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Let Q(N;q,a) denotes the number of squares in the arithmetic progressionqn+a, for n=0, 1,...,N-1, and let Q(N) be the maximum of Q(N;q,a) over allnon-trivial arithmetic progressions qn + a. Rudin's conjecture asserts thatQ(N)=O(Sqrt(N)), and in its stronger form that Q(N)=Q(N;24,1) if N=> 6. Weprove the conjecture above for 6<=N<=52. We even prove that the arithmeticprogression 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 (hence, forN=8,13,16,23,27,36,41 and 52). This allow us to assert, what we have calledSuper-Strong Rudin's Conjecture: let be N=GP_k+1=> 8 for some integer k, whereGP_k is the k-th generalized pentagonal number, then Q(N)=Q(N;q,a) withgcd(q,a) squarefree and q> 0 if and only if (q,a)=(24,1).

On a conjecture of Rudin on squares in Arithmetic Progressions