 12)])
(14, [(149389, 50, 29), (149389, 53, 8)])
(15, [(165464, 48, 38), (165464, 54, 20)])
(16, [(171288, 54, 24), (171288, 55, 17)])
(17, [(195841, 57, 22), (195841, 58, 9)])
(18, [(216027, 59, 22), (216027, 60, 3)])
(19, [(216125, 50, 45), (216125, 60, 5)])
(20, [(262656, 60, 36), (262656, 64, 8)])
(21, [(314496, 66, 30), (314496, 68, 4)])
(22, [(320264, 66, 32), (320264, 68, 18)])
(23, [(327763, 58, 51), (327763, 67, 30)])
(24, [(373464, 60, 54), (373464, 72, 6)])
(25, [(402597, 61, 56), (402597, 69, 42)])
(2000, [(1671816384, 944, 940), (1671816384, 1168, 428)])
(2001, [(1672470592, 1124, 632), (1672470592, 1187, 29)])
(2002, [(1673170856, 1034, 828), (1673170856, 1164, 458)])
(2003, [(1675045225, 1081, 744), (1675045225, 1153, 522)])
(2004, [(1675958167, 1096, 711), (1675958167, 1159, 492)])
(2005, [(1676926719, 1095, 714), (1676926719, 1188, 63)])
Racket
This is the straighforward implementation, so it finds only the first 25 values in a sensible amount of time.
#lang racket
(define (cube x) (* x x x))
;floor of cubic root
(define (cubic-root x)
(let ([aprox (inexact->exact (round (expt x (/ 1 3))))])
(if (> (cube aprox) x)
(- aprox 1)
aprox)))
(let loop ([p 1] [n 1])
(let ()
(define pairs
(for*/list ([j (in-range 1 (add1 (cubic-root (quotient n 2))))]
[k (in-value (cubic-root (- n (cube j))))]
#:when (= n (+ (cube j) (cube k))))
(cons j k)))
(if (>= (length pairs) 2)
(begin
(printf "~a: ~a" p n)
(for ([pair (in-list pairs)])
(printf " = ~a^3 + ~a^3" (car pair) (cdr pair)))
(newline)
(when (< p 25)
(loop (add1 p) (add1 n))))
(loop p (add1 n)))))
{{out}}
1: 1729 = 1^3 + 12^3 = 9^3 + 10^3
2: 4104 = 2^3 + 16^3 = 9^3 + 15^3
3: 13832 = 2^3 + 24^3 = 18^3 + 20^3
4: 20683 = 10^3 + 27^3 = 19^3 + 24^3
5: 32832 = 4^3 + 32^3 = 18^3 + 30^3
6: 39312 = 2^3 + 34^3 = 15^3 + 33^3
7: 40033 = 9^3 + 34^3 = 16^3 + 33^3
8: 46683 = 3^3 + 36^3 = 27^3 + 30^3
9: 64232 = 17^3 + 39^3 = 26^3 + 36^3
10: 65728 = 12^3 + 40^3 = 31^3 + 33^3
11: 110656 = 4^3 + 48^3 = 36^3 + 40^3
12: 110808 = 6^3 + 48^3 = 27^3 + 45^3
13: 134379 = 12^3 + 51^3 = 38^3 + 43^3
14: 149389 = 8^3 + 53^3 = 29^3 + 50^3
15: 165464 = 20^3 + 54^3 = 38^3 + 48^3
16: 171288 = 17^3 + 55^3 = 24^3 + 54^3
17: 195841 = 9^3 + 58^3 = 22^3 + 57^3
18: 216027 = 3^3 + 60^3 = 22^3 + 59^3
19: 216125 = 5^3 + 60^3 = 45^3 + 50^3
20: 262656 = 8^3 + 64^3 = 36^3 + 60^3
21: 314496 = 4^3 + 68^3 = 30^3 + 66^3
22: 320264 = 18^3 + 68^3 = 32^3 + 66^3
23: 327763 = 30^3 + 67^3 = 51^3 + 58^3
24: 373464 = 6^3 + 72^3 = 54^3 + 60^3
25: 402597 = 42^3 + 69^3 = 56^3 + 61^3
REXX
Programming note: to ensure that the taxicab numbers are in order, an extra 10% are generated.
/*REXX program displays the specified first (lowest) taxicab numbers (for three ranges).*/
parse arg L.1 H.1 L.2 H.2 L.3 H.3 . /*obtain optional arguments from the CL*/
if L.1=='' | L.1=="," then L.1= 1 /*L1 is the low part of 1st range. */
if H.1=='' | H.1=="," then H.1= 25 /*H1 " " high " " " " */
if L.2=='' | L.2=="," then L.2= 454 /*L2 " " low " " 2nd " */
if H.2=='' | H.2=="," then H.2= 456 /*H2 " " high " " " " */
if L.3=='' | L.3=="," then L.3=2000 /*L3 " " low " " 3rd " */
if H.3=='' | H.3=="," then H.3=2006 /*H3 " " high " " " " */
mx= max(H.1, H.2, H.3) /*find how many taxicab numbers needed.*/
mx= mx + mx % 10 /*cushion; compensate for the triples.*/
ww= length(mx) * 3; w= ww % 2 /*widths used for formatting the output*/
numeric digits max(9, ww) /*prepare to use some larger numbers. */
@.=.; #= 0; @@. =0; @and= " ──and── " /*set some REXX vars and handy literals*/
$.= /* [↓] generate extra taxicab numbers.*/
do j=1 until #>=mx; C= j**3 /*taxicab numbers may not be in order. */
!.j= C /*use memoization for cube calculation.*/
do k=1 for j-1; s= C + !.k /*define a whole bunch of cube sums. */
if @.s==. then do; @.s= j; b.s= k /*Cube not defined? Then process it. */
iterate /*define @.S and B.S≡sum of 2 cubes*/
end /* [↑] define one cube sum at a time. */
has=@@.s /*has this number been defined before? */
if has then $.s=$.s @and U(j,' +')U(k) /* ◄─ build a display string. [↓] */
else $.s=right(s,ww) '───►' U(@.s," +")U(b.s) @and U(j,' +')U(k)
@@.s= 1 /*mark taxicab number as a sum of cubes*/
if has then iterate /*S is a triple (or sometimes better).*/
#= #+1; #.#= s /*bump taxicab counter; define taxicab#*/
end /*k*/ /* [↑] build the cubes one─at─a─time. */
end /*j*/ /* [↑] complete with overage numbers. */
A.=
do k=1 for mx; _= #.k; A.k= $._ /*re─assign disjoint $. elements to A. */
end /*k*/
call Esort mx /*sort taxicab #s with an exchange sort*/
