Greedy algorithm for Egyptian fractions: Difference between revisions

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An Egyptian fraction is the sum of distinct unit fractions such as:

${\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{16}}\,(={\tfrac {43}{48}})$ .

Each fraction in the expression has a numerator equal to $1$ and a denominator that is a positive integer, and all the denominators are distinct (i.e., no repetitions).

Fibonacci's Greedy algorithm for Egyptian fractions expands the fraction ${\tfrac {x}{y}}$ to be represented by repeatedly performing the replacement

{\frac {x}{y}}={\frac {1}{\lceil y/x\rceil }}+{\frac {(-y)\!\!\!\!\mod x}{y\lceil y/x\rceil }}

(simplifying the 2^nd term in this replacement as necessary, and where $\lceil x\rceil$ is the ceiling function).

Proper and improper fractions must be able to be expressed.

Proper fractions are of the form ${\tfrac {a}{b}}$ where $a$ and $b$ are positive integers, such that $a<b$ , and
improper fractions are of the form ${\tfrac {a}{b}}$ where $a$ and $b$ are positive integers, such that a ≥ b.

(See the 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: ${\tfrac {43}{48}}$ and ${\tfrac {5}{121}}$ and ${\tfrac {2014}{59}}$
for all proper fractions, ${\tfrac {a}{b}}$ where $a$ and $b$ 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 (extra credit), find and show (as above).

Also see

Wolfram MathWorld™ entry: Egyptian fraction

D

Assuming the Python entry is correct, this code is equivalent. This requires the D module of the Arithmetic/Rational task.

Translation of: Python

<lang d>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 .. $]);

}</lang>

Output:

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

FreeBASIC

Library: GMP

<lang freebasic>' 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))

Data "43/48", "5/121", "2014/59" ' 4/121 = 12/363 = 11/263 + 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

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))

' empty keyboard buffer While InKey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>

Output:

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

Haskell

<lang haskell>import Data.Ratio

egiptianFraction :: Integral a => Ratio a -> [Ratio a] egiptianFraction n

 | n < 0 = map negate (egiptianFraction (-n))
 | n == 0 = []
 | x == 1 = [n]
 | x > y = (x `div` y % 1) : egiptianFraction (x `mod` y % y)
 | otherwise = (1 % r) : egiptianFraction ((-y) `mod` x % (y*r))
 where x = numerator n
       y = denominator n
       r = y `div` x + 1</lang>

Testing: <lang haskell>λ> :m Test.QuickCheck λ> quickCheck (\n -> n == (sum $ egiptianFraction n)) +++ OK, passed 100 tests.</lang>

Tasks: <lang haskell>import Data.List import Data.Ord

task1 = mapM_ run [ 43 % 48, 5 % 121, 2014 % 59 ]

 where run x = putStrLn $ show x ++ " = " ++ result x
       result x = intercalate " + " $ show <$> egiptianFraction x

task21 n = maximumBy (comparing snd)

 [ (a % b, length $ egiptianFraction (a % b))
 | a <- [1..n], b <- [1..n], a < b ]

task22 n = maximumBy (comparing snd)

 [ (a % b, maximum $ map denominator $ egiptianFraction (a % b))
 | a <- [1..n], b <- [1..n], a < b]

</lang>

J

Solution:<lang j> ef =: [: (}.~ 0={.) [: (, r2ef)/ 0 1 #: x:

  r2ef =: (<(<0);0) { ((] , -) >:@:<.&.%)^:((~:<.)@:%)@:{:^:a:</lang>

Examples (required):<lang j> (; ef)&> 43r48 5r121 2014r59 +-------+--------------------------------------------------------------+ |43r48 |1r2 1r3 1r16 | +-------+--------------------------------------------------------------+ |5r121 |1r25 1r757 1r763309 1r873960180913 1r1527612795642093418846225| +-------+--------------------------------------------------------------+ |2014r59|34 1r8 1r95 1r14947 1r670223480 | +-------+--------------------------------------------------------------+</lang>

Examples (extended):<lang j> 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 </lang>

Julia

Works with: Julia version 0.6

<lang julia>struct EgyptianFraction{T<:Integer} <: Real

   integer::T
   fractions::Vector{Rational{T}}

end function Base.show(io::IO, fr::EgyptianFraction)

   if !iszero(fr.integer)
       print(io, "[$(fr.integer)] + ")
   end
   print(io, join(fr.fractions, " + "))

end

function Base.convert(::Type{EgyptianFraction}, fr::Rational{T}) where T

   int = 0
   if fr ≥ 1//1
       int = floor(T, fr)
   end
   fr -= int
   vfr = Vector{Rational{T}}(0)
   while !iszero(fr)
       ufr = 1 // cld(denominator(fr), numerator(fr))
       push!(vfr, ufr)
       fr -= ufr
   end
   return EgyptianFraction{T}(int, vfr)

end Base.convert(::Type{Rational}, fr::EgyptianFraction{T}) where T =

   sum(fr.fractions) + fr.integer

Base.length(fr::EgyptianFraction) = length(fr.fractions) + !iszero(fr.integer)

for fr in Rational{BigInt}[43//48, 5//121, 2014//59]

   println("$fr = ", EgyptianFraction(fr))

end

function rosettatask(nmax::Integer)

   frs = collect(a // b for b in big.(1:nmax) for a in 1:b-1)
   efr = EgyptianFraction.(frs)

   len = length.(efr)
   maxlen = maximum(len)
   maxfrs = frs[len .== maxlen]
   println("\nEgyptian fractions with max number of terms ($maxlen): ", join(maxfrs, ", "))

   denom = map(denominator ∘ last, getfield.(efr, :fractions))
   maxdenom = maximum(denom)
   maxfrs = frs[denom .== maxdenom]
   println("\nEgyptian fraction with the maximum denominator: $(join(maxfrs, ", ")) with $maxdenom")

end

rosettatask(99) rosettatask(999)</lang>

Output:

