UPDATE: this solution is wrong(I would delete it but I can't :)

I have thought a bit more on this problem and was able to come up with a partial solution.

I would like to ask for some help in order to generalize it.

(It seems that I can't generalize it because I don't yet know well enough regexes in Perl)

MAIN IDEA: if n is a perfect square then n = p1^2kp2^2kp3^2k...pt^2k(as you can see t is the number of prime divisors of n)

Now, for perfect squares whos prime divisors are only 2 and 3 then n=2^2k3^2k

CODE:

sub test {
        print 
        $_[0] =~ $_[1]
        ? "matched\n"
        : "not matched\n";
}

$p2 = qr/(1{4})*/;
$p3 = qr/(1{9})*/;
$issquare = qr/^(1{4})*(1{9})*$/;

#for square numbers of form 2^2k * 3^2k
test(('1'x36), $issquare);#2^2*3^2 <-- perfect square
test(('1'x37), $issquare);#2^2*3^2+1 <-- not perfect square
test(('1'x324), $issquare);#2^2*3^4 <-- perfect square
test(('1'x325), $issquare);#2^2*3^4+1 <-- not perfect square
test(('1'x9216), $issquare);#2^10*3^4 <-- perfect square
test(('1'x9217), $issquare);#2^10*3^4+1 <-- not perfect square
[download]

The last test takes quite a bit and from now I can tell this is not a very efficient way to test for perfect squares(but this was pretty obvious from the very start, using a regex to check arithmetical properties of a number is not a problem where efficiency is desired, or is it ?)

RATAZONG's prime regex

The OP has provided a regex to check if a number is composite /^(11+)\1+$/

and this is a nice regex that works. However, in order to get a regex to check for primes with this we need to negate the regex(I failed while trying to use zero-width look-ahead look-behind negative in order to negate this,so if anyone has any suggestions or ideas on how to negate this ?). But to continue to a final solution let's suppose such a regex for checking if a string of 1s is of prime length and let's denote it $prime.

(PROBABLY) A GENERAL SOLUTION

A general solution to the problem could be something like this

/^( ((?<prime2>$prime)\k<prime2>)* )*$/

I've used named backreferences because if I used the numbered there would be some confusions between the current captures and the ones withing the $prime regex.

Again, it's still unclear to me(or I don't remember) if + or * mean repeat the same sequence multiple times or repeat a sequence that matches the sub-expression multiple times(that's because I'm still a noob).