do grp=1 for 3; call tell L.grp, H.grp /*display the three grps of numbers. */
end /*grp*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
tell: do t=arg(1) to arg(2); say right(t, 9)':' A.t; end; say; return
U: return right(arg(1), w)'^3'arg(2) /*right─justify a number, append "^3" */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Esort: procedure expose A.; parse arg N; h= N /*Esort when items have blanks.*/
do while h>1; h= h % 2
do i=1 for N-h; k=h + i; j= i
do forever; parse var A.k xk .; parse var A.j xj .; if xk>=xj then leave
_= A.j; A.j= A.k; A.k= _ /*swap two elements of A. array*/
if h>=j then leave; j= j - h; k= k - h
end /*forever*/
end /*i*/
end /*while h>1*/; return
{{out|output|text= when using the default inputs:}}
1: 1729 ───► 10^3 + 9^3 ──and── 12^3 + 1^3
2: 4104 ───► 15^3 + 9^3 ──and── 16^3 + 2^3
3: 13832 ───► 20^3 + 18^3 ──and── 24^3 + 2^3
4: 20683 ───► 24^3 + 19^3 ──and── 27^3 + 10^3
5: 32832 ───► 30^3 + 18^3 ──and── 32^3 + 4^3
6: 39312 ───► 33^3 + 15^3 ──and── 34^3 + 2^3
7: 40033 ───► 33^3 + 16^3 ──and── 34^3 + 9^3
8: 46683 ───► 30^3 + 27^3 ──and── 36^3 + 3^3
9: 64232 ───► 36^3 + 26^3 ──and── 39^3 + 17^3
10: 65728 ───► 33^3 + 31^3 ──and── 40^3 + 12^3
11: 110656 ───► 40^3 + 36^3 ──and── 48^3 + 4^3
12: 110808 ───► 45^3 + 27^3 ──and── 48^3 + 6^3
13: 134379 ───► 43^3 + 38^3 ──and── 51^3 + 12^3
14: 149389 ───► 50^3 + 29^3 ──and── 53^3 + 8^3
15: 165464 ───► 48^3 + 38^3 ──and── 54^3 + 20^3
16: 171288 ───► 54^3 + 24^3 ──and── 55^3 + 17^3
17: 195841 ───► 57^3 + 22^3 ──and── 58^3 + 9^3
18: 216027 ───► 59^3 + 22^3 ──and── 60^3 + 3^3
19: 216125 ───► 50^3 + 45^3 ──and── 60^3 + 5^3
20: 262656 ───► 60^3 + 36^3 ──and── 64^3 + 8^3
21: 314496 ───► 66^3 + 30^3 ──and── 68^3 + 4^3
22: 320264 ───► 66^3 + 32^3 ──and── 68^3 + 18^3
23: 327763 ───► 58^3 + 51^3 ──and── 67^3 + 30^3
24: 373464 ───► 60^3 + 54^3 ──and── 72^3 + 6^3
25: 402597 ───► 61^3 + 56^3 ──and── 69^3 + 42^3
454: 87483968 ───► 363^3 + 341^3 ──and── 440^3 + 132^3
455: 87539319 ───► 414^3 + 255^3 ──and── 423^3 + 228^3 ──and── 436^3 + 167^3
456: 87579037 ───► 370^3 + 333^3 ──and── 444^3 + 37^3
2000: 1671816384 ───► 944^3 + 940^3 ──and── 1168^3 + 428^3
2001: 1672470592 ───► 1124^3 + 632^3 ──and── 1187^3 + 29^3
2002: 1673170856 ───► 1034^3 + 828^3 ──and── 1164^3 + 458^3
2003: 1675045225 ───► 1081^3 + 744^3 ──and── 1153^3 + 522^3
2004: 1675958167 ───► 1096^3 + 711^3 ──and── 1159^3 + 492^3
2005: 1676926719 ───► 1095^3 + 714^3 ──and── 1188^3 + 63^3
2006: 1677646971 ───► 990^3 + 891^3 ──and── 1188^3 + 99^3
Ring
# Project : Taxicab numbers
num = 0
for n = 1 to 500000
nr = 0
tax = []
for m = 1 to 75
for p = m + 1 to 75
if n = pow(m, 3) + pow(p, 3)
add(tax, m)
add(tax, p)
nr = nr + 1
ok
next
next
if nr > 1
num = num + 1
see "" + num + " " + n + " => " + tax[1] + "^3 + " + tax[2] + "^3" + ", "
see "" + tax[3] + "^3 +" + tax[4] + "^3" + nl
if num = 25
exit
ok
ok
next
see "ok" + nl
Output:
1 1729 => 9³ + 10³, 1³ + 12³
2 4104 => 9³ + 15³, 2³ + 16³
3 13832 => 18³ + 20³, 2³ + 24³
4 20683 => 19³ + 24³, 10³ + 27³
5 32832 => 18³ + 30³, 4³ + 32³
6 39312 => 15³ + 33³, 2³ + 34³
7 40033 => 16³ + 33³, 9³ + 34³
8 46683 => 27³ + 30³, 3³ + 36³
9 64232 => 26³ + 36³, 17³ + 39³
10 65728 => 31³ + 33³, 12³ + 40³
11 110656 => 36³ + 40³, 4³ + 48³
12 110808 => 27³ + 45³, 6³ + 48³
13 134379 => 38³ + 43³, 12³ + 51³
14 149389 => 29³ + 50³, 8³ + 53³
15 165464 => 38³ + 48³, 20³ + 54³
16 171288 => 24³ + 54³, 17³ + 55³
17 195841 => 22³ + 57³, 9³ + 58³
18 216027 => 22³ + 59³, 3³ + 60³
19 216125 => 45³ + 50³, 5³ + 60³
20 262656 => 36³ + 60³, 8³ + 64³
21 314496 => 30³ + 66³, 4³ + 68³
22 320264 => 32³ + 66³, 18³ + 68³
23 327763 => 51³ + 58³, 30³ + 67³
24 373464 => 54³ + 60³, 6³ + 72³
25 402597 => 56³ + 61³, 42³ + 69³
ok
Ruby
def taxicab_number(nmax=1200) [*1..nmax].repeated_combination(2).group_by{|x,y| x**3 + y**3}.select{|k,v| v.size>1}.sort end t = [0] + taxicab_number [*1..25, *2000...2007].each do |i| puts "%4d: %10d" % [i, t[i][0]] + t[i][1].map{|a| " = %4d**3 + %4d**3" % a}.join end
{{out}}
1: 1729 = 1**3 + 12**3 = 9**3 + 10**3
2: 4104 = 2**3 + 16**3 = 9**3 + 15**3
3: 13832 = 2**3 + 24**3 = 18**3 + 20**3
4: 20683 = 10**3 + 27**3 = 19**3 + 24**3
5: 32832 = 4**3 + 32**3 = 18**3 + 30**3
6: 39312 = 2**3 + 34**3 = 15**3 + 33**3
7: 40033 = 9**3 + 34**3 = 16**3 + 33**3
8: 46683 = 3**3 + 36**3 = 27**3 + 30**3
9: 64232 = 17**3 + 39**3 = 26**3 + 36**3
10: 65728 = 12**3 + 40**3 = 31**3 + 33**3
11: 110656 = 4**3 + 48**3 = 36**3 + 40**3
12: 110808 = 6**3 + 48**3 = 27**3 + 45**3
13: 134379 = 12**3 + 51**3 = 38**3 + 43**3
14: 149389 = 8**3 + 53**3 = 29**3 + 50**3