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

Egyptian fractions with max number of terms (8): 44//53, 8//97

Egyptian fraction with the maximum denominator: 8//97 with 579504587067542801713103191859918608251030291952195423583529357653899418686342360361798689053273749372615043661810228371898539583862011424993909789665

Egyptian fractions with max number of terms (13): 641//796, 529//914

Egyptian fraction with the maximum denominator: 36//457, 36//457, 529//914 with 839018826833450186636781520007011999269820404906753180244759299287837378895397605613261469995626498719289835112392530430840514102146998625666594756995273418015600023494049208108894185781774002683063204252356172520941088783702738286944210460710059319691268110283467445381026653628599765684739105388642310044785844902157076919003735231543781785073393176144167688252446541416466418608465458502997971425428342769433127784560570193376772878336217849260872114137931351960543608384244009505664253173875705234889570853924105640193619301332776989688248555027054395237907581951261868280899150574360164800187964167274323078311078867593844043149124596271281252530924719121766925749760855109100066731841478262812686642693395896229983745226277793055820609058348269152190083695704685769622011655159174272326647342695589818127126303038171968768650476413027459205291075571637957597356820188031655122749743652301268394542123970892422944335857917641636041892192547135178153602038877677614358281581103685526041329841496863410305888255234495015115912388514981113593387572720476744188169200130515719608747338810136728267784013352396910979904545913458536243327311977805126410065576961237640824852114328884086581542091492600312838425666927627674227053793897767395465326589843035773944346372949759909905561209334216847158156644884281300512699910530092870919061876615770708519243818676366245477462042294267674677954783726990349386117468071932874021023714524610740225814235147693954027910741673103980749749728106483987721602738673173009362802337092908847797499475895347112889339502928407808058670297722175686638678788738689803945574002805677250463286479363670076942509109589495377221095405979217163821481666646160815221224686562530536116613645305335922819524037829878961518170177968768364853399057357772141655622381280196908637031556436461404285930426436983658106288733881761514992109680298995922754466040011586713812553117621857109517258943846004179432521131844156242428351270188803919554398620084668514054504414062276012292497375238210886595006249453460414790147611422121782194848803348777061816460876697945418158442269512987729152441940326466631610424906158237288218706447963113019239557885486647314085357651895226117364760315394354624547919209138539180807829672545924239541758108877100331729470119526373928796447673951888289511964811633025369821156695934557103429921063387965046715070102916811976552584464153981214277622597308113449320462341683055200576571910241686615924531368198770946893858410058348221985603151428153382461711196734214085852523778422630907646235900752317571022131569421231196329080023952364788544301495422061066036911772385739659997665503832444529713544286955548310166168837889046149061296461059432238621602179724809510024772127497080258401694929973105184832214622785679651550368465524821062859837409907538269572622296774545103747438431266995525592705

Kotlin

As the JDK lacks a fraction or rational class, I've included a basic one in the program. <lang scala>// 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")
   }

}</lang>

Output:

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

<lang 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 && l1 < 1, ToString[l1, 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]];</lang>

Output:

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

PARI/GP

<lang parigp> 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) </lang>

Output:

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 6

<lang perl6>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;</lang>

Output:

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:

<lang perl6>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;</lang>

Output:

1/2 + 1/3 + 1/4 + 1/6

The list of terms grows exponentially with the value of the fraction, though.

Python

<lang python>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:]))</lang>

Output:

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

Racket

<lang 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)))</lang>

Output:

(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

<lang 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)</lang> 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. <lang rexx>/*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. */</lang> 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

Translation of: Python

<lang ruby>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</lang>

Output:

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

Translation of: Ruby

<lang ruby>func ef(fr) {

 var ans = []
 if (fr >= 1) {
   return([fr]) if (fr.is_int)
   var intfr = fr.to_i
   ans << intfr
   fr -= intfr
 }
 var (x, y) = fr.parts
 while (x != 1) {
   ans << fr.inv.ceil.inv
   fr = ((-y % x) / y*fr.inv.ceil)
   (x, y) = fr.parts
 }
 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]</lang>

Output:

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

<lang 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

}</lang> Demonstrating:

Works with: Tcl version 8.6

<lang tcl>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]"</lang>

Output:

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 ${\tfrac {44}{53}}$ also has 8 terms.

{\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{13}}+{\tfrac {1}{307}}+{\tfrac {1}{120871}}+{\tfrac {1}{20453597227}}+{\tfrac {1}{697249399186783218655}}+{\tfrac {1}{1458470173998990524806872692984177836808420}}

zkl

Translation of: Tcl

<lang zkl># 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(" + ")

}</lang> <lang zkl>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(_)));

}</lang>

Output:

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). <lang zkl>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,*]));</lang>

Output:

Term max is 97/53 with 9 terms
Denominator max is 8/97 with 150 digits 57950...89665