For natural numbers N, q, and a, the function Q(N,q,a) is equal to the number of perfect squares in the finite arithmetic progression {qn+a | n = 0,1,...N-1}, and Q(N) is the maximum value of Q(N,q,a) for all values of q and a. The stronger version of Rudin's conjecture states that Q(N) = Q(N,24,1) for all N>6.