15: 165464 = 20**3 + 54**3 = 38**3 + 48**3
16: 171288 = 17**3 + 55**3 = 24**3 + 54**3
17: 195841 = 9**3 + 58**3 = 22**3 + 57**3
18: 216027 = 3**3 + 60**3 = 22**3 + 59**3
19: 216125 = 5**3 + 60**3 = 45**3 + 50**3
20: 262656 = 8**3 + 64**3 = 36**3 + 60**3
21: 314496 = 4**3 + 68**3 = 30**3 + 66**3
22: 320264 = 18**3 + 68**3 = 32**3 + 66**3
23: 327763 = 30**3 + 67**3 = 51**3 + 58**3
24: 373464 = 6**3 + 72**3 = 54**3 + 60**3
25: 402597 = 42**3 + 69**3 = 56**3 + 61**3
2000: 1671816384 = 428**3 + 1168**3 = 940**3 + 944**3
2001: 1672470592 = 29**3 + 1187**3 = 632**3 + 1124**3
2002: 1673170856 = 458**3 + 1164**3 = 828**3 + 1034**3
2003: 1675045225 = 522**3 + 1153**3 = 744**3 + 1081**3
2004: 1675958167 = 492**3 + 1159**3 = 711**3 + 1096**3
2005: 1676926719 = 63**3 + 1188**3 = 714**3 + 1095**3
2006: 1677646971 = 99**3 + 1188**3 = 891**3 + 990**3
Rust
use std::collections::HashMap; use itertools::Itertools; fn cubes(n: u64) -> Vec<u64> { let mut cube_vector = Vec::new(); for i in 1..=n { cube_vector.push(i.pow(3)); } cube_vector } fn main() { let c = cubes(1201); let it = c.iter().combinations(2); let mut m = HashMap::new(); for x in it { let sum = x[0] + x[1]; m.entry(sum).or_insert(Vec::new()).push(x) } let mut result = Vec::new(); for (k,v) in m.iter() { if v.len() > 1 { result.push((k,v)); } } result.sort(); for f in result { println!("{:?}", f); } }
{{out}}
(1729, [[1, 1728], [729, 1000]])
(4104, [[8, 4096], [729, 3375]])
(13832, [[8, 13824], [5832, 8000]])
(20683, [[1000, 19683], [6859, 13824]])
(32832, [[64, 32768], [5832, 27000]])
(39312, [[8, 39304], [3375, 35937]])
(40033, [[729, 39304], [4096, 35937]])
(46683, [[27, 46656], [19683, 27000]])
(64232, [[4913, 59319], [17576, 46656]])
(65728, [[1728, 64000], [29791, 35937]])
(110656, [[64, 110592], [46656, 64000]])
(110808, [[216, 110592], [19683, 91125]])
(134379, [[1728, 132651], [54872, 79507]])
(149389, [[512, 148877], [24389, 125000]])
(165464, [[8000, 157464], [54872, 110592]])
(171288, [[4913, 166375], [13824, 157464]])
(195841, [[729, 195112], [10648, 185193]])
(216027, [[27, 216000], [10648, 205379]])
(216125, [[125, 216000], [91125, 125000]])
(262656, [[512, 262144], [46656, 216000]])
(314496, [[64, 314432], [27000, 287496]])
Scala
import scala.math.pow implicit class Pairs[A, B]( p:List[(A, B)]) { def collectPairs: Map[A, List[B]] = p.groupBy(_._1).mapValues(_.map(_._2)).filterNot(_._2.size<2) } // Make a sorted List of Taxi Cab Numbers. Limit it to the cube of 1200 because we know it's high enough. val taxiNums = { (1 to 1200).toList // Start with a sequential list of integers .combinations(2).toList // Find all two number combinations .map { case a :: b :: nil => ((pow(a, 3) + pow(b, 3)).toInt, (a, b)) case _ => 0 ->(0, 0) } // Turn the list into the sum of two cubes and // remember what we started with, eg. 28->(1,3) .collectPairs // Only keep taxi cab numbers with a duplicate .toList.sortBy(_._1) // Sort the results } def output() : Unit = { println( "%20s".format( "Taxi Cab Numbers" ) ) println( "%20s%15s%15s".format( "-"*20, "-"*15, "-"*15 ) ) taxiNums.take(25) foreach { case (p, a::b::Nil) => println( "%20d\t(%d\u00b3 + %d\u00b3)\t\t(%d\u00b3 + %d\u00b3)".format(p,a._1,a._2,b._1,b._2) ) } taxiNums.slice(1999,2007) foreach { case (p, a::b::Nil) => println( "%20d\t(%d\u00b3 + %d\u00b3)\t(%d\u00b3 + %d\u00b3)".format(p,a._1,a._2,b._1,b._2) ) } }
{{out}}
Taxi Cab Numbers
--------------------------------------------------
1729 (1³ + 12³) (9³ + 10³)
4104 (2³ + 16³) (9³ + 15³)
13832 (2³ + 24³) (18³ + 20³)
20683 (10³ + 27³) (19³ + 24³)
32832 (4³ + 32³) (18³ + 30³)
39312 (2³ + 34³) (15³ + 33³)
40033 (9³ + 34³) (16³ + 33³)
46683 (3³ + 36³) (27³ + 30³)
64232 (17³ + 39³) (26³ + 36³)
65728 (12³ + 40³) (31³ + 33³)
110656 (4³ + 48³) (36³ + 40³)
110808 (6³ + 48³) (27³ + 45³)
134379 (12³ + 51³) (38³ + 43³)
149389 (8³ + 53³) (29³ + 50³)
165464 (20³ + 54³) (38³ + 48³)
171288 (17³ + 55³) (24³ + 54³)
195841 (9³ + 58³) (22³ + 57³)
216027 (3³ + 60³) (22³ + 59³)
216125 (5³ + 60³) (45³ + 50³)
262656 (8³ + 64³) (36³ + 60³)
314496 (4³ + 68³) (30³ + 66³)
320264 (18³ + 68³) (32³ + 66³)
327763 (30³ + 67³) (51³ + 58³)
373464 (6³ + 72³) (54³ + 60³)
402597 (42³ + 69³) (56³ + 61³)
1671816384 (428³ + 1168³) (940³ + 944³)
1672470592 (29³ + 1187³) (632³ + 1124³)
1673170856 (458³ + 1164³) (828³ + 1034³)
1675045225 (522³ + 1153³) (744³ + 1081³)
1675958167 (492³ + 1159³) (711³ + 1096³)
1676926719 (63³ + 1188³) (714³ + 1095³)
1677646971 (99³ + 1188³) (891³ + 990³)
Scheme
{{libheader|Scheme/SRFIs}}
(import (scheme base)
