1322314049613223140496 = 36363636364 ^ 2.

The key to solving this puzzle is looking at the basic form of these "twin" numbers, which is some number k = 1 + 10^n multiplied by some number 10^(n-1) <= a < 10^n. If ak is a perfect square, k must have some repeated factor, since a<k. Searching the possible values of k for one with a repeated factor eventually turns up the number 1 + 10^11 = 11^2 star 826446281. So, we set a=826446281 and ak = 9090909091^2 = 82644628100826446281, but this needs leading zeros to fit the pattern. So, we multiply by a suitable small square (in this case 16) to get the above answer.
