Toda primes

TL;DR

This paper introduces Toda primes, a new class of primes related to integers, conjectures their universal existence in pairs, and provides partial proofs along with connections to Bernoulli numbers and Sophie Germain primes.

Contribution

It defines Toda primes, formulates a conjecture about their distribution, and offers partial proof of their existence for all positive integers.

Findings

Partial proof that every positive integer has at least one Toda prime

Discussion of connections to Bernoulli number denominators

Relation to generalized Sophie Germain primes

Abstract

A Toda prime of an integer $n$ is an odd prime $p$ such that $4n=(p-1)k$ with $k$ coprime to $p$. We conjecture that every positive integer admits at least two Toda primes. We give a partial proof that every positive integer admits at least one Toda prime. We conclude by discussing connections to denominators of Bernoulli numbers and a generalization of Sophie Germain primes.