(scheme write)
(srfi 1) ; lists
(srfi 69) ; hash tables
(srfi 132)) ; sorting
(define *max-n* 1500) ; let's go up to here, maximum for x and y
(define *numbers* (make-hash-table eqv?)) ; hash table for total -> list of list of pairs
(define (retrieve key) (hash-table-ref/default *numbers* key '()))
;; add all combinations to the hash table
(do ((i 1 (+ i 1)))
((= i *max-n*) )
(do ((j (+ 1 i) (+ j 1)))
((= j *max-n*) )
(let ((n (+ (* i i i) (* j j j))))
(hash-table-set! *numbers* n
(cons (list i j) (retrieve n))))))
(define (display-number i key)
(display (+ 1 i)) (display ": ")
(display key) (display " -> ")
(display (retrieve key)) (newline))
(let ((sorted-keys (list-sort <
(filter (lambda (key) (> (length (retrieve key)) 1))
(hash-table-keys *numbers*)))))
;; first 25
(for-each (lambda (i) (display-number i (list-ref sorted-keys i)))
(iota 25))
;; 2000-2006
(for-each (lambda (i) (display-number i (list-ref sorted-keys i)))
(iota 7 1999))
)
{{out}}
1: 1729 -> ((9 10) (1 12))
2: 4104 -> ((9 15) (2 16))
3: 13832 -> ((18 20) (2 24))
4: 20683 -> ((19 24) (10 27))
5: 32832 -> ((18 30) (4 32))
6: 39312 -> ((15 33) (2 34))
7: 40033 -> ((16 33) (9 34))
8: 46683 -> ((27 30) (3 36))
9: 64232 -> ((26 36) (17 39))
10: 65728 -> ((31 33) (12 40))
11: 110656 -> ((36 40) (4 48))
12: 110808 -> ((27 45) (6 48))
13: 134379 -> ((38 43) (12 51))
14: 149389 -> ((29 50) (8 53))
15: 165464 -> ((38 48) (20 54))
16: 171288 -> ((24 54) (17 55))
17: 195841 -> ((22 57) (9 58))
18: 216027 -> ((22 59) (3 60))
19: 216125 -> ((45 50) (5 60))
20: 262656 -> ((36 60) (8 64))
21: 314496 -> ((30 66) (4 68))
22: 320264 -> ((32 66) (18 68))
23: 327763 -> ((51 58) (30 67))
24: 373464 -> ((54 60) (6 72))
25: 402597 -> ((56 61) (42 69))
2000: 1671816384 -> ((940 944) (428 1168))
2001: 1672470592 -> ((632 1124) (29 1187))
2002: 1673170856 -> ((828 1034) (458 1164))
2003: 1675045225 -> ((744 1081) (522 1153))
2004: 1675958167 -> ((711 1096) (492 1159))
2005: 1676926719 -> ((714 1095) (63 1188))
2006: 1677646971 -> ((891 990) (99 1188))
Sidef
{{trans|Perl 6}}
var (start=1, end=25) = ARGV.map{.to_i}... func display (h, start, end) { var i = start for n in [h.grep {|_,v| v.len > 1 }.keys.sort_by{.to_i}[start-1 .. end-1]] { printf("%4d %10d =>\t%s\n", i++, n, h{n}.map{ "%4d³ + %-s" % (.first, "#{.last}³") }.join(",\t")) } } var taxi = Hash() var taxis = 0 var terminate = 0 for c1 (1..Inf) { if (0<terminate && terminate<c1) { display(taxi, start, end) break } var c = c1**3 for c2 (1..c1) { var this = (c2**3 + c) taxi{this} := [] << [c2, c1] ++taxis if (taxi{this}.len == 2) if (taxis==end && !terminate) { terminate = taxi.grep{|_,v| v.len > 1 }.keys.map{.to_i}.max.root(3) } } }
{{out}}
1 1729 => 9³ + 10³, 1³ + 12³
2 4104 => 9³ + 15³, 2³ + 16³
3 13832 => 18³ + 20³, 2³ + 24³
4 20683 => 19³ + 24³, 10³ + 27³
5 32832 => 18³ + 30³, 4³ + 32³
6 39312 => 15³ + 33³, 2³ + 34³
7 40033 => 16³ + 33³, 9³ + 34³
8 46683 => 27³ + 30³, 3³ + 36³
9 64232 => 26³ + 36³, 17³ + 39³
10 65728 => 31³ + 33³, 12³ + 40³
11 110656 => 36³ + 40³, 4³ + 48³
12 110808 => 27³ + 45³, 6³ + 48³
13 134379 => 38³ + 43³, 12³ + 51³
14 149389 => 29³ + 50³, 8³ + 53³
15 165464 => 38³ + 48³, 20³ + 54³
16 171288 => 24³ + 54³, 17³ + 55³
17 195841 => 22³ + 57³, 9³ + 58³
18 216027 => 22³ + 59³, 3³ + 60³
19 216125 => 45³ + 50³, 5³ + 60³
20 262656 => 36³ + 60³, 8³ + 64³
21 314496 => 30³ + 66³, 4³ + 68³
22 320264 => 32³ + 66³, 18³ + 68³
23 327763 => 51³ + 58³, 30³ + 67³
24 373464 => 54³ + 60³, 6³ + 72³
25 402597 => 56³ + 61³, 42³ + 69³
With passed parameters 2000 and 2006:
2000 1671816384 => 940³ + 944³, 428³ + 1168³
2001 1672470592 => 632³ + 1124³, 29³ + 1187³
2002 1673170856 => 828³ + 1034³, 458³ + 1164³
2003 1675045225 => 744³ + 1081³, 522³ + 1153³
2004 1675958167 => 711³ + 1096³, 492³ + 1159³
2005 1676926719 => 714³ + 1095³, 63³ + 1188³
2006 1677646971 => 891³ + 990³, 99³ + 1188³
Tcl
{{works with|Tcl|8.6}} {{trans|Python}}
package require Tcl 8.6 proc heappush {heapName item} { upvar 1 $heapName heap set idx [lsearch -bisect -index 0 -integer $heap [lindex $item 0]] set heap [linsert $heap [expr {$idx + 1}] $item] } coroutine cubesum apply {{} { yield set h {} set n 1 while true { while {![llength $h] || [lindex $h 0 0] > $n**3} { heappush h [list [expr {$n**3 + 1}] $n 1] incr n } set h [lassign $h item] yield $item lassign $item s x y if {[incr y] < $x} { heappush h [list [expr {$x**3 + $y**3}] $x $y] } } }} coroutine taxis apply {{} { yield set out {{0 0 0}} while true { set s [cubesum] if {[lindex $s 0] == [lindex $out end 0]} { lappend out $s } else { if {[llength $out] > 1} {yield $out} set out [list $s] } } }} # Put a cache in front for convenience variable taxis {} proc taxi {n} { variable taxis while {$n > [llength $taxis]} {lappend taxis [taxis]} return [lindex $taxis [expr {$n-1}]] } set 3 "\u00b3" for {set n 1} {$n <= 25} {incr n} { puts ${n}:[join [lmap t [taxi $n] {format " %d = %d$3 + %d$3" {*}$t}] ","] } for {set n 2000} {$n <= 2006} {incr n} { puts ${n}:[join [lmap t [taxi $n] {format " %d = %d$3 + %d$3" {*}$t}] ","] }
