Let k be a positive integer.

Prove that there must exist a positive integer n such that the sets A( A={1^2, 2^2, 3^2, . . . })
and B ( B = {1^2+n, 2^2+n, 3^2+n, . . . }) share k identical members.

(In reply to

solution by xdog)

For any pair of distinct odd primes p and q, the value p*q^(k-1) has exactly k distinct factorizations.

Since p and q are odd primes then their factors are all odd which then implies the sum and difference of the factor pairs will both be even.

Let f and g be a factorization pair of n=p*q^(k-1). Then the system a+b=f and a-b=g has integer solutions for each possible factorization of n.