ABC Implies There are Infinitely Many non-Fibonacci-Wieferich Primes - An Application of ABC Conjecture over Number Fields

2015/11/18 by Wayne Peng, Peng, Wayne
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #Coding theory and cryptography #math.NT

paper · pdf · doi:10.48550/arxiv.1511.05645

16 pages

arxiv created 2015/11/18 · arxiv updated 2015/11/19

Abstract

In this paper, we define X-base Fibonacci-Wieferich prime which is a generalized Wieferich prime where X is a finite set of algebraic numbers. We are going to 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 free part of abelian group generated by all elements in X, and we will use the arithmetic point of view and geometric point of view to give heuristic.

Related

• Wall's Conjecture and the ABC Conjecutre
• Non-Wieferich primes in number fields and ABC conjecture
• The abc conjecture implies infinitely many non-Wieferich places for fixed bases in number fields
• Distribution of Non-Wieferich primes in certain algebraic groups under ABC
• Wieferich primes in number fields and the conjectures of Ankeny--Artin--Chowla and Mordell