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{{task}}
An [[wp:Egyptian fraction|Egyptian fraction]] is the sum of distinct unit fractions such as:
::::
Each fraction in the expression has a numerator equal to '''1''' (unity) and a denominator that is a positive integer, and all the denominators are distinct (i.e., no repetitions).
Fibonacci's [[wp:Greedy algorithm for Egyptian fractions|Greedy algorithm for Egyptian fractions]] expands the fraction to be represented by repeatedly performing the replacement
::::
(simplifying the 2nd term in this replacement as necessary, and where is the ''ceiling'' function).
For this task, [[wp:Fraction (mathematics)#Simple.2C_common.2C_or_vulgar_fractions|Proper and improper fractions]] must be able to be expressed.
Proper fractions are of the form where and are positive integers, such that , and
improper fractions are of the form where and are positive integers, such that ''a'' ≥ ''b''.
(See the [[#REXX|REXX programming example]] to view one method of expressing the whole number part of an improper fraction.)
For improper fractions, the integer part of any improper fraction should be first isolated and shown preceding the Egyptian unit fractions, and be surrounded by square brackets [''n''].
;Task requirements:
- show the Egyptian fractions for: and and
- for all proper fractions, where and are positive one-or two-digit (decimal) integers, find and show an Egyptian fraction that has: ::* the largest number of terms, ::* the largest denominator.
- for all one-, two-, and three-digit integers, find and show (as above). {extra credit}
;Also see:
- Wolfram MathWorld™ entry: [http://mathworld.wolfram.com/EgyptianFraction.html Egyptian fraction]
C#
{{trans|Visual Basic .NET}}
using System; using System.Collections.Generic; using System.Linq; using System.Numerics; using System.Text; using System.Threading.Tasks; namespace EgyptianFractions { class Program { class Rational : IComparable<Rational>, IComparable<int> { public BigInteger Num { get; } public BigInteger Den { get; } public Rational(BigInteger n, BigInteger d) { var c = Gcd(n, d); Num = n / c; Den = d / c; if (Den < 0) { Num = -Num; Den = -Den; } } public Rational(BigInteger n) { Num = n; Den = 1; } public override string ToString() { if (Den == 1) { return Num.ToString(); } else { return string.Format("{0}/{1}", Num, Den); } } public Rational Add(Rational rhs) { return new Rational(Num * rhs.Den + rhs.Num * Den, Den * rhs.Den); } public Rational Sub(Rational rhs) { return new Rational(Num * rhs.Den - rhs.Num * Den, Den * rhs.Den); } public int CompareTo(Rational rhs) { var ad = Num * rhs.Den; var bc = Den * rhs.Num; return ad.CompareTo(bc); } public int CompareTo(int rhs) { var ad = Num * rhs; var bc = Den * rhs; return ad.CompareTo(bc); } } static BigInteger Gcd(BigInteger a, BigInteger b) { if (b == 0) { if (a < 0) { return -a; } else { return a; } } else { return Gcd(b, a % b); } } static List<Rational> Egyptian(Rational r) { List<Rational> result = new List<Rational>(); if (r.CompareTo(1) >= 0) { if (r.Den == 1) { result.Add(r); result.Add(new Rational(0)); return result; } result.Add(new Rational(r.Num / r.Den)); r = r.Sub(result[0]); } BigInteger modFunc(BigInteger m, BigInteger n) { return ((m % n) + n) % n; } while (r.Num != 1) { var q = (r.Den + r.Num - 1) / r.Num; result.Add(new Rational(1, q)); r = new Rational(modFunc(-r.Den, r.Num), r.Den * q); } result.Add(r); return result; } static string FormatList<T>(IEnumerable<T> col) { StringBuilder sb = new StringBuilder(); var iter = col.GetEnumerator(); sb.Append('['); if (iter.MoveNext()) { sb.Append(iter.Current); } while (iter.MoveNext()) { sb.AppendFormat(", {0}", iter.Current); } sb.Append(']'); return sb.ToString(); } static void Main() { List<Rational> rs = new List<Rational> { new Rational(43, 48), new Rational(5, 121), new Rational(2014, 59) }; foreach (var r in rs) { Console.WriteLine("{0} => {1}", r, FormatList(Egyptian(r))); } var lenMax = Tuple.Create(0UL, new Rational(0)); var denomMax = Tuple.Create(BigInteger.Zero, new Rational(0)); var query = (from i in Enumerable.Range(1, 100) from j in Enumerable.Range(1, 100) select new Rational(i, j)) .Distinct() .ToList(); foreach (var r in query) { var e = Egyptian(r); ulong eLen = (ulong) e.Count; var eDenom = e.Last().Den; if (eLen > lenMax.Item1) { lenMax = Tuple.Create(eLen, r); } if (eDenom > denomMax.Item1) { denomMax = Tuple.Create(eDenom, r); } } Console.WriteLine("Term max is {0} with {1} terms", lenMax.Item2, lenMax.Item1); var dStr = denomMax.Item1.ToString(); Console.WriteLine("Denominator max is {0} with {1} digits {2}...{3}", denomMax.Item2, dStr.Length, dStr.Substring(0, 5), dStr.Substring(dStr.Length - 5, 5)); } } }
{{out}}
43/48 => [1/2, 1/3, 1/16]
5/121 => [1/25, 1/757, 1/763309, 1/873960180913, 1/1527612795642093418846225]
2014/59 => [34, 1/8, 1/95, 1/14947, 1/670223480]
Term max is 97/53 with 9 terms
Denominator max is 8/97 with 150 digits 57950...89665
Common Lisp
(defun egyption-fractions (x y &optional acc) (let* ((a (/ x y))) (cond ((> (numerator a) (denominator a)) (multiple-value-bind (q r) (floor x y) (if (zerop r) (cons q acc) (egyption-fractions r y (cons q acc))))) ((= (numerator a) 1) (reverse (cons a acc))) (t (let ((b (ceiling y x))) (egyption-fractions (mod (- y) x) (* y b) (cons (/ b) acc))))))) (defun test (n fn) (car (sort (loop for i from 1 to n append (loop for j from 2 to n collect (cons (/ i j) (funcall fn (egyption-fractions i j))))) #'> :key #'cdr)))
{{out}} Basic tests:
(egyption-fractions 43 48)
(egyption-fractions 5 121)
(egyption-fractions 2014 59)
(egyption-fractions 8 97)
(1/2 1/3 1/16)
(1/25 1/757 1/763309 1/873960180913 1/1527612795642093418846225)
(34 1/8 1/95 1/14947 1/670223480)
(1/13 1/181 1/38041 1/1736503177 1/3769304102927363485
1/18943537893793408504192074528154430149
1/538286441900380211365817285104907086347439746130226973253778132494225813153
1/579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665)
Other tests:
(test 999 #'length)
(test 999 (lambda (xs) (loop for x in xs maximizing (denominator x))))
(493/457 . 13)
(36/457
. 839018826833450186636781520007011999269820404906753180244759299287837378895397605613261469995626498719289835112392530430840514102146998625666594756995273418015600023494049208108894185781774002683063204252356172520941088783702738286944210460710059319691268110283467445381026653628599765684739105388642310044785844902157076919003735231543781785073393176144167688252446541416466418608465458502997971425428342769433127784560570193376772878336217849260872114137931351960543608384244009505664253173875705234889570853924105640193619301332776989688248555027054395237907581951261868280899150574360164800187964167274323078311078867593844043149124596271281252530924719121766925749760855109100066731841478262812686642693395896229983745226277793055820609058348269152190083695704685769622011655159174272326647342695589818127126303038171968768650476413027459205291075571637957597356820188031655122749743652301268394542123970892422944335857917641636041892192547135178153602038877677614358281581103685526041329841496863410305888255234495015115912388514981113593387572720476744188169200130515719608747338810136728267784013352396910979904545913458536243327311977805126410065576961237640824852114328884086581542091492600312838425666927627674227053793897767395465326589843035773944346372949759909905561209334216847158156644884281300512699910530092870919061876615770708519243818676366245477462042294267674677954783726990349386117468071932874021023714524610740225814235147693954027910741673103980749749728106483987721602738673173009362802337092908847797499475895347112889339502928407808058670297722175686638678788738689803945574002805677250463286479363670076942509109589495377221095405979217163821481666646160815221224686562530536116613645305335922819524037829878961518170177968768364853399057357772141655622381280196908637031556436461404285930426436983658106288733881761514992109680298995922754466040011586713812553117621857109517258943846004179432521131844156242428351270188803919554398620084668514054504414062276012292497375238210886595006249453460414790147611422121782194848803348777061816460876697945418158442269512987729152441940326466631610424906158237288218706447963113019239557885486647314085357651895226117364760315394354624547919209138539180807829672545924239541758108877100331729470119526373928796447673951888289511964811633025369821156695934557103429921063387965046715070102916811976552584464153981214277622597308113449320462341683055200576571910241686615924531368198770946893858410058348221985603151428153382461711196734214085852523778422630907646235900752317571022131569421231196329080023952364788544301495422061066036911772385739659997665503832444529713544286955548310166168837889046149061296461059432238621602179724809510024772127497080258401694929973105184832214622785679651550368465524821062859837409907538269572622296774545103747438431266995525592705)
D
Assuming the Python entry is correct, this code is equivalent. This requires the D module of the Arithmetic/Rational task. {{trans|Python}}
import std.stdio, std.bigint, std.algorithm, std.range, std.conv, std.typecons, arithmetic_rational: Rat = Rational; Rat[] egyptian(Rat r) pure nothrow { typeof(return) result; if (r >= 1) { if (r.denominator == 1) return [r, Rat(0, 1)]; result = [Rat(r.numerator / r.denominator, 1)]; r -= result[0]; } static enum mod = (in BigInt m, in BigInt n) pure nothrow => ((m % n) + n) % n; while (r.numerator != 1) { immutable q = (r.denominator + r.numerator - 1) / r.numerator; result ~= Rat(1, q); r = Rat(mod(-r.denominator, r.numerator), r.denominator * q); } result ~= r; return result; } void main() { foreach (immutable r; [Rat(43, 48), Rat(5, 121), Rat(2014, 59)]) writefln("%s => %(%s %)", r, r.egyptian); Tuple!(size_t, Rat) lenMax; Tuple!(BigInt, Rat) denomMax; foreach (immutable r; iota(1, 100).cartesianProduct(iota(1, 100)) .map!(nd => nd[].Rat).array.sort().uniq) { immutable e = r.egyptian; immutable eLen = e.length; immutable eDenom = e.back.denominator; if (eLen > lenMax[0]) lenMax = tuple(eLen, r); if (eDenom > denomMax[0]) denomMax = tuple(eDenom, r); } writefln("Term max is %s with %d terms", lenMax[1], lenMax[0]); immutable dStr = denomMax[0].text; writefln("Denominator max is %s with %d digits %s...%s", denomMax[1], dStr.length, dStr[0 .. 5], dStr[$ - 5 .. $]); }
{{out}}
43/48 => 1/2 1/3 1/16
5/121 => 1/25 1/757 1/763309 1/873960180913 1/1527612795642093418846225
2014/59 => 34 1/8 1/95 1/14947 1/670223480
Term max is 97/53 with 9 terms
Denominator max is 8/97 with 150 digits 57950...89665
Factor
USING: backtrack formatting fry kernel locals make math
math.functions math.ranges sequences ;
IN: rosetta-code.egyptian-fractions
: >improper ( r -- str ) >fraction "%d/%d" sprintf ;
: improper ( x y -- a b ) [ /i ] [ [ rem ] [ nip ] 2bi / ] 2bi ;
:: proper ( x y -- a b )
y x / ceiling :> d1 1 d1 / y neg x rem y d1 * / ;
: expand ( a -- b c )
>fraction 2dup > [ improper ] [ proper ] if ;
: egyptian-fractions ( x -- seq )
[ [ expand [ , ] dip dup 0 = not ] loop drop ] { } make ;
: part1 ( -- )
43/48 5/121 2014/59 [
[ >improper ] [ egyptian-fractions ] bi
"%s => %[%u, %]\n" printf
] tri@ ;
: all-longest ( seq -- seq )
dup longest length '[ length _ = ] filter ;
: (largest-denominator) ( seq -- n )
