Rudin's conjecture is a mathematical hypothesis (in additive combinatorics and elementary number theory) concerning an upper bound for the number of squares in finite arithmetic progressions. The conjecture, which has applications in the theory of trigonometric series, was first stated by Walter Rudin in his 1960 paper Trigonometric series with gaps.[1][2][3]
For positive integers \( {\displaystyle N,q,a} \) define the expression \( {\displaystyle Q(N;q,a)} \) to be the number of perfect squares in the arithmetic progression \( {\displaystyle qn+a} \), for \( {\displaystyle n=0,1,\ldots ,N-1} \), and define \( {\displaystyle Q(N)} \) to be the maximum of the set {Q(N; q, a) : q, a ≥ 1} . Rudin’s conjecture asserts (in big O notation) that \( {\displaystyle Q(N)=O({\sqrt {N}})} \) and in its stronger form that, if \( {\displaystyle N>5} \), \( {\displaystyle Q(N)=Q(N;24,1)} \).[3]
References
Cilleruelo, Javier; Granville, Andrew (2007). "Lattice points on circles, squares in arithmetic progressions and sumsets of squares". In Granville, Andrew; Nathanson, Melvyn Bernard; Solymosi, József (eds.). Additive combinatorics. CRM Proceedings & Lecture Notes, vol. 43. American Mathematical Society. pp. 241–262. arXiv.org preprint
Rudin, Walter (1960). "Trigonometric series with gaps". Journal of Mathematics and Mechanics: 203–227. JSTOR 24900534.
González-Jiménez, Enrique; Xarles, Xavier (2014). "On a conjecture of Rudin on squares in arithmetic progressions". LMS Journal of Computation and Mathematics. 17 (1): 58–76. arXiv:1301.5122. doi:10.1112/S1461157013000259. arXiv.org preprint
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