Rudin's conjecture

{{Short description|Mathematical conjecture}}

Rudin's conjecture is a mathematical conjecture in additive combinatorics and elementary number theory about 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.{{cite book|editor=Granville, Andrew|editor2=Nathanson, Melvyn Bernard|editor3=Solymosi, József|editor3-link= József Solymosi |author=Cilleruelo, Javier|author2=Granville, Andrew|chapter=Lattice points on circles, squares in arithmetic progressions and sumsets of squares|title=Additive combinatorics|year=2007|series=CRM Proceedings & Lecture Notes, vol. 43|publisher=American Mathematical Society|pages=241–262|isbn=978-0-8218-7039-6 |chapter-url=https://books.google.com/books?id=9q6_O6AwAhQC&pg=PA241}} [https://arxiv.org/abs/math/0608109 arXiv.org preprint]{{cite journal|jstor=24900534|author=Rudin, Walter|title=Trigonometric series with gaps|journal=Journal of Mathematics and Mechanics|year=1960|volume=9 |issue=2 |pages=203–227}}{{cite journal|journal=LMS Journal of Computation and Mathematics|volume=17|issue=1|year=2014|pages=58–76|title=On a conjecture of Rudin on squares in arithmetic progressions|author=González-Jiménez, Enrique|author2=Xarles, Xavier|doi=10.1112/S1461157013000259|arxiv=1301.5122}}

For positive integers $N,\; q,\; a$ define the expression $Q(N;\; q,\; a)$ to be the number of perfect squares in the arithmetic progression $qn\; +\; a$, for $n\; =\; 0,\; 1,\; \backslash ldots,\; N-1$, and define $Q(N)$ to be the maximum of the set {{math|{Q(N; q, a) : q, a ≥ 1} }}. The conjecture asserts (in big O notation) that $Q(N)\; =\; O(\backslash sqrt\; \{\; N\; \})$ and in its stronger form that, if $N\; >\; 6$, $Q(N)\; =\; Q(N;\; 24,\; 1)$.