  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<=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 (hence, for
N=8,13,16,23,27,36,41 and 52). This allow us to assert, what we have called
Super-Strong Rudin's Conjecture: let be N=GP_k+1=> 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).
