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Büchi’s Problem in Modular Arithmetic for Arbitrary Quadratic Polynomials

Part of: Sequences and sets

Published online by Cambridge University Press:  26 April 2019

Pablo Sáez ,
Xavier Vidaux  and
Maxim Vsemirnov
Pablo Sáez
Affiliation:
Concepción, Chile Email: pablosaezphd@gmail.com
Xavier Vidaux
Affiliation:
Universidad de Concepción, Facultad de Ciencias Físicas y Matemáticas, Departamento de Matemática, Casilla 160 C, Concepción, Chile Email: xvidaux@udec.cl
Maxim Vsemirnov
Affiliation:
St. Petersburg Department of V.A.Steklov Institute of Mathematics, 27 Fontanka, St. Petersburg, 191023, Russia St. Petersburg State University, Department of Mathematics and Mechanics, 28 University prospekt, St. Petersburg, 198504, Russia Email: vsemir@pdmi.ras.ru
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Abstract

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Given a prime $p\geqslant 5$ and an integer $s\geqslant 1$, we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^{s}$, such that $f$ is not a square, if a sequence $(f(1),\ldots ,f(N))$ is a sequence of squares, then $N$ is at most $M$. We also provide some explicit formulas for the optimal $M$.

Keywords

Büchi sequenceHensley sequence

MSC classification

Primary: 11B50: Sequences (mod $m$)
Secondary: 11B83: Special sequences and polynomials

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 876 - 885
Copyright
© Canadian Mathematical Society 2019 

Footnotes

All three authors were partially supported by the grants “Fondecyt research projects 1130134 and 1170315, Chile” to author X. V., from Conicyt. Author M. V. was partially supported by the government of the Russian Federation (grant 14.Z50.31.0030).

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