use Math::Pari 'lcm';my($n,$d)=split/\//,shift;sub p{print"1/$_[0] "}w
+hile(1){my
$u=int$d/$n+1;p$u;my $l=lcm$d,$u;$n=$n*$l/$d-$l/$u;$d=$l;p($d),die$/if
+$n==1}
[download]

($n,$d)=split/\//,shift;sub p{print"1/$_[0] "}while(1){use Math::Pari
lcm;$u=int$d/$n+1;p$u;$l=lcm$d,$u;$n=$n*$l/$d-$l/$u;$d=$l;p($d),die$/i
+f$n==1}
[download]

Hi Limbic~Region,

A couple of small changes will get you down to 142:

($n,$d)=split'/',shift;sub p{print"1/$_[0] "}{use Math::Pari
lcm;$u=int$d/$n+1;p$u;$l=lcm$d,$u;$n=$n*$l/$d-$l/$u;$d=$l;p($d),die$/i
+f$n==1;redo}
[download]

---

s''(q.S:$/9=(T1';s;(..)(..);$..=substr+crypt($1,$2),2,3;eg;print$..$/

Refinement of this particular track at 130:

($n,$d)=split'/',pop;sub p{print"1/@_ "}{use Math::Pari
lcm;p$u=int$d/$n+1;$l=lcm$d,$u;$n=$n*$l/$d-($d=$l)/$u;$n-1?redo:
p$d+die$/}
[download]

81 chars:

$ perl -e '($n,$d)=split"/",pop;{1while++$x<$d/$n;warn"1/$x\n";$n=$n*$
+x-$d;$d*=$x;redo if$n}' 18/20
1/2
1/3
1/15
[download]

Update: changing the entire structure to a C-style for-loop saves 2 chars, so this gives me a 79-char solution:

for(($n,$d)=split"/",pop;$n;){1while++$x<$d/$n;warn"1/$x\n";$n=$n*$x-$
+d;$d*=$x}
[download]

This uses the greedy heuristic, so it doesn't find the "best" Egyptian fractions. $x keeps increasing until 1/$x is <= our current fraction. Then we print 1/$x and subtract from the current fraction. But there is no need to put this fraction in lowest terms, which saves a lot of strokes.
If POSIX::ceil were available, 1while++$x<$d/$n is really just $x=ceil($d/$n) ..
BTW, this 58-char solution would work if it weren't for floating-point error:
perl -e '$f=eval pop;{1while++$x<1/$f;warn"1/$x\n";redo if$f-=1/$x}' 1
+9/20
[download]
I don't have Math::Pari where I am, but it may be possible to use it in the above approach (it treats rationals with absolute precision).

blokhead

Ah, no, Limbic~Region,

your's ain't golf. No modules, lest I would say

use Junk;do
[download]

- 11 chars.. but that could be golfed down to use J;d - 7 chars ;-)

blokhead's solutions look neat, but they hog my cpu... (with 2355/12344 - still not finished in 3/4 hour ;-)

#!/usr/bin/perl -l
pop=~m|/|;($f,$g)=($`,$');sub d{int($_[1]/$_[0]+1)}sub
g{($x,$y)=@_;($x,$y)=($y,$x% $y)while$y;$x} sub re{($p
,$e,$r,$l)=@_;($p,$l)=($p*$l-$e*$r,$e*$l);$g=g($p,$l);
for($p,$l){$_/=$g};($p,$l)}while($f>1){push@o,"1/".d(#
$f,$g);($f,$g)=re($f,$g,1,d($f,$g));}print join' + ',#
@o,"$f/$g";# ungolfed and thus not for production use!
[download]

way too long...

<update> golfed down a bit... (198 chars, counting newlines).

($z,$n)=($_=pop)=~/(.+)\/(.+)/;$s='==';for(;;){$m=int($n/$z+1);$_
.=" $s 1/".($z==1?$n:$m);$z<=1&&last;($z,$n)=($z*$m-$n,$m*$n);($x
,$y)=($z,$n);($x,$y)=($y,$x%$y)while$y;$z/=$x;$n/=$x;$s='+'}print
[download]

It computes egyptian fractions like this

qwurx [shmem] ~> perl -l egy.pl 2355/12344
2355/12344 == 1/6 + 1/42 + 1/3282 + 1/15755059 + 1/744665636525384
[download]

in no time... </update>

--shmem

_($_=" "x(1<<5)."?\n".q·/)Oo.  G°\        /
                              /\_¯/(q    /
----------------------------  \__(m.====·.(_("always off the crowd"))."·
");sub _{s./.($e="'Itrs `mnsgdq Gdbj O`qkdq")=~y/"-y/#-z/;$e.e && print}

A modification of blokhead's idea... what golf can go without regular expressions? 72 strokes:

for($_=pop;/\//,$`;$_=$x*$`-$'.'/'.$'*$x){1while++$x<$'/$`;print"1/$x
"}
[download]

I counted the line break as one character, is that kosher?
Update: I see also shmem had a similar thought, I had missed that before posting somehow.
Update again: Changed the title of this node to not be dumb. Sorry all, thanks for the feedback those who messaged me!
~dewey