[ denominator ] map supremum ;
: most-terms ( seq -- )
all-longest [ [ sum ] map ] [ first length ] bi
"most terms: %[%u, %] => %d\n" printf ;
: largest-denominator ( seq -- )
[ (largest-denominator) ] supremum-by
[ sum ] [ (largest-denominator) ] bi
"largest denominator: %u => %d\n" printf ;
: part2 ( -- )
[
99 [1,b] amb-lazy dup [1,b] amb-lazy swap /
egyptian-fractions
] bag-of [ most-terms ] [ largest-denominator ] bi ;
: egyptian-fractions-demo ( -- ) part1 part2 ;
MAIN: egyptian-fractions-demo
{{out}}
43/48 => { 1/2, 1/3, 1/16 }
5/121 => { 1/25, 1/757, 1/763309, 1/873960180913, 1/1527612795642093418846225 }
2014/59 => { 34, 1/8, 1/95, 1/14947, 1/670223480 }
most terms: { 44/53, 8/97 } => 8
largest denominator: 8/97 => 579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665
FreeBASIC
{{libheader|GMP}}
' version 16-01-2017
' compile with: fbc -s console
#Define max 30
#Include Once "gmp.bi"
Dim Shared As Mpz_ptr num(max), den(max)
Function Egyptian_fraction(fraction As String, ByRef whole As Integer, range As Integer = 0) As Integer
If InStr(fraction,"/") = 0 Then
Print "Not a fraction, program will end"
Sleep 5000, 1
End
End If
Dim As Integer i, count
Dim As Mpz_ptr tmp_num, tmp_den, x, y, q
tmp_num = Allocate(Len(__Mpz_struct)) : Mpz_init(tmp_num)
tmp_den = Allocate(Len(__Mpz_struct)) : Mpz_init(tmp_den)
x = Allocate(Len(__Mpz_struct)) : Mpz_init(x)
y = Allocate(Len(__Mpz_struct)) : Mpz_init(y)
q = Allocate(Len(__Mpz_struct)) : Mpz_init(q)
For i = 1 To max ' clear the list
Mpz_set_ui(num(i), 0)
Mpz_set_ui(den(i), 0)
Next
i = InStr(fraction,"/")
Mpz_set_str(x, Left(fraction, i -1), 10)
Mpz_set_str(y, Right(fraction, Len(fraction) - i), 10)
' if it's a improper fraction make it proper fraction
If Mpz_cmp(x , y) > 0 Then
Mpz_fdiv_q(q, x, y)
whole = Mpz_get_ui(q)
Mpz_fdiv_r(x, x, q)
Else
whole = 0
End If
Mpz_gcd(q, x, y) ' check if reduction is possible
If Mpz_cmp_ui(q, 1) > 0 Then
If range <> 0 Then ' return if we do a range test
Return -1
Else
Mpz_fdiv_q(x, x, q)
Mpz_fdiv_q(y, y, q)
End If
End If
Mpz_set(num(count), x)
Mpz_set(den(count), y)
' Fibonacci's Greedy algorithm for Egyptian fractions
Do
If Mpz_cmp_ui(num(count), 1) = 0 Then Exit Do
Mpz_set(x, num(count))
Mpz_set(y, den(count))
Mpz_cdiv_q(q, y, x)
Mpz_set_ui(num(count), 1)
Mpz_set(den(count), q)
Mpz_mul(tmp_den, y, q)
Mpz_neg(y, y)
Mpz_mod(tmp_num, y, x)
count += 1
Mpz_gcd(q, tmp_num, tmp_den) ' check if reduction is possible
If Mpz_cmp_ui(q, 1) > 0 Then
Mpz_fdiv_q(tmp_num, tmp_num, q)
Mpz_fdiv_q(tmp_den, tmp_den, q)
End If
Mpz_set(num(count), tmp_num)
Mpz_set(den(count), tmp_den)
Loop
Mpz_clear(tmp_num) : Mpz_clear(tmp_den)
Mpz_clear(x) : Mpz_clear(y) :Mpz_clear(q)
Return count
End Function
Sub prt_solution(fraction As String, whole As Integer, count As Integer)
Print fraction; " = ";
If whole <> 0 Then
Print "["; Str(whole); "] + ";
End If
For i As Integer = 0 To count
Gmp_printf("%Zd/%Zd ", num(i), den(i))
If i <> count Then Print "+ ";
Next
Print
End Sub
' ------=< MAIN >=------
Dim As Integer n, d, number, improper, max_term, max_size
Dim As String str_in, max_term_str, max_size_str, m_str
Dim As ZString Ptr gmp_str : gmp_str = Allocate(1000000)
For n = 0 To max
num(n) = Allocate(Len(__Mpz_struct)) : Mpz_init(num(n))
den(n) = Allocate(Len(__Mpz_struct)) : Mpz_init(den(n))
Next
Data "43/48", "5/121", "2014/59"
' 4/121 = 12/363 = 11/363 + 1/363 = 1/33 + 1/363
' 5/121 = 4/121 + 1/121 = 1/33 + 1/121 + 1/363
' 2014/59 = 34 + 8/59
' 8/59 = 1/8 + 5/472 = 1/8 + 4/472 + 1/472 = 1/8 + 1/118 + 1/472
For n = 1 To 3
Read str_in
number = Egyptian_fraction(str_in, improper)
prt_solution(str_in, improper, number)
Print
Next
Dim As Integer a = 1 , b = 99
Do
For d = a To b
For n = 1 To d -1
str_in = Str(n) + "/" + Str(d)
number = Egyptian_fraction(str_in, improper,1)
If number = -1 Then Continue For ' skip
If number > max_term Then
max_term = number
max_term_str = str_in
ElseIf number = max_term Then
max_term_str += ", " & str_in
End If
Mpz_get_str(gmp_str, 10, den(number))
If Len(*gmp_str) > max_size Then
max_size = Len(*gmp_str)
max_size_str = str_in
m_str = *gmp_str
ElseIf max_size = Len(*gmp_str) Then
max_size_str += ", " & str_in
End If
Next
Next
Print
Print "for 1 to"; Len(Str(b)); " digits"
Print "Largest number of terms is"; max_term +1; " for "; max_term_str
Print "Largest size for denominator is"; max_size; " for "; max_size_str
If b = 999 Then Exit Do
a = b +1 : b = b * 10 +9
Loop
For n = 0 To max
Mpz_clear(num(n))
Mpz_clear(den(n))
Next
DeAllocate(gmp_str)
' empty keyboard buffer
While InKey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
{{out}}
43/48 = 1/2 + 1/3 + 1/16
5/121 = 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225
2014/59 = [34] + 1/8 + 1/95 + 1/14947 + 1/670223480
for 1 to 2 digits
Largest number of terms is 8 for 44/53, 8/97
Largest size for denominator is 150 for 8/97
for 1 to 3 digits
Largest number of terms is 13 for 641/796, 529/914
Largest size for denominator is 2847 for 36/457, 529/914
=={{header|Fōrmulæ}}==
In [http://wiki.formulae.org/Egyptian_fractions this] page you can see the solution of this task.
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text ([http://wiki.formulae.org/Editing_F%C5%8Drmul%C3%A6_expressions more info]). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.
The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.
Erlang
-module(egypt). -import(lists, [reverse/1, seq/2]). -export([frac/2, show/2, rosetta/0]). rosetta() -> Fractions = [{N, D, second(frac(N, D))} || N <- seq(2,99), D <- seq(N+1, 99)], {Longest, A1, B1} = findmax(fun length/1, Fractions), io:format("~b/~b has ~b terms.~n", [A1, B1, Longest]), {Largest, A2, B2} = findmax(fun (L) -> hd(reverse(L)) end, Fractions), io:format("~b/~b has a really long denominator. (~b)~n", [A2, B2, Largest]). second({_, B}) -> B. findmax(Fn, L) -> findmax(Fn, L, 0, 0, 0). findmax(_, [], M, A, B) -> {M, A, B}; findmax(Fn, [{A,B,Frac}|Fracs], M, A0, B0) -> Val = Fn(Frac), case Val > M of true -> findmax(Fn, Fracs, Val, A, B); false -> findmax(Fn, Fracs, M, A0, B0) end. show(A, B) -> {W, R} = frac(A, B), case W of 0 -> ok; _ -> io:format("[~b] ", [W]) end, case R of [] -> ok; [D0|Ds] -> io:format("1/~b ", [D0]), [io:format("+ 1/~b ", [D]) || D <- Ds], ok end. frac(A, B) -> {A div B, reverse(proper(A rem B, B, []))}. proper(0, _, L) -> L; proper(1, Y, L) -> [Y|L]; proper(X, Y, L) -> D = ceildiv(Y, X), X2 = mod(-Y, X), Y2 = Y*ceildiv(Y, X), proper(X2, Y2, [D|L]). ceildiv(A, B) -> Q = A div B, case A rem B of 0 -> Q; _ -> Q+1 end. mod(A, M) -> B = A rem M, if B < 0 -> B + M; true -> B end.
{{out}}
129> egypt:show(43,48).
1/2 + 1/3 + 1/16 ok
130> egypt:show(5,121).
1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225 ok
131> egypt:show(2014,59).
[34] 1/8 + 1/95 + 1/14947 + 1/670223480 ok
132> egypt:rosetta().
8/97 has 8 terms.
8/97 has a really long denominator. (579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665)
ok
Go
{{trans|Kotlin}} ... except that Go already has support for arbitrary precision rational numbers in its standard library.
package main import ( "fmt" "math/big" "strings" ) var zero = new(big.Int) var one = big.NewInt(1) func toEgyptianRecursive(br *big.Rat, fracs []*big.Rat) []*big.Rat { if br.Num().Cmp(zero) == 0 { return fracs } iquo := new(big.Int) irem := new(big.Int) iquo.QuoRem(br.Denom(), br.Num(), irem) if irem.Cmp(zero) > 0 { iquo.Add(iquo, one) } rquo := new(big.Rat).SetFrac(one, iquo) fracs = append(fracs, rquo) num2 := new(big.Int).Neg(br.Denom()) num2.Rem(num2, br.Num()) if num2.Cmp(zero) < 0 { num2.Add(num2, br.Num()) } denom2 := new(big.Int) denom2.Mul(br.Denom(), iquo) f := new(big.Rat).SetFrac(num2, denom2) if f.Num().Cmp(one) == 0 { fracs = append(fracs, f) return fracs } fracs = toEgyptianRecursive(f, fracs) return fracs } func toEgyptian(rat *big.Rat) []*big.Rat { if rat.Num().Cmp(zero) == 0 { return []*big.Rat{rat} } var fracs []*big.Rat if rat.Num().CmpAbs(rat.Denom()) >= 0 { iquo := new(big.Int) iquo.Quo(rat.Num(), rat.Denom()) rquo := new(big.Rat).SetFrac(iquo, one) rrem := new(big.Rat) rrem.Sub(rat, rquo) fracs = append(fracs, rquo) fracs = toEgyptianRecursive(rrem, fracs) } else { fracs = toEgyptianRecursive(rat, fracs) } return fracs } func main() { fracs := []*big.Rat{big.NewRat(43, 48), big.NewRat(5, 121), big.NewRat(2014, 59)} for _, frac := range fracs { list := toEgyptian(frac) if list[0].Denom().Cmp(one) == 0 { first := fmt.Sprintf("[%v]", list[0].Num()) temp := make([]string, len(list)-1) for i := 1; i < len(list); i++ { temp[i-1] = list[i].String() } rest := strings.Join(temp, " + ") fmt.Printf("%v -> %v + %s\n", frac, first, rest) } else { temp := make([]string, len(list)) for i := 0; i < len(list); i++ { temp[i] = list[i].String() } all := strings.Join(temp, " + ") fmt.Printf("%v -> %s\n", frac, all) } } for _, r := range [2]int{98, 998} { if r == 98 { fmt.Println("\nFor proper fractions with 1 or 2 digits:") } else { fmt.Println("\nFor proper fractions with 1, 2 or 3 digits:") } maxSize := 0 var maxSizeFracs []*big.Rat maxDen := zero var maxDenFracs []*big.Rat var sieve = make([][]bool, r+1) // to eliminate duplicates for i := 0; i <= r; i++ { sieve[i] = make([]bool, r+2) } for i := 1; i <= r; i++ { for j := i + 1; j <= r+1; j++ { if sieve[i][j] { continue } f := big.NewRat(int64(i), int64(j)) list := toEgyptian(f) listSize := len(list) if listSize > maxSize { maxSize = listSize maxSizeFracs = maxSizeFracs[0:0] maxSizeFracs = append(maxSizeFracs, f) } else if listSize == maxSize { maxSizeFracs = append(maxSizeFracs, f) } listDen := list[len(list)-1].Denom() if listDen.Cmp(maxDen) > 0 { maxDen = listDen maxDenFracs = maxDenFracs[0:0] maxDenFracs = append(maxDenFracs, f) } else if listDen.Cmp(maxDen) == 0 { maxDenFracs = append(maxDenFracs, f) } if i < r/2 { k := 2 for { if j*k > r+1 { break } sieve[i*k][j*k] = true k++ } } } } fmt.Println(" largest number of items =", maxSize) fmt.Println(" fraction(s) with this number :", maxSizeFracs) md := maxDen.String() fmt.Print(" largest denominator = ", len(md), " digits, ") fmt.Print(md[0:20], "...", md[len(md)-20:], "\b\n") fmt.Println(" fraction(s) with this denominator :", maxDenFracs) } }
{{out}}
43/48 -> 1/2 + 1/3 + 1/16
5/121 -> 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225
2014/59 -> [34] + 1/8 + 1/95 + 1/14947 + 1/670223480
For proper fractions with 1 or 2 digits:
largest number of items = 8
fraction(s) with this number : [8/97 44/53]
largest denominator = 150 digits, 57950458706754280171...62011424993909789665
fraction(s) with this denominator : [8/97]
For proper fractions with 1, 2 or 3 digits:
largest number of items = 13
fraction(s) with this number : [529/914 641/796]
largest denominator = 2847 digits, 83901882683345018663...38431266995525592705
fraction(s) with this denominator : [36/457 529/914]
Haskell
import Data.Ratio (Ratio, (%), denominator, numerator) egyptianFraction :: Integral a => Ratio a -> [Ratio a] egyptianFraction n | n < 0 = map negate (egyptianFraction (-n)) | n == 0 = [] | x == 1 = [n] | x > y = (x `div` y % 1) : egyptianFraction (x `mod` y % y) | otherwise = (1 % r) : egyptianFraction ((-y) `mod` x % (y * r)) where x = numerator n y = denominator n r = y `div` x + 1
'''Testing''':
λ> :m Test.QuickCheck λ> quickCheck (\n -> n == (sum $ egyptianFraction n)) +++ OK, passed 100 tests.