{{out}}
1: 1729 = 10³ + 9³, 1729 = 12³ + 1³
2: 4104 = 15³ + 9³, 4104 = 16³ + 2³
3: 13832 = 20³ + 18³, 13832 = 24³ + 2³
4: 20683 = 24³ + 19³, 20683 = 27³ + 10³
5: 32832 = 30³ + 18³, 32832 = 32³ + 4³
6: 39312 = 33³ + 15³, 39312 = 34³ + 2³
7: 40033 = 33³ + 16³, 40033 = 34³ + 9³
8: 46683 = 30³ + 27³, 46683 = 36³ + 3³
9: 64232 = 36³ + 26³, 64232 = 39³ + 17³
10: 65728 = 33³ + 31³, 65728 = 40³ + 12³
11: 110656 = 40³ + 36³, 110656 = 48³ + 4³
12: 110808 = 45³ + 27³, 110808 = 48³ + 6³
13: 134379 = 43³ + 38³, 134379 = 51³ + 12³
14: 149389 = 50³ + 29³, 149389 = 53³ + 8³
15: 165464 = 48³ + 38³, 165464 = 54³ + 20³
16: 171288 = 54³ + 24³, 171288 = 55³ + 17³
17: 195841 = 57³ + 22³, 195841 = 58³ + 9³
18: 216027 = 59³ + 22³, 216027 = 60³ + 3³
19: 216125 = 50³ + 45³, 216125 = 60³ + 5³
20: 262656 = 60³ + 36³, 262656 = 64³ + 8³
21: 314496 = 66³ + 30³, 314496 = 68³ + 4³
22: 320264 = 66³ + 32³, 320264 = 68³ + 18³
23: 327763 = 58³ + 51³, 327763 = 67³ + 30³
24: 373464 = 60³ + 54³, 373464 = 72³ + 6³
25: 402597 = 61³ + 56³, 402597 = 69³ + 42³
2000: 1671816384 = 944³ + 940³, 1671816384 = 1168³ + 428³
2001: 1672470592 = 1124³ + 632³, 1672470592 = 1187³ + 29³
2002: 1673170856 = 1034³ + 828³, 1673170856 = 1164³ + 458³
2003: 1675045225 = 1081³ + 744³, 1675045225 = 1153³ + 522³
2004: 1675958167 = 1096³ + 711³, 1675958167 = 1159³ + 492³
2005: 1676926719 = 1095³ + 714³, 1676926719 = 1188³ + 63³
2006: 1677646971 = 990³ + 891³, 1677646971 = 1188³ + 99³
VBA
Public Type tuple
i As Variant
j As Variant
sum As Variant
End Type
Public Type tuple3
i1 As Variant
j1 As Variant
i2 As Variant
j2 As Variant
i3 As Variant
j3 As Variant
sum As Variant
End Type
Sub taxicab_numbers()
Dim i As Variant, j As Variant
Dim k As Long
Const MAX = 2019
Dim p(MAX) As Variant
Const bigMAX = (MAX + 1) * (MAX / 2)
Dim big(1 To bigMAX) As tuple
Const resMAX = 4400
Dim res(1 To resMAX) As tuple3
For i = 1 To MAX
p(i) = CDec(i * i * i) 'convert Variant to Decimal
Next i 'wich hold numbers upto 10^28
k = 1
For i = 1 To MAX
For j = i To MAX
big(k).i = CDec(i)
big(k).j = CDec(j)
big(k).sum = CDec(p(i) + p(j))
k = k + 1
Next j
Next i
n = 1
Quicksort big, LBound(big), UBound(big)
For i = 1 To bigMAX - 1
If big(i).sum = big(i + 1).sum Then
res(n).i1 = CStr(big(i).i)
res(n).j1 = CStr(big(i).j)
res(n).i2 = CStr(big(i + 1).i)
res(n).j2 = CStr(big(i + 1).j)
If big(i + 1).sum = big(i + 2).sum Then
res(n).i3 = CStr(big(i + 2).i)
res(n).j3 = CStr(big(i + 2).j)
i = i + 1
End If
res(n).sum = CStr(big(i).sum)
n = n + 1
i = i + 1
End If
Next i
Debug.Print n - 1; " taxis"
For i = 1 To 25
With res(i)
Debug.Print String$(4 - Len(CStr(i)), " "); i;
Debug.Print String$(11 - Len(.sum), " "); .sum; " = ";
Debug.Print String$(4 - Len(.i1), " "); .i1; "^3 +";
Debug.Print String$(4 - Len(.j1), " "); .j1; "^3 = ";
Debug.Print String$(4 - Len(.i2), " "); .i2; "^3 +";
Debug.Print String$(4 - Len(.j2), " "); .j2; "^3"
End With
Next i
Debug.Print
For i = 2000 To 2006
With res(i)
Debug.Print String$(4 - Len(CStr(i)), " "); i;
Debug.Print String$(11 - Len(.sum), " "); .sum; " = ";
Debug.Print String$(4 - Len(.i1), " "); .i1; "^3 +";
Debug.Print String$(4 - Len(.j1), " "); .j1; "^3 = ";
Debug.Print String$(4 - Len(.i2), " "); .i2; "^3 +";
Debug.Print String$(4 - Len(.j2), " "); .j2; "^3"
End With
Next i
Debug.Print
For i = 1 To resMAX
If res(i).i3 <> "" Then
With res(i)
Debug.Print String$(4 - Len(CStr(i)), " "); i;
Debug.Print String$(11 - Len(.sum), " "); .sum; " = ";
Debug.Print String$(4 - Len(.i1), " "); .i1; "^3 +";
Debug.Print String$(4 - Len(.j1), " "); .j1; "^3 = ";
Debug.Print String$(4 - Len(.i2), " "); .i2; "^3 +";
Debug.Print String$(4 - Len(.j2), " "); .j2; "^3";
Debug.Print String$(4 - Len(.i3), " "); .i3; "^3 +";
Debug.Print String$(4 - Len(.j3), " "); .j3; "^3"
End With
End If
Next i
End Sub
Sub Quicksort(vArray() As tuple, arrLbound As Long, arrUbound As Long)
'https://wellsr.com/vba/2018/excel/vba-quicksort-macro-to-sort-arrays-fast/
'Sorts a one-dimensional VBA array from smallest to largest
'using a very fast quicksort algorithm variant.
