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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). 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