'''Tasks''':
import Data.List (intercalate, maximumBy) import Data.Ord (comparing) task1 = mapM_ run [43 % 48, 5 % 121, 2014 % 59] where run x = putStrLn $ show x ++ " = " ++ result x result x = intercalate " + " $ show <$> egyptianFraction x task21 n = maximumBy (comparing snd) [ (a % b, length $ egyptianFraction (a % b)) | a <- [1 .. n] , b <- [1 .. n] , a < b ] task22 n = maximumBy (comparing snd) [ (a % b, maximum $ map denominator $ egyptianFraction (a % b)) | a <- [1 .. n] , b <- [1 .. n] , a < b ]
λ> task1 43 % 48 = 1 % 2 + 1 % 3 + 1 % 16 5 % 121 = 1 % 25 + 1 % 757 + 1 % 763309 + 1 % 873960180913 + 1 % 1527612795642093418846225 2014 % 59 = 34 % 1 + 1 % 8 + 1 % 95 + 1 % 14947 + 1 % 670223480 λ> task21 99 (44 % 53, 8) λ> task22 99 (8 % 97, 579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665) λ> task21 999 (641 % 796,13) λ> task22 999 (529 % 914, 839018826833450186636781520007011999269820404906753180244759299287837378895397605613261469995626498719289835112392530430840514102146998625666594756995273418015600023494049208108894185781774002683063204252356172520941088783702738286944210460710059319691268110283467445381026653628599765684739105388642310044785844902157076919003735231543781785073393176144167688252446541416466418608465458502997971425428342769433127784560570193376772878336217849260872114137931351960543608384244009505664253173875705234889570853924105640193619301332776989688248555027054395237907581951261868280899150574360164800187964167274323078311078867593844043149124596271281252530924719121766925749760855109100066731841478262812686642693395896229983745226277793055820609058348269152190083695704685769622011655159174272326647342695589818127126303038171968768650476413027459205291075571637957597356820188031655122749743652301268394542123970892422944335857917641636041892192547135178153602038877677614358281581103685526041329841496863410305888255234495015115912388514981113593387572720476744188169200130515719608747338810136728267784013352396910979904545913458536243327311977805126410065576961237640824852114328884086581542091492600312838425666927627674227053793897767395465326589843035773944346372949759909905561209334216847158156644884281300512699910530092870919061876615770708519243818676366245477462042294267674677954783726990349386117468071932874021023714524610740225814235147693954027910741673103980749749728106483987721602738673173009362802337092908847797499475895347112889339502928407808058670297722175686638678788738689803945574002805677250463286479363670076942509109589495377221095405979217163821481666646160815221224686562530536116613645305335922819524037829878961518170177968768364853399057357772141655622381280196908637031556436461404285930426436983658106288733881761514992109680298995922754466040011586713812553117621857109517258943846004179432521131844156242428351270188803919554398620084668514054504414062276012292497375238210886595006249453460414790147611422121782194848803348777061816460876697945418158442269512987729152441940326466631610424906158237288218706447963113019239557885486647314085357651895226117364760315394354624547919209138539180807829672545924239541758108877100331729470119526373928796447673951888289511964811633025369821156695934557103429921063387965046715070102916811976552584464153981214277622597308113449320462341683055200576571910241686615924531368198770946893858410058348221985603151428153382461711196734214085852523778422630907646235900752317571022131569421231196329080023952364788544301495422061066036911772385739659997665503832444529713544286955548310166168837889046149061296461059432238621602179724809510024772127497080258401694929973105184832214622785679651550368465524821062859837409907538269572622296774545103747438431266995525592705)
J
'''Solution''':
ef =: [: (}.~ 0={.) [: (, r2ef)/ 0 1 #: x:
r2ef =: (<(<0);0) { ((] , -) >:@:<.&.%)^:((~:<.)@:%)@:{:^:a:
'''Examples''' (''required''):
(; ef)&> 43r48 5r121 2014r59
+-------+--------------------------------------------------------------+
|43r48 |1r2 1r3 1r16 |
+-------+--------------------------------------------------------------+
|5r121 |1r25 1r757 1r763309 1r873960180913 1r1527612795642093418846225|
+-------+--------------------------------------------------------------+
|2014r59|34 1r8 1r95 1r14947 1r670223480 |
+-------+--------------------------------------------------------------+
'''Examples''' (''extended''):
NB. ef for all 1- and 2-digit fractions
EF2 =: ef :: _1:&.> (</~ * %/~) i. 10^2x
NB. longest ef for 1- or 2-digit fraction
($ #: (i. >./)@:,)#&>EF2
8 97
# ef 8r97
8
NB. largest denom among for 1- and 2-digit fractions
($ #: (i. <./)@:|@:(<./&>)@:,) EF2
8 97
_80 ]\ ": % <./ ef 8r97
57950458706754280171310319185991860825103029195219542358352935765389941868634236
0361798689053273749372615043661810228371898539583862011424993909789665
NB. ef for all 1-,2-, and 3-digit fractions
EF3 =: ef :: _1:&.> (</~ * %/~) i. 10^3x
NB. longest ef for 1-, 2-,or 3-digit fraction
($ #: (i. >./)@:,)#&>EF3
529 914
# ef 529r914
13
NB. largest denom among for 1-, 2-, and 3-digit fractions
($ #: (i. <./)@:|@:(<./&>)@:,) EF3
36 457
_80 ]\ ": % <./ ef 36r457
83901882683345018663678152000701199926982040490675318024475929928783737889539760
56132614699956264987192898351123925304308405141021469986256665947569952734180156
00023494049208108894185781774002683063204252356172520941088783702738286944210460
71005931969126811028346744538102665362859976568473910538864231004478584490215707
69190037352315437817850733931761441676882524465414164664186084654585029979714254
28342769433127784560570193376772878336217849260872114137931351960543608384244009
50566425317387570523488957085392410564019361930133277698968824855502705439523790
75819512618682808991505743601648001879641672743230783110788675938440431491245962
71281252530924719121766925749760855109100066731841478262812686642693395896229983
74522627779305582060905834826915219008369570468576962201165515917427232664734269
55898181271263030381719687686504764130274592052910755716379575973568201880316551
22749743652301268394542123970892422944335857917641636041892192547135178153602038
87767761435828158110368552604132984149686341030588825523449501511591238851498111
35933875727204767441881692001305157196087473388101367282677840133523969109799045
45913458536243327311977805126410065576961237640824852114328884086581542091492600
31283842566692762767422705379389776739546532658984303577394434637294975990990556
12093342168471581566448842813005126999105300928709190618766157707085192438186763
66245477462042294267674677954783726990349386117468071932874021023714524610740225
81423514769395402791074167310398074974972810648398772160273867317300936280233709
29088477974994758953471128893395029284078080586702977221756866386787887386898039
45574002805677250463286479363670076942509109589495377221095405979217163821481666
64616081522122468656253053611661364530533592281952403782987896151817017796876836
48533990573577721416556223812801969086370315564364614042859304264369836581062887
33881761514992109680298995922754466040011586713812553117621857109517258943846004
17943252113184415624242835127018880391955439862008466851405450441406227601229249
73752382108865950062494534604147901476114221217821948488033487770618164608766979
45418158442269512987729152441940326466631610424906158237288218706447963113019239
55788548664731408535765189522611736476031539435462454791920913853918080782967254
59242395417581088771003317294701195263739287964476739518882895119648116330253698
21156695934557103429921063387965046715070102916811976552584464153981214277622597
30811344932046234168305520057657191024168661592453136819877094689385841005834822
19856031514281533824617111967342140858525237784226309076462359007523175710221315
69421231196329080023952364788544301495422061066036911772385739659997665503832444
52971354428695554831016616883788904614906129646105943223862160217972480951002477
21274970802584016949299731051848322146227856796515503684655248210628598374099075
38269572622296774545103747438431266995525592705
Java
{{trans|Kotlin}} {{works with|Java|9}}
import java.math.BigDecimal; import java.math.BigInteger; import java.math.MathContext; import java.util.ArrayList; import java.util.Collections; import java.util.List; public class EgyptianFractions { private static BigInteger gcd(BigInteger a, BigInteger b) { if (b.equals(BigInteger.ZERO)) { return a; } return gcd(b, a.mod(b)); } private static class Frac implements Comparable<Frac> { private BigInteger num, denom; public Frac(BigInteger n, BigInteger d) { if (d.equals(BigInteger.ZERO)) { throw new IllegalArgumentException("Parameter d may not be zero."); } BigInteger nn = n; BigInteger dd = d; if (nn.equals(BigInteger.ZERO)) { dd = BigInteger.ONE; } else if (dd.compareTo(BigInteger.ZERO) < 0) { nn = nn.negate(); dd = dd.negate(); } BigInteger g = gcd(nn, dd).abs(); if (g.compareTo(BigInteger.ZERO) > 0) { nn = nn.divide(g); dd = dd.divide(g); } num = nn; denom = dd; } public Frac(int n, int d) { this(BigInteger.valueOf(n), BigInteger.valueOf(d)); } public Frac plus(Frac rhs) { return new Frac( num.multiply(rhs.denom).add(denom.multiply(rhs.num)), rhs.denom.multiply(denom) ); } public Frac unaryMinus() { return new Frac(num.negate(), denom); } public Frac minus(Frac rhs) { return plus(rhs.unaryMinus()); } @Override public int compareTo(Frac rhs) { BigDecimal diff = this.toBigDecimal().subtract(rhs.toBigDecimal()); if (diff.compareTo(BigDecimal.ZERO) < 0) { return -1; } if (BigDecimal.ZERO.compareTo(diff) < 0) { return 1; } return 0; } @Override public boolean equals(Object obj) { if (null == obj || !(obj instanceof Frac)) { return false; } Frac rhs = (Frac) obj; return compareTo(rhs) == 0; } @Override public String toString() { if (denom.equals(BigInteger.ONE)) { return num.toString(); } return String.format("%s/%s", num, denom); } public BigDecimal toBigDecimal() { BigDecimal bdn = new BigDecimal(num); BigDecimal bdd = new BigDecimal(denom); return bdn.divide(bdd, MathContext.DECIMAL128); } public List<Frac> toEgyptian() { if (num.equals(BigInteger.ZERO)) { return Collections.singletonList(this); } List<Frac> fracs = new ArrayList<>(); if (num.abs().compareTo(denom.abs()) >= 0) { Frac div = new Frac(num.divide(denom), BigInteger.ONE); Frac rem = this.minus(div); fracs.add(div); toEgyptian(rem.num, rem.denom, fracs); } else { toEgyptian(num, denom, fracs); } return fracs; } public void toEgyptian(BigInteger n, BigInteger d, List<Frac> fracs) { if (n.equals(BigInteger.ZERO)) { return; } BigDecimal n2 = new BigDecimal(n); BigDecimal d2 = new BigDecimal(d); BigDecimal[] divRem = d2.divideAndRemainder(n2, MathContext.UNLIMITED); BigInteger div = divRem[0].toBigInteger(); if (divRem[1].compareTo(BigDecimal.ZERO) > 0) { div = div.add(BigInteger.ONE); } fracs.add(new Frac(BigInteger.ONE, div)); BigInteger n3 = d.negate().mod(n); if (n3.compareTo(BigInteger.ZERO) < 0) { n3 = n3.add(n); } BigInteger d3 = d.multiply(div); Frac f = new Frac(n3, d3); if (f.num.equals(BigInteger.ONE)) { fracs.add(f); return; } toEgyptian(f.num, f.denom, fracs); } } public static void main(String[] args) { List<Frac> fracs = List.of( new Frac(43, 48), new Frac(5, 121), new Frac(2014, 59) ); for (Frac frac : fracs) { List<Frac> list = frac.toEgyptian(); Frac first = list.get(0); if (first.denom.equals(BigInteger.ONE)) { System.out.printf("%s -> [%s] + ", frac, first); } else { System.out.printf("%s -> %s", frac, first); } for (int i = 1; i < list.size(); ++i) { System.out.printf(" + %s", list.get(i)); } System.out.println(); } for (Integer r : List.of(98, 998)) { if (r == 98) { System.out.println("\nFor proper fractions with 1 or 2 digits:"); } else { System.out.println("\nFor proper fractions with 1, 2 or 3 digits:"); } int maxSize = 0; List<Frac> maxSizeFracs = new ArrayList<>(); BigInteger maxDen = BigInteger.ZERO; List<Frac> maxDenFracs = new ArrayList<>(); boolean[][] sieve = new boolean[r + 1][]; for (int i = 0; i < r + 1; ++i) { sieve[i] = new boolean[r + 2]; } for (int i = 1; i < r; ++i) { for (int j = i + 1; j < r + 1; ++j) { if (sieve[i][j]) continue; Frac f = new Frac(i, j); List<Frac> list = f.toEgyptian(); int listSize = list.size(); if (listSize > maxSize) { maxSize = listSize; maxSizeFracs.clear(); maxSizeFracs.add(f); } else if (listSize == maxSize) { maxSizeFracs.add(f); } BigInteger listDen = list.get(list.size() - 1).denom; if (listDen.compareTo(maxDen) > 0) { maxDen = listDen; maxDenFracs.clear(); maxDenFracs.add(f); } else if (listDen.equals(maxDen)) { maxDenFracs.add(f); } if (i < r / 2) { int k = 2; while (true) { if (j * k > r + 1) break; sieve[i * k][j * k] = true; k++; } } } } System.out.printf(" largest number of items = %s\n", maxSize); System.out.printf("fraction(s) with this number : %s\n", maxSizeFracs); String md = maxDen.toString(); System.out.printf(" largest denominator = %s digits, ", md.length()); System.out.printf("%s...%s\n", md.substring(0, 20), md.substring(md.length() - 20, md.length())); System.out.printf("fraction(s) with this denominator : %s\n", maxDenFracs); } } }
