On the 𝑝-adic zeros of the Tribonacci sequence

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation Pub Date : 2023-08-31 DOI:10.1090/mcom/3893

Yuri Bilu, Florian Luca, Joris Nieuwveld, Joël Ouaknine, James Worrell

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For <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p equals 2\"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">p=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Marques and Lengyel found some formulas relating <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"nu Subscript p Baseline left-parenthesis upper T Subscript n Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\nu _p(T_n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"nu Subscript p Baseline left-parenthesis f left-parenthesis n right-parenthesis right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\nu _p(f(n))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f left-parenthesis n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">f(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is some linear function of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (which might be constant) according to the residue class of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> modulo <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"32\"> <mml:semantics> <mml:mn>32</mml:mn> <mml:annotation encoding=\"application/x-tex\">32</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and asked if similar formulas exist for other primes <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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引用次数: 0

Abstract

Let ( T n ) n ∈ Z (T_n)_{n\in {\mathbb Z}} be the Tribonacci sequence and for a prime p p and an integer m m let ν p ( m ) \nu _p(m) be the exponent of p p in the factorization of m m . For p = 2 p=2 Marques and Lengyel found some formulas relating ν p ( T n ) \nu _p(T_n) with ν p ( f ( n ) ) \nu _p(f(n)) where f ( n ) f(n) is some linear function of n n (which might be constant) according to the residue class of n n modulo 32 32 and asked if similar formulas exist for other primes p p . In this paper, we give an algorithm which tests whether for a given prime p p such formulas exist or not. When they exist, our algorithm computes these formulas. Some numerical results are presented.

查看原文本刊更多论文

在Tribonacci数列的𝑝-adic零点上

设(tn) n∈Z (T_n)_{n\in {\mathbb Z}}为Tribonacci序列对于素数p p和整数m m，设ν p(m) \nu _p(m)为p p在m m分解中的指数。对于p=2 p=2, Marques和Lengyel发现了一些关于ν p(tn) \nu _p(T_n)与ν p(f(n)) \nu _p(f(n))的公式，其中f(n) f(n)是n n的某个线性函数(可能是常数)根据n n模32 32的剩余类，并询问是否存在其他素数p p的类似公式。在本文中，我们给出了一个算法来检验对于给定素数p p是否存在这样的公式。当它们存在时，我们的算法计算这些公式。给出了一些数值结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematics of Computation 数学-应用数学

CiteScore

3.90

自引率

5.00%

发文量

审稿时长

7.0 months

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include numerical analysis, computational discrete mathematics, including number theory, algebra and combinatorics, and related fields such as stochastic numerical methods. Articles must be of significant computational interest and contain original and substantial mathematical analysis or development of computational methodology.