factoring

(x+y)(x-y)

The prime factor of the decomposition of 7 is 1×7, the parity is the same, and the equation is listed, so x=4 and y=3.

But 6 can be decomposed into 1×6 or 2×3.

Because both X and Y are integers, the parity of x+y and x-y is the same, but no matter 1 and 6, 2 and 3, the parity is different, so 6 does not belong to this set.

So 7 belongs and 6 doesn't.