Originally posted by FabianFnas My solution is something like this:

Def: R is a positive number not started with a zero. Then RR is R repeated twice, we call it a 'repeat'. The number of figures in RR is always twice the number of R.
Ex: If R = 2006 then RR = 20062006.

Def: 1(n)1, when n is a non-negative number, is a number starting with a one followed by n zeroes, ending with a ...[text shortened]... he middle it can never be a square.

And that proves that a repeat can never be a square.

Why do people do this? An example has been posted so unless you can find a mistake in the example then your proof that no example exists must be incorrect.

But there's a smaller example than the one given above.

It all depends on whether or not we can find a square factor of 10^n + 1 for some n.

In fact

10^11 + 1 = 11^2 x 23 x 4093 x 8779

and so (10^11 + 1) x 23 x 4093 x 8779 is square. Unfortunately this square is not the repeat of 23 x 4093 x 8779, which has too few digits, but we can simply multiply it by 4^2 to get it in the right range.

In fact 23 x 4093 x 8779 x 4^2 = 13223140496 and

1322314049613223140496 = 36363636364^2

where of course 36363636364 = 11 x 23 x 4093 x 8779 x 4.