{{out}}
43/48 -> 1/2 + 1/3 + 1/16
5/121 -> 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225
2014/59 -> [34] + + 1/8 + 1/95 + 1/14947 + 1/670223480
For proper fractions with 1 or 2 digits:
largest number of items = 8
fraction(s) with this number : [8/97, 44/53]
largest denominator = 150 digits, 57950458706754280171...62011424993909789665
fraction(s) with this denominator : [8/97]
For proper fractions with 1, 2 or 3 digits:
largest number of items = 13
fraction(s) with this number : [529/914, 641/796]
largest denominator = 2847 digits, 83901882683345018663...38431266995525592705
fraction(s) with this denominator : [36/457, 529/914]
Julia
{{works with|Julia|0.6}}
struct EgyptianFraction{T<:Integer} <: Real int::T frac::NTuple{N,Rational{T}} where N end Base.show(io::IO, ef::EgyptianFraction) = println(io, "[", ef.int, "] ", join(ef.frac, " + ")) Base.length(ef::EgyptianFraction) = !iszero(ef.int) + length(ef.frac) function Base.convert(::Type{EgyptianFraction{T}}, fr::Rational) where T fr, int::T = modf(fr) fractions = Vector{Rational{T}}(0) x::T, y::T = numerator(fr), denominator(fr) iszero(x) && return EgyptianFraction{T}(int, (x // y,)) while x != one(x) push!(fractions, one(T) // cld(y, x)) x, y = mod1(-y, x), y * cld(y, x) d = gcd(x, y) x ÷= d y ÷= d end push!(fractions, x // y) return EgyptianFraction{T}(int, tuple(fractions...)) end Base.convert(::Type{EgyptianFraction}, fr::Rational{T}) where T = convert(EgyptianFraction{T}, fr) Base.convert(::Type{EgyptianFraction{T}}, fr::EgyptianFraction) where T = EgyptianFraction{T}(convert(T, fr.int), convert.(Rational{T}, fr.frac)) Base.convert(::Type{Rational{T}}, fr::EgyptianFraction) where T = T(fr.int) + sum(convert.(Rational{T}, fr.frac)) Base.convert(::Type{Rational}, fr::EgyptianFraction{T}) where T = convert(Rational{T}, fr) @show EgyptianFraction(43 // 48) @show EgyptianFraction{BigInt}(5 // 121) @show EgyptianFraction(2014 // 59) function task(fractions::AbstractVector) fracs = convert(Vector{EgyptianFraction{BigInt}}, fractions) local frlenmax::EgyptianFraction{BigInt} local lenmax = 0 local frdenmax::EgyptianFraction{BigInt} local denmax = 0 for f in fracs if length(f) ≥ lenmax lenmax = length(f) frlenmax = f end if denominator(last(f.frac)) ≥ denmax denmax = denominator(last(f.frac)) frdenmax = f end end return frlenmax, lenmax, frdenmax, denmax end fr = collect((x // y) for x in 1:100 for y in 1:100 if x != y) |> unique frlenmax, lenmax, frdenmax, denmax = task(fr) println("Longest fraction, with length $lenmax: \n", Rational(frlenmax), "\n = ", frlenmax) println("Fraction with greatest denominator\n(that is $denmax):\n", Rational(frdenmax), "\n = ", frdenmax) println("\n# For 1 digit-integers:") fr = collect((x // y) for x in 1:10 for y in 1:10 if x != y) |> unique frlenmax, lenmax, frdenmax, denmax = task(fr) println("Longest fraction, with length $lenmax: \n", Rational(frlenmax), "\n = ", frlenmax) println("Fraction with greatest denominator\n(that is $denmax):\n", Rational(frdenmax), "\n = ", frdenmax) println("# For 3 digit-integers:") fr = collect((x // y) for x in 1:1000 for y in 1:1000 if x != y) |> unique frlenmax, lenmax, frdenmax, denmax = task(fr) println("Longest fraction, with length $lenmax: \n", Rational(frlenmax), "\n = ", frlenmax) println("Fraction with greatest denominator\n(that is $denmax):\n", Rational(frdenmax), "\n = ", frdenmax)
{{out}}
EgyptianFraction(43 // 48) = [0] 1//2 + 1//3 + 1//16
EgyptianFraction{BigInt}(5 // 121) = [0] 1//25 + 1//757 + 1//763309 + 1//873960180913 + 1//1527612795642093418846225
EgyptianFraction(2014 // 59) = [34] 1//8 + 1//95 + 1//14947 + 1//670223480
Longest fraction, with length 9:
97//53
= [1] 1//2 + 1//4 + 1//13 + 1//307 + 1//120871 + 1//20453597227 + 1//697249399186783218655 + 1//1458470173998990524806872692984177836808420
Fraction with greatest denominator
(that is 5795045870675428...424993909789665):
8//97
= [0] 1//13 + 1//181 + 1//38041 + 1//1736503177 + 1//3769304102927363485 + 1//18943537893793408504192074528154430149 + [...]
# For 1 digit-integers:
Longest fraction, with length 4:
10//7
= [1] 1//3 + 1//11 + 1//231
Fraction with greatest denominator
(that is 231):
10//7
= [1] 1//3 + 1//11 + 1//231
# For 3 digit-integers:
Longest fraction, with length 13:
950//457
= [2] 1//13 + 1//541 + 1//321409 + 1//114781617793 + 1//14821672255960844346913 + ...
Fraction with greatest denominator
(that is 8390188268334501866367815200...[2847 digits]):
950//457
= [2] 1//13 + 1//541 + 1//321409 + 1//114781617793 + 1//14821672255960844346913...
Kotlin
As the JDK lacks a fraction or rational class, I've included a basic one in the program.
// version 1.2.10 import java.math.BigInteger import java.math.BigDecimal import java.math.MathContext val bigZero = BigInteger.ZERO val bigOne = BigInteger.ONE val bdZero = BigDecimal.ZERO val context = MathContext.UNLIMITED fun gcd(a: BigInteger, b: BigInteger): BigInteger = if (b == bigZero) a else gcd(b, a % b) class Frac : Comparable<Frac> { val num: BigInteger val denom: BigInteger constructor(n: BigInteger, d: BigInteger) { require(d != bigZero) var nn = n var dd = d if (nn == bigZero) { dd = bigOne } else if (dd < bigZero) { nn = -nn dd = -dd } val g = gcd(nn, dd).abs() if (g > bigOne) { nn /= g dd /= g } num = nn denom = dd } constructor(n: Int, d: Int) : this(n.toBigInteger(), d.toBigInteger()) operator fun plus(other: Frac) = Frac(num * other.denom + denom * other.num, other.denom * denom) operator fun unaryMinus() = Frac(-num, denom) operator fun minus(other: Frac) = this + (-other) override fun compareTo(other: Frac): Int { val diff = this.toBigDecimal() - other.toBigDecimal() return when { diff < bdZero -> -1 diff > bdZero -> +1 else -> 0 } } override fun equals(other: Any?): Boolean { if (other == null || other !is Frac) return false return this.compareTo(other) == 0 } override fun toString() = if (denom == bigOne) "$num" else "$num/$denom" fun toBigDecimal() = num.toBigDecimal() / denom.toBigDecimal() fun toEgyptian(): List<Frac> { if (num == bigZero) return listOf(this) val fracs = mutableListOf<Frac>() if (num.abs() >= denom.abs()) { val div = Frac(num / denom, bigOne) val rem = this - div fracs.add(div) toEgyptian(rem.num, rem.denom, fracs) } else { toEgyptian(num, denom, fracs) } return fracs } private tailrec fun toEgyptian( n: BigInteger, d: BigInteger, fracs: MutableList<Frac> ) { if (n == bigZero) return val n2 = n.toBigDecimal() val d2 = d.toBigDecimal() var divRem = d2.divideAndRemainder(n2, context) var div = divRem[0].toBigInteger() if (divRem[1] > bdZero) div++ fracs.add(Frac(bigOne, div)) var n3 = (-d) % n if (n3 < bigZero) n3 += n val d3 = d * div val f = Frac(n3, d3) if (f.num == bigOne) { fracs.add(f) return } toEgyptian(f.num, f.denom, fracs) } } fun main(args: Array<String>) { val fracs = listOf(Frac(43, 48), Frac(5, 121), Frac(2014,59)) for (frac in fracs) { val list = frac.toEgyptian() if (list[0].denom == bigOne) { val first = "[${list[0]}]" println("$frac -> $first + ${list.drop(1).joinToString(" + ")}") } else { println("$frac -> ${list.joinToString(" + ")}") } } for (r in listOf(98, 998)) { if (r == 98) println("\nFor proper fractions with 1 or 2 digits:") else println("\nFor proper fractions with 1, 2 or 3 digits:") var maxSize = 0 var maxSizeFracs = mutableListOf<Frac>() var maxDen = bigZero var maxDenFracs = mutableListOf<Frac>() val sieve = List(r + 1) { BooleanArray(r + 2) } // to eliminate duplicates for (i in 1..r) { for (j in (i + 1)..(r + 1)) { if (sieve[i][j]) continue val f = Frac(i, j) val list = f.toEgyptian() val listSize = list.size if (listSize > maxSize) { maxSize = listSize maxSizeFracs.clear() maxSizeFracs.add(f) } else if (listSize == maxSize) { maxSizeFracs.add(f) } val listDen = list[list.lastIndex].denom if (listDen > maxDen) { maxDen = listDen maxDenFracs.clear() maxDenFracs.add(f) } else if (listDen == maxDen) { maxDenFracs.add(f) } if (i < r / 2) { var k = 2 while (true) { if (j * k > r + 1) break sieve[i * k][j * k] = true k++ } } } } println(" largest number of items = $maxSize") println(" fraction(s) with this number : $maxSizeFracs") val md = maxDen.toString() print(" largest denominator = ${md.length} digits, ") println("${md.take(20)}...${md.takeLast(20)}") println(" fraction(s) with this denominator : $maxDenFracs") } }
{{out}}
43/48 -> 1/2 + 1/3 + 1/16
5/121 -> 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225
2014/59 -> [34] + 1/8 + 1/95 + 1/14947 + 1/670223480
For proper fractions with 1 or 2 digits:
largest number of items = 8
fraction(s) with this number : [8/97, 44/53]
largest denominator = 150 digits, 57950458706754280171...62011424993909789665
fraction(s) with this denominator : [8/97]
For proper fractions with 1, 2 or 3 digits:
largest number of items = 13
fraction(s) with this number : [529/914, 641/796]
largest denominator = 2847 digits, 83901882683345018663...38431266995525592705
fraction(s) with this denominator : [36/457, 529/914]
Mathematica
frac[n_] /; IntegerQ[1/n] := frac[n] = {n};
frac[n_] :=
frac[n] =
With[{p = Numerator[n], q = Denominator[n]},
Prepend[frac[Mod[-q, p]/(q Ceiling[1/n])], 1/Ceiling[1/n]]];
disp[f_] :=
StringRiffle[
SequenceCases[f,
l : {_, 1 ...} :>
If[Length[l] == 1 && l[[1]] < 1, ToString[l[[1]], InputForm],
"[" <> ToString[Length[l]] <> "]"]], " + "] <> " = " <>
ToString[Numerator[Total[f]]] <> "/" <>
ToString[Denominator[Total[f]]];
Print[disp[frac[43/48]]];
Print[disp[frac[5/121]]];
Print[disp[frac[2014/59]]];
fracs = Flatten[Table[frac[p/q], {q, 99}, {p, q}], 1];
Print[disp[MaximalBy[fracs, Length@*Union][[1]]]];
Print[disp[MaximalBy[fracs, Denominator@*Last][[1]]]];
fracs = Flatten[Table[frac[p/q], {q, 999}, {p, q}], 1];
Print[disp[MaximalBy[fracs, Length@*Union][[1]]]];
Print[disp[MaximalBy[fracs, Denominator@*Last][[1]]]];
{{out}}
1/2 + 1/3 + 1/16 = 43/48
1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225 = 5/121
[34] + 1/8 + 1/95 + 1/14947 + 1/670223480 = 2014/59
1/2 + 1/4 + 1/13 + 1/307 + 1/120871 + 1/20453597227 + 1/697249399186783218655 + 1/1458470173998990524806872692984177836808420 = 44/53
1/13 + 1/181 + 1/38041 + 1/1736503177 + 1/3769304102927363485 + 1/18943537893793408504192074528154430149 + 1/538286441900380211365817285104907086347439746130226973253778132494225813153 + 1/579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665 = 8/97
