ABC implies there are infinitely many non-Fibonacci-Wieferich primes

Wayne Peng

doi:10.1016/j.jnt.2019.11.010

ABC implies there are infinitely many non-Fibonacci-Wieferich primes

Wayne Peng

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We define X-base Fibonacci-Wieferich primes, which generalize Wieferich primes, where X is a finite set of algebraic numbers. We show that there are infinitely many non-X-base Fibonacci-Wieferich primes, assuming the abc-conjecture of Masser-Oesterlé-Szpiro for number fields. We also provide a new conjecture concerning the rank of the free part of the abelian group generated by all elements in X and give some heuristics that support the conjecture.

Original language	English
Pages (from-to)	354-375
Number of pages	22
Journal	Journal of Number Theory
Volume	212
DOIs	https://doi.org/10.1016/j.jnt.2019.11.010
State	Published - Jul 2020

Keywords

abc Conjecture
Arithmetic dynamical system
Fibonacci-Wieferich primes
Wall's conjecture
Wall-Sun-Sun primes
Wieferich criteria
Wieferich primes

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10.1016/j.jnt.2019.11.010

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title = "ABC implies there are infinitely many non-Fibonacci-Wieferich primes",

abstract = "We define X-base Fibonacci-Wieferich primes, which generalize Wieferich primes, where X is a finite set of algebraic numbers. We show that there are infinitely many non-X-base Fibonacci-Wieferich primes, assuming the abc-conjecture of Masser-Oesterl{\'e}-Szpiro for number fields. We also provide a new conjecture concerning the rank of the free part of the abelian group generated by all elements in X and give some heuristics that support the conjecture.",

keywords = "abc Conjecture, Arithmetic dynamical system, Fibonacci-Wieferich primes, Wall's conjecture, Wall-Sun-Sun primes, Wieferich criteria, Wieferich primes",

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N2 - We define X-base Fibonacci-Wieferich primes, which generalize Wieferich primes, where X is a finite set of algebraic numbers. We show that there are infinitely many non-X-base Fibonacci-Wieferich primes, assuming the abc-conjecture of Masser-Oesterlé-Szpiro for number fields. We also provide a new conjecture concerning the rank of the free part of the abelian group generated by all elements in X and give some heuristics that support the conjecture.

AB - We define X-base Fibonacci-Wieferich primes, which generalize Wieferich primes, where X is a finite set of algebraic numbers. We show that there are infinitely many non-X-base Fibonacci-Wieferich primes, assuming the abc-conjecture of Masser-Oesterlé-Szpiro for number fields. We also provide a new conjecture concerning the rank of the free part of the abelian group generated by all elements in X and give some heuristics that support the conjecture.

KW - abc Conjecture

KW - Arithmetic dynamical system

KW - Fibonacci-Wieferich primes

KW - Wall's conjecture

KW - Wall-Sun-Sun primes

KW - Wieferich criteria

KW - Wieferich primes

UR - https://www.scopus.com/pages/publications/85077140795

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DO - 10.1016/j.jnt.2019.11.010

M3 - 期刊論文

AN - SCOPUS:85077140795

SN - 0022-314X

VL - 212

SP - 354

EP - 375

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

ABC implies there are infinitely many non-Fibonacci-Wieferich primes

Abstract

Keywords

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