'Adapted to multidimensions/typedef
Dim pivotVal As Variant
Dim vSwap As tuple
Dim tmpLow As Long
Dim tmpHi As Long
tmpLow = arrLbound
tmpHi = arrUbound
pivotVal = vArray((arrLbound + arrUbound) \ 2).sum
While (tmpLow <= tmpHi) 'divide
While (vArray(tmpLow).sum < pivotVal And tmpLow < arrUbound)
tmpLow = tmpLow + 1
Wend
While (pivotVal < vArray(tmpHi).sum And tmpHi > arrLbound)
tmpHi = tmpHi - 1
Wend
If (tmpLow <= tmpHi) Then
vSwap.i = vArray(tmpLow).i
vSwap.j = vArray(tmpLow).j
vSwap.sum = vArray(tmpLow).sum
vArray(tmpLow).i = vArray(tmpHi).i
vArray(tmpLow).j = vArray(tmpHi).j
vArray(tmpLow).sum = vArray(tmpHi).sum
vArray(tmpHi).i = vSwap.i
vArray(tmpHi).j = vSwap.j
vArray(tmpHi).sum = vSwap.sum
tmpLow = tmpLow + 1
tmpHi = tmpHi - 1
End If
Wend
If (arrLbound < tmpHi) Then Quicksort vArray, arrLbound, tmpHi 'conquer
If (tmpLow < arrUbound) Then Quicksort vArray, tmpLow, arrUbound 'conquer
End Sub
{{out}}
4399 taxis
1 1729 = 9^3 + 10^3 = 1^3 + 12^3
2 4104 = 2^3 + 16^3 = 9^3 + 15^3
3 13832 = 2^3 + 24^3 = 18^3 + 20^3
4 20683 = 19^3 + 24^3 = 10^3 + 27^3
5 32832 = 18^3 + 30^3 = 4^3 + 32^3
6 39312 = 15^3 + 33^3 = 2^3 + 34^3
7 40033 = 16^3 + 33^3 = 9^3 + 34^3
8 46683 = 27^3 + 30^3 = 3^3 + 36^3
9 64232 = 26^3 + 36^3 = 17^3 + 39^3
10 65728 = 31^3 + 33^3 = 12^3 + 40^3
11 110656 = 4^3 + 48^3 = 36^3 + 40^3
12 110808 = 27^3 + 45^3 = 6^3 + 48^3
13 134379 = 12^3 + 51^3 = 38^3 + 43^3
14 149389 = 29^3 + 50^3 = 8^3 + 53^3
15 165464 = 38^3 + 48^3 = 20^3 + 54^3
16 171288 = 24^3 + 54^3 = 17^3 + 55^3
17 195841 = 9^3 + 58^3 = 22^3 + 57^3
18 216027 = 22^3 + 59^3 = 3^3 + 60^3
19 216125 = 45^3 + 50^3 = 5^3 + 60^3
20 262656 = 36^3 + 60^3 = 8^3 + 64^3
21 314496 = 4^3 + 68^3 = 30^3 + 66^3
22 320264 = 32^3 + 66^3 = 18^3 + 68^3
23 327763 = 51^3 + 58^3 = 30^3 + 67^3
24 373464 = 54^3 + 60^3 = 6^3 + 72^3
25 402597 = 56^3 + 61^3 = 42^3 + 69^3
2000 1671816384 = 940^3 + 944^3 = 428^3 +1168^3
2001 1672470592 = 29^3 +1187^3 = 632^3 +1124^3
2002 1673170856 = 828^3 +1034^3 = 458^3 +1164^3
2003 1675045225 = 744^3 +1081^3 = 522^3 +1153^3
2004 1675958167 = 492^3 +1159^3 = 711^3 +1096^3
2005 1676926719 = 714^3 +1095^3 = 63^3 +1188^3
2006 1677646971 = 99^3 +1188^3 = 891^3 + 990^3
455 87539319 = 167^3 + 436^3 = 228^3 + 423^3 255^3 + 414^3
535 119824488 = 90^3 + 492^3 = 346^3 + 428^3 11^3 + 493^3
588 143604279 = 408^3 + 423^3 = 359^3 + 460^3 111^3 + 522^3
655 175959000 = 70^3 + 560^3 = 315^3 + 525^3 198^3 + 552^3
888 327763000 = 300^3 + 670^3 = 339^3 + 661^3 510^3 + 580^3
1299 700314552 = 334^3 + 872^3 = 456^3 + 846^3 510^3 + 828^3
1398 804360375 = 15^3 + 930^3 = 295^3 + 920^3 198^3 + 927^3
1515 958595904 = 22^3 + 986^3 = 180^3 + 984^3 692^3 + 856^3
1660 1148834232 = 718^3 + 920^3 = 816^3 + 846^3 222^3 +1044^3
1837 1407672000 = 140^3 +1120^3 = 396^3 +1104^3 630^3 +1050^3
2100 1840667192 = 681^3 +1151^3 = 372^3 +1214^3 225^3 +1223^3
2143 1915865217 = 9^3 +1242^3 = 484^3 +1217^3 969^3 +1002^3
2365 2363561613 = 501^3 +1308^3 = 684^3 +1269^3 765^3 +1242^3
2480 2622104000 = 1020^3 +1160^3 = 600^3 +1340^3 678^3 +1322^3
2670 3080802816 = 904^3 +1328^3 = 81^3 +1455^3 456^3 +1440^3
2732 3235261176 = 33^3 +1479^3 = 270^3 +1476^31038^3 +1284^3
2845 3499524728 = 116^3 +1518^3 = 350^3 +1512^31169^3 +1239^3
2895 3623721192 = 348^3 +1530^3 = 761^3 +1471^31098^3 +1320^3
2979 3877315533 = 1224^3 +1269^3 = 1077^3 +1380^3 333^3 +1566^3
3293 4750893000 = 210^3 +1680^3 = 945^3 +1575^3 594^3 +1656^3
3562 5544709352 = 207^3 +1769^3 = 1076^3 +1626^3 842^3 +1704^3
3589 5602516416 = 912^3 +1692^3 = 1020^3 +1656^3 668^3 +1744^3
3826 6434883000 = 590^3 +1840^3 = 30^3 +1860^3 396^3 +1854^3
4162 7668767232 = 44^3 +1972^3 = 1384^3 +1712^3 360^3 +1968^3
4359 8849601000 = 1017^3 +1983^3 = 1530^3 +1740^3 900^3 +2010^3
zkl
{{trans|D}} An array of bytes is used to hold n, where array[n³+m³]==n.
