UPDATE

Since the poster requested a single regex that matches against strings like "36/270", but says it doesn’t matter how legible it is, that regex is:

my $reducible_rx = qr{^(\d+)/(\d+)$(?(?{(1x$1."/".1x$2)=~m{^(?|1+/(1)|(11+)\1*/\1+)$}})|^)};

But, if like me, you believe that an illegible regex is absolutely unacceptable, you will write that more legibly as:

my $reducible_rx = qr{ # first match a fraction: ^ ( \d+ ) / ( \d+ ) $ # now for the hard part: (?(?{ ( 1 x $1 . "/" . 1 x $2 ) =~ m{ ^ (?| 1+ / (1) # trivial case: GCD=1 | (11+) \1* / \1+ # find the GCD ) $ }x }) # more portable version of (*PASS) | ^ # more portable version of (*FAIL) ) }x;

You can improve maintainability by splitting out the version that matches the unary version from the one that matches the decimal version like this:

# this one assumes unary notation my $unary_rx = qr{ ^ (?| 1+ / (1) | (11+) \1* / \1+ ) $ }x; # this one assumes decimal notation and converts internally my $decimal_rx = qr{ # first match a fraction: ^ ( \d+ ) / ( \d+ ) $ # now for the hard part: (?(?{( 1 x $1 . "/" . 1 x $2 ) =~ $unary_rx}) # more portable version of (*PASS) | ^ # more portable version of (*FAIL) ) }x;

Isn’t that much easier by separating it into two named regexes? That would now make $reducible_rx the same as $decimal_rx , but the unary version is its own thing. That’s how I would do it, but the original poster wanted a single regex, so you’d have to interpolate the nested one for that as I first present above.

Either way, you can plug into the test harness below using:

if ($frac =~ $reducible_rx) { cmp_ok($frac, "ne", reduce($i, $j), "$i/$j is $test"); } else { cmp_ok($frac, "eq", reduce($i, $j), "$i/$j is $test"); }

And you will see that it is a correct regex that passes all tests, and does so moreover using a single regex, wherefore having now passed all requirements of the original question, I declare Qᴜᴏᴅ ᴇʀᴀᴛ ᴅᴇᴍᴏɴsᴛʀᴀɴᴅᴜᴍ: “Quit, enough done.” 😇

And you’re welcome.

The answer is to match the regex ^(?|1+/(1)|(11+)\1*/\1+)$ against the fraction once it has been converted from decimal to unary notation, at which point the greatest common factor will be found in $1 on a match; otherwise they are coprimes. If you are using Perl 5.14 or better, you can even do this in one step:

use 5.014; my $reg = qr{^(?|1+/(1)|(11+)\1*/\1+)$}; my $frac = "36/270"; # for example if ($frac =~ s/(\d+)/1 x $1/reg =~ /$reg/) { say "$frac can be reduced by ", length $1; } else { say "$frac is irreducible"; }

Which will correctly report that:

36/270 can be reduced by 18

(And of course, reducing by 1 means there is no longer a denominator.)

If you wanted to have a bit of punning fun with your readers, you could even do it this way:

use 5.014; my $regex = qr{^(?|1+/(1)|(11+)\1*/\1+)$}; my $frac = "36/270"; # for example if ($frac =~ s/(\d+)/"1 x $1"/regex =~ /$regex/) { say "$frac can be reduced by ", length $1; } else { say "$frac is irreducible"; }

Here is the code that demonstrates how to do this. Furthermore, it constructs a test suite that tests its algorithm using all (positive) numerators and denominators up to its argument, or 30 by default. To run it under a test harness, put it in a file named coprimes and do this:

$ perl -MTest::Harness -e 'runtests("coprimes")' coprimes .. ok All tests successful. Files=1, Tests=900, 1 wallclock secs ( 0.13 usr 0.02 sys + 0.33 cusr 0.02 csys = 0.50 CPU) Result: PASS

Here is an example of its output when run without the test harness:

