Toda primes

TL;DR

This work defines Toda primes for a given integer $n$ as odd primes $p$ with $p-1\mid 4n$ and $\gcd(p,\frac{4n}{p-1})=1$, and studies the sets $T(n)$ and $t(n)=|T(n)|$. It conjectures that every positive integer $n$ has at least two Toda primes, while proving a partial result that every $n$ admits at least one, and develops an inductive framework on the number of distinct prime factors $\omega(n)$ to approach the conjecture. The authors provide heuristic arguments and computational evidence suggesting $t(n)\ge4$ under the conjecture for many $n$, and connect the Toda prime structure to deeper number-theoretic objects, notably Bernoulli denominators $D_{2m}$ and germane primes (primes of the form $p(q-1)+1$). They propose general Bernoulli-denominator–Toda-prime conjectures and illustrate a rich two-wavefront pattern in germane primes, with implications for understanding divisibility properties and potential simplifications in stable homotopy calculations via Toda-prime nonexistence results. Overall, the paper blends elementary divisibility lemmas, inductive arguments, and data-driven heuristics to illuminate the arithmetic landscape surrounding Toda primes and their connections to classical number theory.

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.

Figures (3)

• Figure 1: $t(n)$ for $n\leq 100000$
• Figure 2: Ratio of primes germane to $p$
• Figure 3: Width versus length of germane primes

Theorems & Definitions (31)

• Definition 1.1
• Conjecture 1.3
• Conjecture 1.4
• Theorem 1.5
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• proof