On a conjecture of Rudin on squares in Arithmetic Progressions

Let Q(N;q,a) denotes 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 asserts that Q(N)=O(Sqrt(N)), and in its stronger form that Q(N)=Q(N;24,1) if N=> 6. We prove the conjecture above for 6 8 for some integer k, where GP_k is the k-th generalized pentagonal number, then Q(N)=Q(N;q,a) with gcd(q,a) squarefree and q> 0 if and only if (q,a)=(24,1).