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Rudin's conjecture — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Rudin%27s_conjecture
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T17:23:19.605Z
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Rudin's conjecture — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Rudin%27s_conjecture (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_51bbddebfb3b (Metraipolis archive dc_51bbddebfb3b, sha256 b7f7ee0797d552e596edaf582e891c0c2dfef62b71444aecb0d00c53942277be)

BibTeX

@misc{dc51bbddebfb3b,
  title = {Rudin's conjecture — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Rudin%27s_conjecture},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_51bbddebfb3b; sha256 b7f7ee0797d552e596edaf582e891c0c2dfef62b71444aecb0d00c53942277be},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Rudin%27s_conjecture. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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. For positive integers N , q , a {\displaystyle N,q,a} define the expression Q ( N ; q , a ) {\displaystyle Q(N;q,a)} to be the number of perfect squares in the arithmetic progression q n + a {\displaystyle qn+a} , for n = 0 , 1 , … , N − 1 {\displaystyle n=0,1,\ldots ,N-1} , and define Q ( N ) {\displaystyle Q(N)} to be the maximum of the set {Q(N; q, a) : q, a ≥ 1} . The conjecture asserts (in big O notation) that Q ( N ) = O ( N ) {\displaystyle Q(N)=O({\sqrt {N}})} and in its stronger form that, if N > 6 {\displaystyle N>6} , Q ( N ) = Q ( N ; 24 , 1 ) {\displaystyle Q(N)=Q(N;24,1)} . This progression begins with 1 , 25 , 49 , 73 , 97 , 121 , . . . {\displaystyle 1,25,49,73,97,121,...} and its square terms correspond to indices that are generalized pentagonal numbers. For N = 5 {\displaystyle N=5} , the arithmetic progression 120 n + 49 {\displaystyle 120n+49} contains four squares ( 49 , 169 , 289 , 529 ) {\displaystyle (49,169,289,529)} , so Q ( 5 ) = 4 {\displaystyle Q(5)=4} , whereas 24 n + 1 {\displaystyle 24n+1} only contains three squares in its first five terms.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; categoria «Combinatorics»).
Fonte: Wikipedia in inglese, voce «Rudin's conjecture» — https://en.wikipedia.org/wiki/Rudin%27s_conjecture
Wikidata Q55632080 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q55632080
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Rudin%27s_conjecture?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1308188802, consultata il 2026-10-07T16:54:30.508Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: c1d69fbe401cbcbeea72fdebfa731e26149ec9da2698320cb143ce971ed2912a