1/2 + 1/4 + 1/19 + 1/379 + 1/159223 + 1/28520799973 + 1/929641178371338400861 + 1/1008271507277592391123742528036634174730681 + 1/1219933718865393655364635368068124756713122928811333803786753398211072842948484537833 + 1/1860297848030936654742608399135821395565274404917258533393305147319524009551744684579405649080712180254407780735949179513154143641842892458088536544987153757401025882029 + 1/4614277444518045184646591832326467411359277711335974416082881814986405515888533562332069783067894981850924485553345190160771506460024406127868096951360637582674289834858262576425271895218431296391169922044160278696744025988461165811212428548328350795432691637759392474030879286312785400132190057899968737693594392669884878193448874327093 + 1/31937334502481972335865307630139228000187060941658399518862518849553429993133277230560087986574331290756232125775998863890963263813589266879406694561350952988662850757053371133819179770003609046815203982179108798005308113258134895569927488690118483730232440575942894680942308888321353318333183158977270294582315388855860989819894602178852719674244639951777398683083694723999674418435726557523519535770015019287382321071804865681731226989916286199314883016472947639367666251368202759691810399195092598892275413777035275182318485652713871000041272524440519262054008953943029365257325370839037761555465335452562216651250516983405134378252470216494582635109781712938341456418881 + 1/2039986670246850822853427080268636607703538330430958135006350872460188775376402385474575383380701179275926633909293920375037781006938834602683282504456671345800481611955974906577358109966753513899436209725756764159504134559394933538420714469300931804842468643272796657406808805007786178371184391663721349034183315512035012402176731111044506314978549915206516847224339930494935465558632905912262959736737614637514921726288403470224139024425700070180324623265095949577758695292697562554242228453440276043742370033993859881981612938703208463591285870376619588297958810138295747858827756577616148419423031480258559516303907719233914603343421735341220080271152090557188286289527661792734931298102513902518914250419121432886312102736349552224188669212688846219382874287241971706387850290821170997846726526589069990513808709560793139660289273086403155344460608865436195352720549406793512677065107181955781264579349071905411393100989250722104770801720673437692418988638492506057962758754921169589084980707251205329924087857682559921447010465898318288868258062129919867004394488124710647843586978379399594154917914477913086776811741840849911967039211773201428676384229432761943488196359561416605048969002045397348240530911560634680322446588472763785839765588633770016209055874572792498932175778494089116461654628549726895871636209026849103988563732410165441 = 641/796
1/13 + 1/541 + 1/321409 + 1/114781617793 + 1/14821672255960844346913 + 1/251065106814993628596500876449600804290086881 + 1/73539302503361520198362339236500915390885795679264404865887253300925727812630083326272641 + 1/6489634815217096741758907148982381236931234341288936993640630568353888026513046373352130623124225014404918014072680355409470797372507720812828610332359154836067922616607391865217 + 1/52644200043597301715163084170074049765863371513744701000308778672552161021188727897845435784419167578097570308323221316037189809321236639774156001218848770417914304730719451756764847141999454715415348579218576135692260706546084789833559164567239198064491721524233401718052341737694961761810858726456915514545036448002629051435498625211733293978125476206145 + 1/3695215730973720191743335450900515442837964059737103132125137784392340041085824276783333540815086968586494259680343732030671448522298751008735945486795776365973142745077411841504712940444458881229478108614230774637316342940593842925604630011475333378620376362943942755446627099104200059416153812858633723638212819657597061963458758259287734950993940819872945202809437805131650984566124057319228963533088559443909352453788455968978250113376533423265233637558939144535732287317303130488802163512444658441011602922480039143050047663394967808639154754442570791381496210122415541628843804495020590646687354364355396925939868087995781911240513904752765014910531863571167632659092232428610030201325032663259931238141889 + 1/20481928947653467858867964360215698922460866349989714221296388791180533521147068328398292448571350580917144516243144419767021450972552458770890215041236338405232471846144964422722088363577942656244304369314740680337368003341749927848292268159627280776486153786277410225081205358330757686606252814923029488556248114378465151886875778980493919811102286892641254175976181063891774788890129279669791215911728886439002027991447164421080590166911130116483359749418047307595497010369457711350953018694479942850146580996402187310635505278301929397030213544531068769667892360925519410013180703331321321833900350008776368272790481252519169303988218210095146759870287941250090204506960847016059468728275311477613271084474766715488264771177830115028195215223644336345646870679050787515340804351339449474385172464387868299006904638274425855008729765086091731260299397062138670321522563954731398813138738073326593694555049353805161855854036423870334342280080335804850998490793742536882308453307029152812821729798744074167237835462214043679643723245065093600037959124662392297413473130606861784229249604290090458912391096328362137163951398211801143455350336317188806956746282700489013366856863803112203078858200161688528939040348825835610989725020068306497091337571398894447440161081470240965873628208205669354804691958270783090585006358905094926094885655359774269830169287513005586562246433405044654325439410730648108371520856384706590593 + 1/839018826833450186636781520007011999269820404906753180244759299287837378895397605613261469995626498719289835112392530430840514102146998625666594756995273418015600023494049208108894185781774002683063204252356172520941088783702738286944210460710059319691268110283467445381026653628599765684739105388642310044785844902157076919003735231543781785073393176144167688252446541416466418608465458502997971425428342769433127784560570193376772878336217849260872114137931351960543608384244009505664253173875705234889570853924105640193619301332776989688248555027054395237907581951261868280899150574360164800187964167274323078311078867593844043149124596271281252530924719121766925749760855109100066731841478262812686642693395896229983745226277793055820609058348269152190083695704685769622011655159174272326647342695589818127126303038171968768650476413027459205291075571637957597356820188031655122749743652301268394542123970892422944335857917641636041892192547135178153602038877677614358281581103685526041329841496863410305888255234495015115912388514981113593387572720476744188169200130515719608747338810136728267784013352396910979904545913458536243327311977805126410065576961237640824852114328884086581542091492600312838425666927627674227053793897767395465326589843035773944346372949759909905561209334216847158156644884281300512699910530092870919061876615770708519243818676366245477462042294267674677954783726990349386117468071932874021023714524610740225814235147693954027910741673103980749749728106483987721602738673173009362802337092908847797499475895347112889339502928407808058670297722175686638678788738689803945574002805677250463286479363670076942509109589495377221095405979217163821481666646160815221224686562530536116613645305335922819524037829878961518170177968768364853399057357772141655622381280196908637031556436461404285930426436983658106288733881761514992109680298995922754466040011586713812553117621857109517258943846004179432521131844156242428351270188803919554398620084668514054504414062276012292497375238210886595006249453460414790147611422121782194848803348777061816460876697945418158442269512987729152441940326466631610424906158237288218706447963113019239557885486647314085357651895226117364760315394354624547919209138539180807829672545924239541758108877100331729470119526373928796447673951888289511964811633025369821156695934557103429921063387965046715070102916811976552584464153981214277622597308113449320462341683055200576571910241686615924531368198770946893858410058348221985603151428153382461711196734214085852523778422630907646235900752317571022131569421231196329080023952364788544301495422061066036911772385739659997665503832444529713544286955548310166168837889046149061296461059432238621602179724809510024772127497080258401694929973105184832214622785679651550368465524821062859837409907538269572622296774545103747438431266995525592705 = 36/457
Microsoft Small Basic
Small Basic but large (not huge) integers.
'Egyptian fractions - 26/07/2018
xx=2014
yy=59
x=xx
y=yy
If x>=y Then
q=Math.Floor(x/y)
tt="+("+q+")"
x=Math.Remainder(x,y)
EndIf
If x<>0 Then
While x<>1
'i=modulo(-y,x)
u=-y
v=x
modulo()
i=ret
k=Math.Ceiling(y/x)
m=m+1
tt=tt+"+1/"+k
j=y*k
If i=1 Then
tt=tt+"+1/"+j
EndIf
'n=gcd(i,j)
x=i
y=j
gcd()
n=ret
x=i/n
y=j/n
EndWhile
EndIf
TextWindow.WriteLine(xx+"/"+yy+"="+Text.GetSubTextToEnd(tt,2))
Sub modulo
wr=Math.Remainder(u,v)
While wr<0
wr=wr+v
EndWhile
ret=wr
EndSub
Sub gcd
wx=i
wy=j
wr=1
While wr<>0
wr=Math.Remainder(wx,wy)
wx=wy
wy=wr
EndWhile
ret=wx
EndSub
{{out}} 43/48=1/2+1/3 5/121=1/25+1/757+1/763309+1/873960180913+1/1527612795642093418846225 2014/59=(34)+1/8+1/95+1/14947+1/670223480
PARI/GP
efrac(f)=my(v=List());while(f,my(x=numerator(f),y=denominator(f));listput(v,ceil(y/x));f=(-y)%x/y/v[#v]);Vec(v);
show(f)=my(n=f\1,v=efrac(f-n)); print1(f" = ["n"; "v[1]); for(i=2,#v,print1(", "v[i])); print("]");
best(n)=my(denom,denomAt,term,termAt,v); for(a=1,n-1,for(b=a+1,n, v=efrac(a/b); if(#v>term, termAt=a/b; term=#v); if(v[#v]>denom, denomAt=a/b; denom=v[#v]))); print("Most terms is "termAt" with "term); print("Biggest denominator is "denomAt" with "denom)
apply(show, [43/48, 5/121, 2014/59]);
best(9)
best(99)
best(999)
{{out}}
43/48 = [0; 2, 3, 16]
5/121 = [0; 25, 757, 763309, 873960180913, 1527612795642093418846225]
2014/59 = [34; 8, 95, 14947, 670223480]
Most terms is 3/7 with 3
Biggest denominator is 3/7 with 231
Most terms is 8/97 with 8
Biggest denominator is 8/97 with 579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665
Most terms is 529/914 with 13
Biggest denominator is 36/457 with 839...705
Perl
use strict; use warnings; use bigint; sub isEgyption{ my $nr = int($_[0]); my $de = int($_[1]); if($nr == 0 or $de == 0){ #Invalid input return; } if($de % $nr == 0){ # They divide so print printf "1/" . int($de/$nr); return; } if($nr % $de == 0){ # Invalid fraction printf $nr/$de; return; } if($nr > $de){ printf int($nr/$de) . " + "; isEgyption($nr%$de, $de); return; } # Floor to find ceiling and print as fraction my $tmp = int($de/$nr) + 1; printf "1/" . $tmp . " + "; isEgyption($nr*$tmp-$de, $de*$tmp); } my $nrI = 2014; my $deI = 59; printf "\nEgyptian Fraction Representation of " . $nrI . "/" . $deI . " is: \n\n"; isEgyption($nrI,$deI);
{{out}}
Egyptian Fraction Representation of 2014/59 is:
34 + 1/8 + 1/95 + 1/14947 + 1/670223480
Perl 6
role Egyptian {
method gist {
join ' + ',
("[{self.floor}]" if self.abs >= 1),
map {"1/$_"}, self.denominators;
}
method denominators {
my ($x, $y) = self.nude;
$x %= $y;
my @denom = gather ($x, $y) = -$y % $x, $y * take ($y / $x).ceiling
while $x;
}
}
say .nude.join('/'), " = ", $_ but Egyptian for 43/48, 5/121, 2014/59;
my @sample = map { $_ => .denominators },
grep * < 1,
map {$_ but Egyptian},
(2 .. 99 X/ 2 .. 99);
say .key.nude.join("/"),
" has max denominator, namely ",
.value.max
given max :by(*.value.max), @sample;
say .key.nude.join("/"),
" has max number of denominators, namely ",
.value.elems
given max :by(*.value.elems), @sample;
{{out}}
43/48 = 1/2 + 1/3 + 1/16
5/121 = 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225
2014/59 = [34] + 1/8 + 1/95 + 1/14947 + 1/670223480
8/97 has max denominator, namely 579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665
8/97 has max number of denominators, namely 8
Because the harmonic series diverges (albeit very slowly), it is possible to write even improper fractions as a sum of distinct unit fractions. Here is a code to do that:
role Egyptian {
method gist { join ' + ', map {"1/$_"}, self.list }
method list {
my $sum = 0;
gather for 2 .. * {
last if $sum == self;
$sum += 1 / .take unless $sum + 1 / $_ > self;
}
}
}
say 5/4 but Egyptian;
{{out}}
1/2 + 1/3 + 1/4 + 1/6
The list of terms grows exponentially with the value of the fraction, though.