$ perl coprimes 10 1..100 ok 1 - 1/1 is 1 ok 2 - 1/2 is 1/2 ok 3 - 1/3 is 1/3 ok 4 - 1/4 is 1/4 ok 5 - 1/5 is 1/5 ok 6 - 1/6 is 1/6 ok 7 - 1/7 is 1/7 ok 8 - 1/8 is 1/8 ok 9 - 1/9 is 1/9 ok 10 - 1/10 is 1/10 ok 11 - 2/1 is 2 ok 12 - 2/2 is 1 ok 13 - 2/3 is 2/3 ok 14 - 2/4 is 1/2 ok 15 - 2/5 is 2/5 ok 16 - 2/6 is 1/3 ok 17 - 2/7 is 2/7 ok 18 - 2/8 is 1/4 ok 19 - 2/9 is 2/9 ok 20 - 2/10 is 1/5 ok 21 - 3/1 is 3 ok 22 - 3/2 is 3/2 ok 23 - 3/3 is 1 ok 24 - 3/4 is 3/4 ok 25 - 3/5 is 3/5 ok 26 - 3/6 is 1/2 ok 27 - 3/7 is 3/7 ok 28 - 3/8 is 3/8 ok 29 - 3/9 is 1/3 ok 30 - 3/10 is 3/10 ok 31 - 4/1 is 4 ok 32 - 4/2 is 2 ok 33 - 4/3 is 4/3 ok 34 - 4/4 is 1 ok 35 - 4/5 is 4/5 ok 36 - 4/6 is 2/3 ok 37 - 4/7 is 4/7 ok 38 - 4/8 is 1/2 ok 39 - 4/9 is 4/9 ok 40 - 4/10 is 2/5 ok 41 - 5/1 is 5 ok 42 - 5/2 is 5/2 ok 43 - 5/3 is 5/3 ok 44 - 5/4 is 5/4 ok 45 - 5/5 is 1 ok 46 - 5/6 is 5/6 ok 47 - 5/7 is 5/7 ok 48 - 5/8 is 5/8 ok 49 - 5/9 is 5/9 ok 50 - 5/10 is 1/2 ok 51 - 6/1 is 6 ok 52 - 6/2 is 3 ok 53 - 6/3 is 2 ok 54 - 6/4 is 3/2 ok 55 - 6/5 is 6/5 ok 56 - 6/6 is 1 ok 57 - 6/7 is 6/7 ok 58 - 6/8 is 3/4 ok 59 - 6/9 is 2/3 ok 60 - 6/10 is 3/5 ok 61 - 7/1 is 7 ok 62 - 7/2 is 7/2 ok 63 - 7/3 is 7/3 ok 64 - 7/4 is 7/4 ok 65 - 7/5 is 7/5 ok 66 - 7/6 is 7/6 ok 67 - 7/7 is 1 ok 68 - 7/8 is 7/8 ok 69 - 7/9 is 7/9 ok 70 - 7/10 is 7/10 ok 71 - 8/1 is 8 ok 72 - 8/2 is 4 ok 73 - 8/3 is 8/3 ok 74 - 8/4 is 2 ok 75 - 8/5 is 8/5 ok 76 - 8/6 is 4/3 ok 77 - 8/7 is 8/7 ok 78 - 8/8 is 1 ok 79 - 8/9 is 8/9 ok 80 - 8/10 is 4/5 ok 81 - 9/1 is 9 ok 82 - 9/2 is 9/2 ok 83 - 9/3 is 3 ok 84 - 9/4 is 9/4 ok 85 - 9/5 is 9/5 ok 86 - 9/6 is 3/2 ok 87 - 9/7 is 9/7 ok 88 - 9/8 is 9/8 ok 89 - 9/9 is 1 ok 90 - 9/10 is 9/10 ok 91 - 10/1 is 10 ok 92 - 10/2 is 5 ok 93 - 10/3 is 10/3 ok 94 - 10/4 is 5/2 ok 95 - 10/5 is 2 ok 96 - 10/6 is 5/3 ok 97 - 10/7 is 10/7 ok 98 - 10/8 is 5/4 ok 99 - 10/9 is 10/9 ok 100 - 10/10 is 1

And here is the program: