Call n the number that is to be squared.
n must be 32 to 99
n cannot be divisible by 3 or 11 or the reversal wont be prime
also n^2 cannot begin with 2,4,5,6,or 8 or the reversal wont be prime
[we are down to 19 possibilities]
when a number is added to its reversal the result is divisible by 11, to be a semiprime it cannot be divisible by 2 or 5.
So the first and last digits of n^2 cannot be both odd, both even, or add to 5.
This leaves 7 possibilities for n
{34, 40, 56, 58, 62, 86, 98}
Checking which n^2 are prime when reversed rules out five of these. So the 2 left are
{40, 62}
You must be allowing leading zeros, since these are the two solutions:
40^2 = 1600. 0061 = 61 which is prime, and 1600+61 = 1661 = 11*151
62^2 = 3844. 4483 is prime and 3844+4483 = 8327 = 11*757

Posted by Jer
on 20101214 17:17:43 