Phix
{{trans|tcl}} {{libheader|mpfr}} The sieve copied from Go
include mpfr.e
function egyptian(integer num, denom)
mpz n = mpz_init(num),
d = mpz_init(denom),
t = mpz_init()
sequence result = {}
while mpz_cmp_si(n,0)!=0 do
mpz_cdiv_q(t, d, n)
result = append(result,"1/"&mpz_get_str(t))
mpz_neg(d,d)
mpz_mod(n,d,n)
mpz_neg(d,d)
mpz_mul(d,d,t)
end while
{n,d} = mpz_free({n,d})
return result
end function
procedure efrac(integer num, denom)
string fraction = sprintf("%d/%d",{num,denom}),
prefix = ""
if num>=denom then
integer whole = floor(num/denom)
num -= whole*denom
prefix = sprintf("[%d] + ",whole)
end if
string e = join(egyptian(num, denom)," + ")
printf(1,"%s -> %s%s\n",{fraction,prefix,e})
end procedure
efrac(43,48)
efrac(5,121)
efrac(2014,59)
integer maxt = 0,
maxd = 0
string maxts = "",
maxds = "",
maxda = ""
for r=98 to 998 by 900 do -- (iterates just twice!)
sequence sieve = repeat(repeat(false,r+1),r) -- to eliminate duplicates
for n=1 to r do
for d=n+1 to r+1 do
if sieve[n][d]=false then
string term = sprintf("%d/%d",{n,d})
sequence terms = egyptian(n,d)
integer nterms = length(terms)
if nterms>maxt then
maxt = nterms
maxts = term
elsif nterms=maxt then
maxts &= ", " & term
end if
integer mlen = length(terms[$])-2
if mlen>maxd then
maxd = mlen
maxds = term
maxda = terms[$]
elsif mlen=maxd then
maxds &= ", " & term
end if
if n<r/2 then
for k=2 to 9999 do
if d*k > r+1 then exit end if
sieve[n*k][d*k] = true
end for
end if
end if
end for
end for
printf(1,"\nfor proper fractions with 1 to %d digits\n",{length(sprint(r))})
printf(1,"Largest number of terms is %d for %s\n",{maxt,maxts})
maxda = maxda[3..$] -- (strip the "1/")
maxda[6..-6]="..." -- (show only first/last 5 digits)
printf(1,"Largest size for denominator is %d digits (%s) for %s\n",{maxd,maxda,maxds})
end for
{{out}}
43/48 -> 1/2 + 1/3 + 1/16
5/121 -> 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225
2014/59 -> [34] + 1/8 + 1/95 + 1/14947 + 1/670223480
for proper fractions with 1 to 2 digits
Largest number of terms is 8 for 8/97, 44/53
Largest size for denominator is 150 digits (57950...89665) for 8/97
for proper fractions with 1 to 3 digits
Largest number of terms is 13 for 529/914, 641/796
Largest size for denominator is 2847 digits (83901...92705) for 36/457, 529/914
Python
Procedural
from fractions import Fraction from math import ceil class Fr(Fraction): def __repr__(self): return '%s/%s' % (self.numerator, self.denominator) def ef(fr): ans = [] if fr >= 1: if fr.denominator == 1: return [[int(fr)], Fr(0, 1)] intfr = int(fr) ans, fr = [[intfr]], fr - intfr x, y = fr.numerator, fr.denominator while x != 1: ans.append(Fr(1, ceil(1/fr))) fr = Fr(-y % x, y* ceil(1/fr)) x, y = fr.numerator, fr.denominator ans.append(fr) return ans if __name__ == '__main__': for fr in [Fr(43, 48), Fr(5, 121), Fr(2014, 59)]: print('%r ─► %s' % (fr, ' '.join(str(x) for x in ef(fr)))) lenmax = denommax = (0, None) for fr in set(Fr(a, b) for a in range(1,100) for b in range(1, 100)): e = ef(fr) #assert sum((f[0] if type(f) is list else f) for f in e) == fr, 'Whoops!' elen, edenom = len(e), e[-1].denominator if elen > lenmax[0]: lenmax = (elen, fr, e) if edenom > denommax[0]: denommax = (edenom, fr, e) print('Term max is %r with %i terms' % (lenmax[1], lenmax[0])) dstr = str(denommax[0]) print('Denominator max is %r with %i digits %s...%s' % (denommax[1], len(dstr), dstr[:5], dstr[-5:]))
{{out}}
43/48 ─► 1/2 1/3 1/16
5/121 ─► 1/25 1/757 1/763309 1/873960180913 1/1527612795642093418846225
2014/59 ─► [34] 1/8 1/95 1/14947 1/670223480
Term max is 97/53 with 9 terms
Denominator max is 8/97 with 150 digits 57950...89665
Composition of pure functions
The derivation of a sequence of unit fractions from a single fraction is a classic case of an anamorphism or '''unfold''' abstraction – dual to a fold or catamorphism. Rather than reducing, collapsing or summarizing a structure '''to''' a single value, it builds a structure '''from''' a single value.
See the '''unfoldr''' function below: {{Works with|Python|3.7}}
'''Egyptian fractions''' from fractions import Fraction from functools import reduce from operator import neg # eqyptianFraction :: Ratio Int -> Ratio Int def eqyptianFraction(nd): '''The rational number nd as a sum of the series of unit fractions obtained by application of the greedy algorithm.''' def go(x): n, d = x.numerator, x.denominator r = 1 + d // n if n else None return Just((0, x) if 1 == n else ( (fr(n % d, d), fr(n // d, 1)) if n > d else ( fr(-d % n, d * r), fr(1, r) ) )) if n else Nothing() fr = Fraction f = unfoldr(go) return list(map(neg, f(-nd))) if 0 > nd else f(nd) # TESTS --------------------------------------------------- # maxEqyptianFraction :: Int -> (Ratio Int -> a) # -> (Ratio Int, a) def maxEqyptianFraction(nDigits): '''An Egyptian Fraction, representing a proper fraction with numerators and denominators of up to n digits each, which returns a maximal value for the supplied function f.''' # maxVals :: ([Ratio Int], a) -> (Ratio Int, a) # -> ([Ratio Int], a) def maxima(xsv, ndfx): xs, v = xsv nd, fx = ndfx return ([nd], fx) if fx > v else ( xs + [nd], v ) if fx == v and nd not in xs else xsv # go :: (Ratio Int -> a) -> ([Ratio Int], a) def go(f): iLast = int(nDigits * '9') fs, mx = reduce( maxima, [ (nd, f(eqyptianFraction(nd))) for nd in [ Fraction(n, d) for n in enumFromTo(1)(iLast) for d in enumFromTo(1 + n)(iLast) ] ], ([], 0) ) return f.__name__ + ' -> [' + ', '.join( map(str, fs) ) + '] -> ' + str(mx) return lambda f: go(f) # main :: IO () def main(): '''Tests''' ef = eqyptianFraction fr = Fraction print('Three values as Eqyptian fractions:') print('\n'.join([ str(fr(*nd)) + ' -> ' + ' + '.join(map(str, ef(fr(*nd)))) for nd in [(43, 48), (5, 121), (2014, 59)] ])) # maxDenominator :: [Ratio Int] -> Int def maxDenominator(ef): return max(map(lambda nd: nd.denominator, ef)) # maxTermCount :: [Ratio Int] -> Int def maxTermCount(ef): return len(ef) for i in [1, 2, 3]: print( '\nMaxima for proper fractions with up to ' + ( str(i) + ' digit(s):' ) ) for f in [maxTermCount, maxDenominator]: print(maxEqyptianFraction(i)(f)) # GENERIC ------------------------------------------------- # Just :: a -> Maybe a def Just(x): '''Constructor for an inhabited Maybe (option type) value.''' return {'type': 'Maybe', 'Nothing': False, 'Just': x} # Nothing :: Maybe a def Nothing(): '''Constructor for an empty Maybe (option type) value.''' return {'type': 'Maybe', 'Nothing': True} # enumFromTo :: (Int, Int) -> [Int] def enumFromTo(m): '''Integer enumeration from m to n.''' return lambda n: list(range(m, 1 + n)) # unfoldr :: (b -> Maybe (b, a)) -> b -> [a] def unfoldr(f): '''Dual to reduce or foldr. Where catamorphism reduces a list to a summary value, the anamorphic unfoldr builds a list from a seed value. As long as f returns Just(a, b), a is prepended to the list, and the residual b is used as the argument for the next application of f. When f returns Nothing, the completed list is returned.''' def go(xr): mb = f(xr[0]) if mb.get('Nothing'): return [] else: y, r = mb.get('Just') return [r] + go((y, r)) return lambda x: go((x, x)) # MAIN --- if __name__ == '__main__': main()
{{Out}}
Three values as Eqyptian fractions:
43/48 -> 1/2 + 1/3 + 1/16
5/121 -> 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225
2014/59 -> 34 + 1/8 + 1/95 + 1/14947 + 1/670223480
Maxima for proper fractions with up to 1 digit(s):
maxTermCount -> [3/7, 4/5, 5/7, 6/7, 7/8, 7/9, 8/9] -> 3
maxDenominator -> [3/7] -> 231
Maxima for proper fractions with up to 2 digit(s):
maxTermCount -> [8/97, 44/53] -> 8
maxDenominator -> [8/97] -> 579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665
Maxima for proper fractions with up to 3 digit(s):
maxTermCount -> [529/914, 641/796] -> 13
maxDenominator -> [36/457, 529/914] -> 839018826833450186636781520007011999269820404906753180244759299287837378895397605613261469995626498719289835112392530430840514102146998625666594756995273418015600023494049208108894185781774002683063204252356172520941088783702738286944210460710059319691268110283467445381026653628599765684739105388642310044785844902157076919003735231543781785073393176144167688252446541416466418608465458502997971425428342769433127784560570193376772878336217849260872114137931351960543608384244009505664253173875705234889570853924105640193619301332776989688248555027054395237907581951261868280899150574360164800187964167274323078311078867593844043149124596271281252530924719121766925749760855109100066731841478262812686642693395896229983745226277793055820609058348269152190083695704685769622011655159174272326647342695589818127126303038171968768650476413027459205291075571637957597356820188031655122749743652301268394542123970892422944335857917641636041892192547135178153602038877677614358281581103685526041329841496863410305888255234495015115912388514981113593387572720476744188169200130515719608747338810136728267784013352396910979904545913458536243327311977805126410065576961237640824852114328884086581542091492600312838425666927627674227053793897767395465326589843035773944346372949759909905561209334216847158156644884281300512699910530092870919061876615770708519243818676366245477462042294267674677954783726990349386117468071932874021023714524610740225814235147693954027910741673103980749749728106483987721602738673173009362802337092908847797499475895347112889339502928407808058670297722175686638678788738689803945574002805677250463286479363670076942509109589495377221095405979217163821481666646160815221224686562530536116613645305335922819524037829878961518170177968768364853399057357772141655622381280196908637031556436461404285930426436983658106288733881761514992109680298995922754466040011586713812553117621857109517258943846004179432521131844156242428351270188803919554398620084668514054504414062276012292497375238210886595006249453460414790147611422121782194848803348777061816460876697945418158442269512987729152441940326466631610424906158237288218706447963113019239557885486647314085357651895226117364760315394354624547919209138539180807829672545924239541758108877100331729470119526373928796447673951888289511964811633025369821156695934557103429921063387965046715070102916811976552584464153981214277622597308113449320462341683055200576571910241686615924531368198770946893858410058348221985603151428153382461711196734214085852523778422630907646235900752317571022131569421231196329080023952364788544301495422061066036911772385739659997665503832444529713544286955548310166168837889046149061296461059432238621602179724809510024772127497080258401694929973105184832214622785679651550368465524821062859837409907538269572622296774545103747438431266995525592705
Racket
#lang racket
(define (real->egyptian-list R)
(define (inr r rv)
(match* ((exact-floor r) (numerator r) (denominator r))
[(0 0 1) (reverse rv)]
[(0 1 d) (reverse (cons (/ d) rv))]
[(0 x y) (let ((^y/x (exact-ceiling (/ y x))))
(inr (/ (modulo (- y) x) (* y ^y/x)) (cons (/ ^y/x) rv)))]
[(flr _ _) (inr (- r flr) (cons flr rv))]))
(inr R null))
(define (real->egyptian-string f)
(define e.f.-list (real->egyptian-list f))
(define fmt-part
(match-lambda
[(? integer? (app number->string s)) s]
[(app (compose number->string /) s) (format "/~a"s)]))
(string-join (map fmt-part e.f.-list) " + "))
(define (stat-egyptian-fractions max-b+1)
(define-values (max-l max-l-f max-d max-d-f)
(for*/fold ((max-l 0) (max-l-f #f) (max-d 0) (max-d-f #f))
((b (in-range 1 max-b+1)) (a (in-range 1 b)) #:when (= 1 (gcd a b)))
(define f (/ a b))
(define e.f (real->egyptian-list (/ a b)))
(define l (length e.f))
(define d (denominator (last e.f)))
(values (max max-l l) (if (> l max-l) f max-l-f)
(max max-d d) (if (> d max-d) f max-d-f))))
(printf #<<EOS
max #terms: ~a has ~a
[~.a]
max denominator: ~a has ~a
[~.a]
EOS
max-l-f max-l (real->egyptian-string max-l-f)
max-d-f max-d (real->egyptian-string max-d-f)))
(displayln (real->egyptian-string 43/48))
(displayln (real->egyptian-string 5/121))
(displayln (real->egyptian-string 2014/59))
(newline)
(stat-egyptian-fractions 100)
(newline)
(stat-egyptian-fractions 1000)
(module+ test (require tests/eli-tester)
(test (real->egyptian-list 43/48) => '(1/2 1/3 1/16)))
{{out}} (Line continuations have been manually added to this "post-production")
/2 + /3 + /16
/25 + /757 + /763309 + /873960180913 + /1527612795642093418846225
34 + /8 + /95 + /14947 + /670223480
max #terms: 44/53 has 8
[/2 + /4 + /13 + /307 + /120871 + /20453597227 + /697249399186783218655 + /1458\
470173998990524806872692984177836808420]
max denominator: 8/97 has 57950458706754280171310319185991860825103029195219542\
3583529357653899418686342360361798689053273749372615043661810228371898539583862\
011424993909789665
[/13 + /181 + /38041 + /1736503177 + /3769304102927363485 + /189435378937934085\
04192074528154430149 + /5382864419003802113658172851049070863474397461302269732\
53778132494225813153 + /5795045870675428017131031918599186082510302919521954235\
83529357653...]