fcn taxiCabNumbers{
const HeapSZ=0d5_000_000;
iCubes:=[1..120].apply("pow",3);
sum2cubes:=Data(HeapSZ).fill(0); // BFheap of 1 byte zeros
taxiNums:=List();
foreach i,i3 in ([1..].zip(iCubes)){
foreach j,j3 in ([i+1..].zip(iCubes[i,*])){
ij3:=i3+j3;
if(z:=sum2cubes[ij3]){
taxiNums.append(T(ij3,
z,(ij3-z.pow(3)).toFloat().pow(1.0/3).round().toInt(),
i,j));
}
else sum2cubes[ij3]=i;
}
}
taxiNums.sort(fcn([(a,_)],[(b,_)]){ a<b })
}
fcn print(n,taxiNums){
[n..].zip(taxiNums).pump(Console.println,fcn([(n,t)]){
"%4d: %10,d = %2d\u00b3 + %2d\u00b3 = %2d\u00b3 + %2d\u00b3".fmt(n,t.xplode())
})
}
taxiNums:=taxiCabNumbers(); // 63 pairs
taxiNums[0,25]:print(1,_);
{{out}}
1: 1,729 = 1³ + 12³ = 9³ + 10³
2: 4,104 = 2³ + 16³ = 9³ + 15³
3: 13,832 = 2³ + 24³ = 18³ + 20³
4: 20,683 = 10³ + 27³ = 19³ + 24³
5: 32,832 = 4³ + 32³ = 18³ + 30³
6: 39,312 = 2³ + 34³ = 15³ + 33³
7: 40,033 = 9³ + 34³ = 16³ + 33³
8: 46,683 = 3³ + 36³ = 27³ + 30³
9: 64,232 = 17³ + 39³ = 26³ + 36³
10: 65,728 = 12³ + 40³ = 31³ + 33³
11: 110,656 = 4³ + 48³ = 36³ + 40³
12: 110,808 = 6³ + 48³ = 27³ + 45³
13: 134,379 = 12³ + 51³ = 38³ + 43³
14: 149,389 = 8³ + 53³ = 29³ + 50³
15: 165,464 = 20³ + 54³ = 38³ + 48³
16: 171,288 = 17³ + 55³ = 24³ + 54³
17: 195,841 = 9³ + 58³ = 22³ + 57³
18: 216,027 = 3³ + 60³ = 22³ + 59³
19: 216,125 = 5³ + 60³ = 45³ + 50³
20: 262,656 = 8³ + 64³ = 36³ + 60³
21: 314,496 = 4³ + 68³ = 30³ + 66³
22: 320,264 = 18³ + 68³ = 32³ + 66³
23: 327,763 = 30³ + 67³ = 51³ + 58³
24: 373,464 = 6³ + 72³ = 54³ + 60³
25: 402,597 = 42³ + 69³ = 56³ + 61³
{{trans|Python}} Using a binary heap:
fcn cubeSum{
heap,n:=Heap(fcn([(a,_)],[(b,_)]){ a<=b }), 1; // heap cnt maxes out @ 244
while(1){
while(heap.empty or heap.top[0]>n.pow(3)){ # could also pre-calculate cubes
heap.push(T(n.pow(3) + 1, n,1));
n+=1;
}
s,x,y:= sxy:=heap.pop();
vm.yield(sxy);
y+=1;
if(y<x) # should be y <= x?
heap.push(T(x.pow(3) + y.pow(3), x,y));
}
}
fcn taxis{
out:=List(T(0,0,0));
foreach s in (Utils.Generator(cubeSum)){
if(s[0]==out[-1][0]) out.append(s);
else{
if(out.len()>1) vm.yield(out);
out.clear(s)
}
}
}
n:=0;
foreach x in (Utils.Generator(taxis)){
n += 1;
if(n >= 2006) break;
if(n <= 25 or n >= 2000) println(n,": ",x);
}
And a quickie heap implementation:
class Heap{ // binary heap
fcn init(lteqFcn='<=){
var [const, private] heap=List().pad(64,Void); // a power of 2
var cnt=0, cmp=lteqFcn;
}
fcn push(v){
// Resize the heap if it is too small to hold another item
if (cnt==heap.len()) heap.pad(cnt*2,Void);
index:=cnt; cnt+=1; while(index){ // Find out where to put the element
parent:=(index - 1)/2;
if(cmp(heap[parent],v)) break;
heap[index] = heap[parent];
index = parent;
}
heap[index] = v;
}
fcn pop{ // Remove the biggest element and return it
if(not cnt) return(Void);
v,temp:=heap[0], heap[cnt-=1];
// Reorder the elements
index:=0; while(1){ // Find the child to swap with
swap:=index*2 + 1;
if (swap>=cnt) break; // If there are no children, the heap is reordered
other:=swap + 1;
if(other<cnt and cmp(heap[other],heap[swap])) swap = other;
if(cmp(temp,heap[swap])) break; // If the bigger child is less than or equal to its parent, the heap is reordered
heap[index]=heap[swap];
index = swap;
}
heap[index] = temp;
v
}
var [proxy] top=fcn { if(cnt==0) Void else heap[0] };
var [proxy] empty=fcn{ (not cnt) };
}
{{out}}
1: L(L(1729,10,9),L(1729,12,1))
...
