Originally Posted by coryallen2

*cough*

This list includes the squares of 3, 5, 7, 9, 13, 19, and 63.) These
numbers are the result of a progressive sieve, analogous to the prime
sieve. For example, every term greater than 10 must not be divisible
by 2, because otherwise it would give an integer N for (3K-10)/2 based
on the pair 1,-3. Likewise from the pair 2,-2 we see that every term
greater than 8 must not be congruent to 2 modulo 3, because otherwise
it would give an integer N for (4K-8)/3. Here is a short table of
the expressions that must not be integers for sufficiently large
"prime K" values

-1 -2 -3 -4 -5

1 - (2K-5)/1 (3K-10)/2 ( 4K-17)/ 3 ( 5K-26)/ 4
2 (4K-8)/3 (6K-13)/5 ( 8K-20)/ 7 (10K-29)/ 9
3 (9K-18)/8 (12K-25)/11 (15K-34)/14
4 (16K-32)/15 (20K-41)/19
5 (25K-50)/24

In each case the expression (AK-B)/(A-1) implies that for K values
greater than B we must exclude those such that K = B (mod A-1). In
other words, the sieve excludes every number greater than q = x^2 + y^2
congruent to q mod (xy-1).

*cough*