max #terms: 641/796 has 13
[/2 + /4 + /19 + /379 + /159223 + /28520799973 + /929641178371338400861 + /1008\
271507277592391123742528036634174730681 + /121993371886539365536463536806812475\
6713122928811333803786753398211072842948484537833 + /18602978480309366547426083\
99135821395...]
max denominator: 36/457 has 839018826833450186636781520007011999269820404906753\
1802447592992878373788953976056132614699956264987192898351123925304308405141021\
4699862566659475699527341801560002349404920810889418578177400268306320425235617\
2520941088783702738286944210460710059319691268110283467445381026653628599765684\
7391053886423100447858449021570769190037352315437817850733931761441676882524465\
4141646641860846545850299797142542834276943312778456057019337677287833621784926\
0872114137931351960543608384244009505664253173875705234889570853924105640193619\
3013327769896882485550270543952379075819512618682808991505743601648001879641672\
7432307831107886759384404314912459627128125253092471912176692574976085510910006\
6731841478262812686642693395896229983745226277793055820609058348269152190083695\
7046857696220116551591742723266473426955898181271263030381719687686504764130274\
5920529107557163795759735682018803165512274974365230126839454212397089242294433\
5857917641636041892192547135178153602038877677614358281581103685526041329841496\
8634103058882552344950151159123885149811135933875727204767441881692001305157196\
0874733881013672826778401335239691097990454591345853624332731197780512641006557\
6961237640824852114328884086581542091492600312838425666927627674227053793897767\
3954653265898430357739443463729497599099055612093342168471581566448842813005126\
9991053009287091906187661577070851924381867636624547746204229426767467795478372\
6990349386117468071932874021023714524610740225814235147693954027910741673103980\
7497497281064839877216027386731730093628023370929088477974994758953471128893395\
0292840780805867029772217568663867878873868980394557400280567725046328647936367\
0076942509109589495377221095405979217163821481666646160815221224686562530536116\
6136453053359228195240378298789615181701779687683648533990573577721416556223812\
8019690863703155643646140428593042643698365810628873388176151499210968029899592\
2754466040011586713812553117621857109517258943846004179432521131844156242428351\
2701888039195543986200846685140545044140622760122924973752382108865950062494534\
6041479014761142212178219484880334877706181646087669794541815844226951298772915\
2441940326466631610424906158237288218706447963113019239557885486647314085357651\
8952261173647603153943546245479192091385391808078296725459242395417581088771003\
3172947011952637392879644767395188828951196481163302536982115669593455710342992\
1063387965046715070102916811976552584464153981214277622597308113449320462341683\
0552005765719102416866159245313681987709468938584100583482219856031514281533824\
6171119673421408585252377842263090764623590075231757102213156942123119632908002\
3952364788544301495422061066036911772385739659997665503832444529713544286955548\
3101661688378890461490612964610594322386216021797248095100247721274970802584016\
9492997310518483221462278567965155036846552482106285983740990753826957262229677\
4545103747438431266995525592705
[/13 + /541 + /321409 + /114781617793 + /14821672255960844346913 + /25106510681\
4993628596500876449600804290086881 + /73539302503361520198362339236500915390885\
795679264404865887253300925727812630083326272641 + /648963481521709674175890714\
89823812369...]
1 test passed
REXX
/*REXX program converts a fraction (can be improper) to an Egyptian fraction. */
parse arg fract '' -1 t; z=$egyptF(fract) /*compute the Egyptian fraction. */
if t\==. then say fract ' ───► ' z /*show Egyptian fraction from C.L.*/
return z /*stick a fork in it, we're done.*/
/*────────────────────────────────$EGYPTF subroutine──────────────────────────*/
$egyptF: parse arg z 1 zn '/' zd,,$; if zd=='' then zd=1 /*whole number ?*/
if z='' then call erx "no fraction was specified."
if zd==0 then call erx "denominator can't be zero:" zd
if zn==0 then call erx "numerator can't be zero:" zn
if zd<0 | zn<0 then call erx "fraction can't be negative" z
if \datatype(zn,'W') then call erx "numerator must be an integer:" zn
if \datatype(zd,'W') then call erx "denominator must be an integer:" zd
_=zn%zd /*check if it's an improper fraction. */
if _>=1 then do /*if improper fraction, then append it.*/
$='['_"]" /*append the whole # part of fraction. */
zn=zn-_*zd /*now, just use the proper fraction. */
if zn==0 then return $ /*Is there no fraction? Then we're done*/
end
if zd//zn==0 then do; zd=zd%zn; zn=1; end
do forever
if zn==1 & datatype(zd,'W') then return $ "1/"zd /*append Egyptian fract.*/
nd=zd%zn+1; $=$ '1/'nd /*add unity to integer fraction, append*/
z=$fractSub(zn'/'zd, "-", 1'/'nd) /*go and subtract the two fractions. */
parse var z zn '/' zd /*extract the numerator and denominator*/
L=2*max(length(zn),length(zd)) /*calculate if need more decimal digits*/
if L>=digits() then numeric digits L+L /*yes, then bump the decimal digits*/
end /*forever*/ /* [↑] the DO forever ends when zn==1.*/
/*────────────────────────────────$FRACTSUB subroutine────────────────────────*/
$fractSub: procedure; parse arg z.1,,z.2 1 zz.2; arg ,op
do j=1 for 2; z.j=translate(z.j,'/',"_"); end
if z.1=='' then z.1=(op\=="+" & op\=='-') /*unary +,- first fraction.*/
if z.2=='' then z.2=(op\=="+" & op\=='-') /*unary +.- second fraction.*/
do j=1 for 2 /*process both of the fractions*/
if pos('/',z.j)==0 then z.j=z.j"/1"; parse var z.j n.j '/' d.j
if \datatype(n.j,'N') then call erx "numerator isn't an integer:" n.j
if \datatype(d.j,'N') then call erx "denominator isn't an integer:" d.j
n.j=n.j/1; d.j=d.j/1 /*normalize numerator/denominator.*/
do while \datatype(n.j,'W'); n.j=n.j*10/1; d.j=d.j*10/1; end /*while*/
/* [↑] normalize both numbers. */
if d.j=0 then call erx "denominator can't be zero:" z.j
g=gcd(n.j,d.j); if g=0 then iterate; n.j=n.j/g; d.j=d.j/g
end /*j*/
l=lcm(d.1 d.2); do j=1 for 2; n.j=l*n.j/d.j; d.j=l; end /*j*/
if op=='-' then n.2=-n.2
t=n.1+n.2; u=l; if t==0 then return 0
g=gcd(t,u); t=t/g; u=u/g; if u==1 then return t
return t'/'u
/*─────────────────────────────general 1─line subs────────────────────────────*/
erx: say; say '***error!***' arg(1); say; exit 13
gcd:procedure;$=;do i=1 for arg();$=$ arg(i);end;parse var $ x z .;if x=0 then x=z;x=abs(x);do j=2 to words($);y=abs(word($,j));if y=0 then iterate;do until _==0;_=x//y;x=y;y=_;end;end;return x
lcm:procedure;y=;do j=1 for arg();y=y arg(j);end;x=word(y,1);do k=2 to words(y);!=abs(word(y,k));if !=0 then return 0;x=x*!/gcd(x,!);end;return x
p: return word(arg(1),1)
'''output''' when the input used is: 43/48
43/48 ───► 1/2 1/3 1/16
'''output''' when the input used is: 5/121
5/121 ───► 1/25 1/757 1/763309 1/873960180913 1/1527612795642093418846225
'''output''' when the input used is: 2014/59
2014/59 ───► [34] 1/8 1/95 1/14947 1/670223480
The following is a driver program to address the requirements to find the largest number of terms for a
1- or 2-digit integer, and the largest denominator.
Also, the same program is used for the 1-, 2-, and 3-digit extra credit task.