23: L(L(327763,67,30),L(327763,58,51))
24: L(L(373464,60,54),L(373464,72,6))
25: L(L(402597,61,56),L(402597,69,42))
2000: L(L(1671816384,944,940),L(1671816384,1168,428))
2001: L(L(1672470592,1124,632),L(1672470592,1187,29))
2002: L(L(1673170856,1034,828),L(1673170856,1164,458))
2003: L(L(1675045225,1153,522),L(1675045225,1081,744))
2004: L(L(1675958167,1096,711),L(1675958167,1159,492))
2005: L(L(1676926719,1188,63),L(1676926719,1095,714))
ZX Spectrum Basic
This will, in the strictest sense, work. Don't hold your breath though; after six hours on an emulator at full speed it had generated the first 10 numbers. Getting to 2006 may take a while longer.
You cannot fit the whole 1625-entry table of cubes (and this program on top) into the 16K ZX Spectrum. Replace all 1625s with 1200s to resolve; numerically unjustified as an exhaustive search, but we know this will be sufficient to find the 2006th number. Eventually.
10 DIM f(1625): REM populating a cube table at the start will be faster than computing the cubes on the fly
20 FOR x=1 TO 1625
30 LET f(x)=x*x*x: REM x*x*x rather than x^3 as the ZX Spectrum's exponentiation function is legendarily slow
40 NEXT x
50 LET c=0
60 FOR x=1 TO 4294967295: REM the highest number the ZX Spectrum Basic can accurately hold internally; floor (cuberoot max)=1625, hence the table limit
70 LET k=0
80 FOR m=1 TO 1625
90 FOR n=m+1 TO 1625
100 IF f(m)+f(n)=x THEN GOTO 160
110 IF f(n)>=x THEN LET n=1625: REM overshot, break out of the loop
120 IF f(m)>=x THEN LET m=1625
130 NEXT n
140 NEXT m
150 NEXT x
160 IF k=1 THEN LET q=m: LET r=n: GO TO 230: REM got one!
170 LET o=m
180 LET p=n
190 LET k=1
200 NEXT n
210 NEXT m
220 NEXT x
230 LET c=c+1
240 IF c>25 AND c<2000 THEN GO TO 330
250 LET t$="": REM convert number to string; while ZX Spectrum Basic can store all the digits of integers up to 2^32-1...
260 LET t=INT (x/100000): REM ...it will resort to scientific notation trying to display any more than eight digits
270 LET b=x-t*100000
280 IF t=0 THEN GO TO 300: REM omit leading zero
290 LET t$=STR$ t
300 LET t$=t$+STR$ b
310 PRINT c;":";t$;"=";q;"^3+";r;"^3=";o;"^3+";p;"^3"
320 POKE 23692,10: REM suppress "scroll?" prompt when screen fills up at c=22
330 IF c=2006 THEN LET x=4294967295: LET n=1625: LET m=1625
340 NEXT n
350 NEXT m
360 NEXT x
{{out}}
1:1729=9^3+10^3=1^3+12^3
2:4104=9^3+15^3=2^3+16^3
3:13832=18^3+20^3=2^3+24^3
4:20683=19^3+24^3=10^3+27^3
5:32832=18^3+30^3=4^3+32^3
6:39312=15^3+33^3=2^3+34^3
7:40033=16^3+33^3=9^3+34^3
8:46683=27^3+30^3=3^3+36^3
9:64232=26^3+36^3=17^3+39^3
10:65728=31^3+33^3=12^3+40^3
D BREAK into program, 100:1
This program produces the first 25 Taxicab numbers. It is written with speed in mind. The runtime is about 45 minutes on a ZX Spectrum (3.5 Mhz).
10 LET T=0: DIM F(72): LET D=0: LET S=0: LET B=0: LET A=0: LET C=0
20 DIM H(50): DIM Y(50,2): FOR D=1 TO 72: LET F(D)=D*D*D: NEXT D
30 FOR A=1 TO 58: FOR B=A+1 TO 72: LET S=F(A)+F(B): FOR D=B-1 TO A STEP -1
40 LET T=S-F(D): IF T>F(D) THEN NEXT B: NEXT A: GO TO 90
45 IF s>405224 THEN GO TO 70
50 IF F(INT (EXP (LN (T)/3)+.5))=T THEN GO TO 80
60 NEXT D
70 NEXT B: NEXT A: GO TO 90
80 PRINT S,: LET C=C+1: LET H(C)=S: LET Y(C,1)=A*65536+B: LET Y(C,2)=INT (EXP (LN (T)/3)+.5)*65536+D: GO TO 70
90 LET S=INT (C/2)
100 LET T=0: FOR A=1 TO C-S: IF H(A)>H(A+S) THEN LET T=H(A): LET H(A)=H(A+S): LET H(A+S)=T: LET T=Y(A,1): LET Y(A,1)=Y(A+S,1): LET Y(A+S,1)=T: LET T=Y(A,2): LET Y(A,2)=Y(A+S,2): LET Y(A+S,2)=T
110 NEXT A: IF T<>0 THEN GO TO 100
120 IF S<>1 THEN LET S=INT (S/2): GO TO 100
130 CLS : FOR A=1 TO 25: PRINT A;":";H(A);"=";
131 LPRINT A;":";H(A);"=";:
140 LET T=INT (Y(A,1)/65536): PRINT T;"^3+";Y(A,1)-T*65536;"^3=";
141 LPRINT T;"^3+";Y(A,1)-T*65536;"^3=";
150 LET T=INT (Y(A,2)/65536): PRINT T;"^3+";Y(A,2)-T*65536;"^3"
151 LPRINT T;"^3+";Y(A,2)-T*65536;"^3"
160 NEXT A: PRINT
170 STOP
{{out}}
1:1729=1^3+12^3=9^3+10^3
2:4104=2^3+16^3=9^3+15^3
3:13832=2^3+24^3=18^3+20^3
4:20683=10^3+27^3=19^3+24^3
5:32832=4^3+32^3=18^3+30^3
6:39312=2^3+34^3=15^3+33^3
7:40033=9^3+34^3=16^3+33^3
8:46683=3^3+36^3=27^3+30^3
9:64232=17^3+39^3=26^3+36^3
10:65728=12^3+40^3=31^3+33^3
11:110656=4^3+48^3=36^3+40^3
12:110808=6^3+48^3=27^3+45^3
13:134379=12^3+51^3=38^3+43^3
14:149389=8^3+53^3=29^3+50^3
15:165464=20^3+54^3=38^3+48^3
16:171288=17^3+55^3=24^3+54^3
17:195841=9^3+58^3=22^3+57^3
18:216027=3^3+60^3=22^3+59^3
19:216125=5^3+60^3=45^3+50^3
20:262656=8^3+64^3=36^3+60^3
21:314496=4^3+68^3=30^3+66^3
22:320264=18^3+68^3=32^3+66^3
23:327763=30^3+67^3=51^3+58^3
24:373464=6^3+72^3=54^3+60^3
25:402597=42^3+69^3=56^3+61^3