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Fermat-Catalan and Tijdeman-Zagier conjectures for products

Adam S. Sikora

Abstract

We propose conjectural generalizations of the Fermat-Catalan conjecture, the Tijdeman-Zagier conjecture, and of the Fermat Last Theorem, in which powers are replaced by products of integers. We also formulate a new explicit version of the abc conjecture.

Fermat-Catalan and Tijdeman-Zagier conjectures for products

Abstract

We propose conjectural generalizations of the Fermat-Catalan conjecture, the Tijdeman-Zagier conjecture, and of the Fermat Last Theorem, in which powers are replaced by products of integers. We also formulate a new explicit version of the abc conjecture.

Paper Structure

This paper contains 6 sections, 4 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Fermat-Catalan and Tijdeman-Zagier conjectures
  3. The Fermat-Catalan and Pillai's conjectures for products
  4. A version of the Tijdeman-Zagier conjecture for products
  5. Conjectural Version of Fermat's Last Theorem for products
  6. Relation to the abc conjecture.

Key Result

Proposition 3

If $n,m$ are not divisible by $3$ and $gcd(n,m)\leq 2$ then Eq. e.FC has infinitely many non-maxgcd solutions.

Theorems & Definitions (15)

  • Conjecture 1: Fermat-Catalan Conjecture for Products
  • Conjecture 2
  • Proposition 3
  • proof
  • Conjecture 4
  • Conjecture 5
  • Remark 6
  • Remark 7
  • Conjecture 9
  • Conjecture 10
  • ...and 5 more