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On a theorem of Kanold on odd perfect numbers

Tomohiro Yamada

Abstract

We shall prove that if $N=p^αq_1^{2β_1} q_2^{2β_2} \cdots q_{r-1}^{2β_{r-1}}$ is an odd perfect number such that $p, q_1, \ldots, q_{r-1}$ are distinct primes, $p\equivα\equiv 1\mod{4}$ and $t$ divides $2β_i+1$ for all $i=1, 2, \ldots, r-1$, then $t^5$ divides $N$, improving an eighty-year old result of Kanold.

On a theorem of Kanold on odd perfect numbers

Abstract

We shall prove that if is an odd perfect number such that are distinct primes, and divides for all , then divides , improving an eighty-year old result of Kanold.
Paper Structure (3 sections, 15 theorems, 32 equations)

This paper contains 3 sections, 15 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the theorem

Key Result

Theorem 1

If $N$ is an odd perfect number in the form eq11 and an integer $t$ divides $2\beta_i+1$ for $i=1, 2, \ldots, r-1$, then $t^5$ divides $N$.

Theorems & Definitions (25)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Corollary 1
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 15 more