Is Rudin's conjecture true?

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For natural numbers N, q, and a, the function Q(N,q,a) is equal to the number of perfect squares in the finite arithmetic progression {qn+a | n = 0,1,...N-1}, and Q(N) is the maximum value of Q(N,q,a) for all values of q and a. Rudin's conjecture states that Q(N) = O(√N).

Note: The O is big O notation, not a function.

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