On a conjecture of Rudin on squares in Arithmetic Progressions

TL;DR

This paper proves Rudin's conjecture on the maximum number of squares in arithmetic progressions for N up to 52, showing the progression 24n+1 is uniquely optimal for certain N, and proposes a super-strong version of the conjecture.

Contribution

It verifies Rudin's conjecture for 6<=N<=52 and establishes the uniqueness of the progression 24n+1 in containing the maximum number of squares for these N, introducing a new super-strong conjecture.

Findings

Proved Rudin's conjecture for N up to 52.

Identified 24n+1 as the unique progression with maximum squares for certain N.

Formulated the super-strong Rudin's conjecture based on generalized pentagonal numbers.

Abstract

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).