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Regular primes, non-Wieferich primes, and finite multiple zeta values of level $N$

Shin-ichiro Seki

Abstract

We introduce finite multiple zeta values of general level and discuss the relationship between the non-zeroness of these values and regular or non-Wieferich primes. Because it's challenging to prove the infinitude of these types of primes, we suggest tackling several related problems more promptly.

Regular primes, non-Wieferich primes, and finite multiple zeta values of level $N$

Abstract

We introduce finite multiple zeta values of general level and discuss the relationship between the non-zeroness of these values and regular or non-Wieferich primes. Because it's challenging to prove the infinitude of these types of primes, we suggest tackling several related problems more promptly.
Paper Structure (8 sections, 13 theorems, 29 equations)

This paper contains 8 sections, 13 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Finite multiple zeta values of level $N$
  3. Finite multiple zeta values
  4. Finite multiple zeta values of level $N$
  5. Non-zeroness
  6. Bernoulli--Seki numbers
  7. Fermat quotients
  8. Problems

Key Result

Theorem 1.3

If $p$ is a regular prime, then Fermat's last theorem for the exponent $p$ is correct.

Theorems & Definitions (29)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1.3: Kummer Kummer
  • Theorem 1.4: Wieferich Wieferich
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 19 more