On a conjecture of Rudin on squares in arithmetic progressions

Let Q(N;q, a) be the number of squares in the arithmetic progression qn + a, for n = 0, 1,..., N - 1, and let Q(N) be the maximum of Q(N;q, a) over all non-trivial arithmetic progressions qn + a. Rudin's conjecture claims that Q(N) = O(root N), and in its stronger form that Q(N) = Q(N;24, 1) if N >= 6. We prove the conjecture above for 6 <= N <= 52. We even prove that the arithmetic progression 24n + 1 is the only one, up to equivalence, that contains Q(N) squares for the values of N such that Q(N) increases, for 7 <= N <= 52 (N = 8, 13, 16, 23, 27, 36, 41 and 52).