Greedy algorithm for Egyptian fractions

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

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>

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.abs >= 1 ?? "[{self.floor}]" !! Nil), map {"1/$_"}, self.denominators;

   }
   method denominators {

my ($x, $y) = self.nude; $x %= $y; gather ($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[] }
   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 pgm converts a fraction (can be improper) to an Egyptian fraction*/ parse arg fract; z=$egyptF(fract) /*compute the Egyptian fraction. */ say fract ' ───► ' z /*show Egyptian fraction from CL.*/ return z /*stick a fork in it, we're done.*/ /*────────────────────────────────$EGYPTF subroutine────────────────────*/ $egyptF: parse arg z 1 zn '/' zd,,$; if zd== then zd=1 /*whole #?*/ 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 fract*/ if _>=1 then do /*if improper fract, then append.*/

             $='['_"]"                /*append the whole# part of fract*/
             zn=zn-_*zd               /*now, just use the proper fract.*/
             if zn==0  then return $  /*if no fraction, 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 E. fract.*/
 nd=zd%zn+1;      $=$ '1/'nd          /*add unity to int fract., append*/
 z=$fractSub(zn'/'zd,  "-",  1'/'nd)  /*go and subtract the two fracts.*/
 parse var z zn '/' zd                /*extract the numerator & denom. */
 L=2*max(length(zn),length(zd))       /*calculate if we need more digs.*/
 if L>=digits()  then numeric digits L*2   /*yes, then bump the digits.*/
 end   /*forever*/                    /* [↑]  loop 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 fract.*/ if z.2== then z.2=(op\=="+" & op\=='-') /*unary +.- second fract.*/

 do j=1  for 2                             /*process both 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/denom.*/

      do  while \datatype(n.j,'W');   n.j=n.j*10/1;   d.j=d.j*10/1;   end
                                           /* [↑] normalize both nums. */
 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 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 bigest denominator & #terms*/ parse arg top . /*get optional parameter from CL.*/ if top== then top=99 /*Not specified? Then use default*/ bigD=; bigT=; maxT=0; maxD=0 /*initialize some REXX variables.*/

                                      /* [↓]  determine biggest,longest*/
     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 other REXX program. */
         t=words(y)                   /*find out how many terms in E.F.*/
         if t>maxT  then bigT=fract   /*is this a new high for # terms?*/
         maxT=max(maxT,T)             /*find the maximum number terms. */
         b=substr(word(y,t),3)        /*get the denominator from the EF*/
         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  1/n  fractions.  */
                                      /* [↑]  display longest, biggest.*/

@='in the Egyptian fractions used is' /*literal to make a shorter SAY.*/ say 'largest number of terms' @ maxT "terms for" bigT say 'highest denominator' @ length(maxD) "digits is for" bigD':' say maxD /*stick a fork in it, we're 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

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}}