/*REXX pgm runs the EGYPTIAN program to find biggest denominator & # of terms.*/
parse arg top . /*get optional parameter from the C.L. */
if top=='' then top=99 /*Not specified? Then use the default.*/
oTop=top; top=abs(top) /*oTop used as a flag to display maxD. */
maxT=0; maxD=0; bigD=; bigT= /*initialize some REXX variables. */
/* [↓] determine biggest andlongest. */
do n=2 to top /*traipse through the numerators. */
do d=n+1 to top /* " " " denominators */
fract=n'/'d /*create the fraction to be used. */
y='EGYPTIAN'(fract||.) /*invoke the REXX program EGYPTIAN.REX*/
t=words(y) /*number of terms in Egyptian fraction.*/
if t>maxT then bigT=fract /*is this a new high for number terms? */
maxT=max(maxT,T) /*find the maximum number of terms. */
b=substr(word(y,t),3) /*get denominator from Egyptian fract. */
if b>maxD then bigD=fract /*is this a new denominator high ? */
maxD=max(maxD,b) /*find the maximum denominator. */
end /*d*/ /* [↑] only use proper fractions. */
end /*n*/ /* [↑] ignore the 1/n fractions. */
/* [↑] display the longest and biggest*/
@= 'in the Egyptian fractions used is' /*literal is used to make a shorter SAY*/
say 'largest number of terms' @ maxT "terms for" bigT
say
say 'highest denominator' @ length(maxD) "digits for" bigD
if oTop>0 then say maxD /*stick a fork in it, we're all done. */
'''output''' for all 1- and 2-digit integers when using the default input:
largest number of terms in the Egyptian fractions used is 8 terms for 8/97
largest denominator in the Egyptian fractions is 150 digits is for 8/97
579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665
'''output''' for all 1-, 2-, and 3-digit integers when using for input: -999
largest number of terms in the Egyptian fractions used is 13 terms for 529/914
largest denominator in the Egyptian fractions is 2847 digits is for 36/457
Ruby
{{trans|Python}}
def ef(fr) ans = [] if fr >= 1 return [[fr.to_i], Rational(0, 1)] if fr.denominator == 1 intfr = fr.to_i ans, fr = [intfr], fr - intfr end x, y = fr.numerator, fr.denominator while x != 1 ans << Rational(1, (1/fr).ceil) fr = Rational(-y % x, y * (1/fr).ceil) x, y = fr.numerator, fr.denominator end ans << fr end for fr in [Rational(43, 48), Rational(5, 121), Rational(2014, 59)] puts '%s => %s' % [fr, ef(fr).join(' + ')] end lenmax = denommax = [0] for b in 2..99 for a in 1...b fr = Rational(a,b) e = ef(fr) elen, edenom = e.length, e[-1].denominator lenmax = [elen, fr] if elen > lenmax[0] denommax = [edenom, fr] if edenom > denommax[0] end end puts 'Term max is %s with %i terms' % [lenmax[1], lenmax[0]] dstr = denommax[0].to_s puts 'Denominator max is %s with %i digits' % [denommax[1], dstr.size], dstr
{{out}}
43/48 => 1/2 + 1/3 + 1/16
5/121 => 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225
2014/59 => 34 + 1/8 + 1/95 + 1/14947 + 1/670223480
Term max is 44/53 with 8 terms
Denominator max is 8/97 with 150 digits
579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665
Sidef
{{trans|Ruby}}
func ef(fr) { var ans = [] if (fr >= 1) { return([fr]) if (fr.is_int) var intfr = fr.int ans << intfr fr -= intfr } var (x, y) = fr.nude while (x != 1) { ans << fr.inv.ceil.inv fr = ((-y % x) / y*fr.inv.ceil) (x, y) = fr.nude } ans << fr return ans } for fr in [43/48, 5/121, 2014/59] { "%s => %s\n".printf(fr.as_rat, ef(fr).map{.as_rat}.join(' + ')) } var lenmax = (var denommax = [0]) for b in range(2, 99) { for a in range(1, b-1) { var fr = a/b var e = ef(fr) var (elen, edenom) = (e.length, e[-1].denominator) lenmax = [elen, fr] if (elen > lenmax[0]) denommax = [edenom, fr] if (edenom > denommax[0]) } } "Term max is %s with %i terms\n".printf(lenmax[1].as_rat, lenmax[0]) "Denominator max is %s with %i digits\n".printf(denommax[1].as_rat, denommax[0].size) say denommax[0]
{{out}}
43/48 => 1/2 + 1/3 + 1/16
5/121 => 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225
2014/59 => 34 + 1/8 + 1/95 + 1/14947 + 1/670223480
Term max is 44/53 with 8 terms
Denominator max is 8/97 with 150 digits
579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665
Tcl
# Just compute the denominator terms, as the numerators are always 1 proc egyptian {num denom} { set result {} while {$num} { # Compute ceil($denom/$num) without floating point inaccuracy set term [expr {$denom / $num + ($denom/$num*$num < $denom)}] lappend result $term set num [expr {-$denom % $num}] set denom [expr {$denom * $term}] } return $result }
Demonstrating: {{works with|Tcl|8.6}}
package require Tcl 8.6 proc efrac {fraction} { scan $fraction "%d/%d" x y set prefix "" if {$x > $y} { set whole [expr {$x / $y}] set x [expr {$x - $whole*$y}] set prefix "\[$whole\] + " } return $prefix[join [lmap y [egyptian $x $y] {format "1/%lld" $y}] " + "] } foreach f {43/48 5/121 2014/59} { puts "$f = [efrac $f]" } set maxt 0 set maxtf {} set maxd 0 set maxdf {} for {set d 1} {$d < 100} {incr d} { for {set n 1} {$n < $d} {incr n} { set e [egyptian $n $d] if {[llength $e] >= $maxt} { set maxt [llength $e] set maxtf $n/$d } if {[lindex $e end] > $maxd} { set maxd [lindex $e end] set maxdf $n/$d } } } puts "$maxtf has maximum number of terms = [efrac $maxtf]" puts "$maxdf has maximum denominator = [efrac $maxdf]"
{{out}}
43/48 = 1/2 + 1/3 + 1/16
5/121 = 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225
2014/59 = [34] + 1/8 + 1/95 + 1/14947 + 1/670223480
8/97 has maximum number of terms = 1/13 + 1/181 + 1/38041 + 1/1736503177 + 1/3769304102927363485 + 1/18943537893793408504192074528154430149 + 1/538286441900380211365817285104907086347439746130226973253778132494225813153 + 1/579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665
8/97 has maximum denominator = 1/13 + 1/181 + 1/38041 + 1/1736503177 + 1/3769304102927363485 + 1/18943537893793408504192074528154430149 + 1/538286441900380211365817285104907086347439746130226973253778132494225813153 + 1/579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665
Note also that also has 8 terms. :
Visual Basic .NET
{{trans|D}}
Imports System.Numerics
Imports System.Text
Module Module1
Function Gcd(a As BigInteger, b As BigInteger) As BigInteger
If b = 0 Then
If a < 0 Then
Return -a
Else
Return a
End If
Else
Return Gcd(b, a Mod b)
End If
End Function
Function Lcm(a As BigInteger, b As BigInteger) As BigInteger
Return a / Gcd(a, b) * b
End Function
Public Class Rational
Dim num As BigInteger
Dim den As BigInteger
Public Sub New(n As BigInteger, d As BigInteger)
Dim c = Gcd(n, d)
num = n / c
den = d / c
If den < 0 Then
num = -num
den = -den
End If
End Sub
Public Sub New(n As BigInteger)
num = n
den = 1
End Sub
Public Function Numerator() As BigInteger
Return num
End Function
Public Function Denominator() As BigInteger
Return den
End Function
Public Overrides Function ToString() As String
If den = 1 Then
Return num.ToString()
Else
Return String.Format("{0}/{1}", num, den)
End If
End Function
'Arithmetic operators
Public Shared Operator +(lhs As Rational, rhs As Rational) As Rational
Return New Rational(lhs.num * rhs.den + rhs.num * lhs.den, lhs.den * rhs.den)
End Operator
Public Shared Operator -(lhs As Rational, rhs As Rational) As Rational
Return New Rational(lhs.num * rhs.den - rhs.num * lhs.den, lhs.den * rhs.den)
End Operator
'Comparison operators
Public Shared Operator =(lhs As Rational, rhs As Rational) As Boolean
Return lhs.num = rhs.num AndAlso lhs.den = rhs.den
End Operator
Public Shared Operator <>(lhs As Rational, rhs As Rational) As Boolean
Return lhs.num <> rhs.num OrElse lhs.den <> rhs.den
End Operator
Public Shared Operator <(lhs As Rational, rhs As Rational) As Boolean
'a/b < c/d
'ad < bc
Dim ad = lhs.num * rhs.den
Dim bc = lhs.den * rhs.num
Return ad < bc
End Operator
Public Shared Operator >(lhs As Rational, rhs As Rational) As Boolean
'a/b > c/d
'ad > bc
Dim ad = lhs.num * rhs.den
Dim bc = lhs.den * rhs.num
Return ad > bc
End Operator
Public Shared Operator <=(lhs As Rational, rhs As Rational) As Boolean
Return lhs < rhs OrElse lhs = rhs
End Operator
Public Shared Operator >=(lhs As Rational, rhs As Rational) As Boolean
Return lhs > rhs OrElse lhs = rhs
End Operator
'Conversion operators
Public Shared Widening Operator CType(ByVal bi As BigInteger) As Rational
Return New Rational(bi)
End Operator
Public Shared Widening Operator CType(ByVal lo As Long) As Rational
Return New Rational(lo)
End Operator
End Class
Function Egyptian(r As Rational) As List(Of Rational)
Dim result As New List(Of Rational)
If r >= 1 Then
If r.Denominator() = 1 Then
result.Add(r)
result.Add(New Rational(0))
Return result
End If
result.Add(New Rational(r.Numerator / r.Denominator))
r -= result(0)
End If
Dim modFunc = Function(m As BigInteger, n As BigInteger)
Return ((m Mod n) + n) Mod n
End Function
While r.Numerator() <> 1
Dim q = (r.Denominator() + r.Numerator() - 1) / r.Numerator()
result.Add(New Rational(1, q))
r = New Rational(modFunc(-r.Denominator(), r.Numerator()), r.Denominator * q)
End While
result.Add(r)
Return result
End Function
Function FormatList(Of T)(col As List(Of T)) As String
Dim iter = col.GetEnumerator()
Dim sb As New StringBuilder
sb.Append("[")
If iter.MoveNext() Then
sb.Append(iter.Current)
End If
While iter.MoveNext()
sb.Append(", ")
sb.Append(iter.Current)
End While
sb.Append("]")
Return sb.ToString()
End Function
Sub Main()
Dim rs = {New Rational(43, 48), New Rational(5, 121), New Rational(2014, 59)}
For Each r In rs
Console.WriteLine("{0} => {1}", r, FormatList(Egyptian(r)))
Next
Dim lenMax As Tuple(Of ULong, Rational) = Tuple.Create(0UL, New Rational(0))
Dim denomMax As Tuple(Of BigInteger, Rational) = Tuple.Create(New BigInteger(0), New Rational(0))
Dim query = (From i In Enumerable.Range(1, 100)
From j In Enumerable.Range(1, 100)
Select New Rational(i, j)).Distinct().ToList()
For Each r In query
Dim e = Egyptian(r)
Dim eLen As ULong = e.Count
Dim eDenom = e.Last().Denominator()
If eLen > lenMax.Item1 Then
lenMax = Tuple.Create(eLen, r)
End If
If eDenom > denomMax.Item1 Then
denomMax = Tuple.Create(eDenom, r)
End If
Next
Console.WriteLine("Term max is {0} with {1} terms", lenMax.Item2, lenMax.Item1)
Dim dStr = denomMax.Item1.ToString()
Console.WriteLine("Denominator max is {0} with {1} digits {2}...{3}", denomMax.Item2, dStr.Length, dStr.Substring(0, 5), dStr.Substring(dStr.Length - 5, 5))
End Sub
End Module
{{out}}
43/48 => [1/2, 1/3, 1/16]
5/121 => [1/25, 1/757, 1/763309, 1/873960180913, 1/1527612795642093418846225]
2014/59 => [34, 1/8, 1/95, 1/14947, 1/670223480]
Term max is 97/53 with 9 terms
Denominator max is 8/97 with 150 digits 57950...89665
zkl
{{trans|Tcl}}
# Just compute the denominator terms, as the numerators are always 1
fcn egyptian(num,denom){
result,t := List(),Void;
t,num=num.divr(denom); // reduce fraction
if(t) result.append(T(t)); // signal t isn't a denominator
while(num){
# Compute ceil($denom/$num) without floating point inaccuracy
term:=denom/num + (denom/num*num < denom);
result.append(term);
z:=denom%num;
num=(if(z) num-z else 0);
denom*=term;
}
result
}
fcn efrac(fraction){ // list to string, format list of denominators
fraction.pump(List,fcn(denom){
if(denom.isType(List)) denom[0]
else String("1/",denom);
}).concat(" + ")
}
foreach n,d in (T(T(43,48), T(5,121), T(2014,59))){
println("%s/%s --> %s".fmt(n,d, egyptian(n,d):efrac(_)));
}
{{out}}
43/48 --> 1/2 + 1/3 + 1/16
5/121 --> 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1025410058030422033
2014/59 --> 34 + 1/8 + 1/95 + 1/14947 + 1/670223480
For the big denominators, use GMP (Gnu Multi Precision).
var [const] BN=Import("zklBigNum"); // libGMP
lenMax,denomMax := List(0,Void),List(0,Void);
foreach n,d in (Walker.cproduct([1..99],[1..99])){ // 9801 fractions
e,eLen,eDenom := egyptian(BN(n),BN(d)), e.len(), e[-1];
if(eDenom.isType(List)) eDenom=1;
if(eLen >lenMax[0]) lenMax.clear(eLen,T(n,d));
if(eDenom>denomMax[0]) denomMax.clear(eDenom,T(n,d));
}
println("Term max is %s/%s with %d terms".fmt(lenMax[1].xplode(), lenMax[0]));
dStr:=denomMax[0].toString();
println("Denominator max is %s/%s with %d digits %s...%s"
.fmt(denomMax[1].xplode(), dStr.len(), dStr[0,5], dStr[-5,*]));
{{out}}
Term max is 97/53 with 9 terms
Denominator max is 8/97 with 150 digits 57950...89665