TY - JOUR

T1 - On a conjecture of Rudin on squares in arithmetic progressions

AU - González-Jiménez, Enrique

AU - Xarles, Xavier

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

U2 - https://doi.org/10.1112/S1461157013000259

DO - https://doi.org/10.1112/S1461157013000259

M3 - Article

VL - 17

SP - 58

EP - 76

JO - LMS Journal of Computation and Mathematics

JF - LMS Journal of Computation and Mathematics

SN - 1461-1570

ER -