On a conjecture of Rudin on squares in arithmetic progressions

Enrique González-Jiménez, Xavier Xarles

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Resumen

Let Q(N;q,a) be 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 claims that Q(N)=O({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\le 52 (N=8,13,16,23,27,36,41 and 52). © 2014 Authors.
Idioma original Inglés 58-76 LMS Journal of Computation and Mathematics 17 https://doi.org/10.1112/S1461157013000259 Publicada - 1 ene 2014

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