Cantor Primes as Prime-Valued Cyclotomic Polynomials

TL;DR

This paper explores Cantor primes, primes related to the middle-third Cantor set, and establishes their connection to prime-valued cyclotomic polynomials, revealing new insights into their structure and open problems about their infinitude.

Contribution

The paper proves that all Cantor primes greater than 3 are exactly the prime-valued cyclotomic polynomials of a specific form, linking number theory and fractal sets.

Findings

Cantor primes > 3 satisfy a specific exponential equation.

They are characterized as prime-valued cyclotomic polynomials of a certain form.

Open problems remain about the infinitude of these primes.

Abstract

Cantor primes are primes p such that 1/p belongs to the middle-third Cantor set. One way to look at them is as containing the base-3 analogues of the famous Mersenne primes, which encompass all base-2 repunit primes, i.e., primes consisting of a contiguous sequence of 1's in base 2 and satisfying an equation of the form p + 1 = 2^q. The Cantor primes encompass all base-3 repunit primes satisfying an equation of the form 2p + 1 = 3^q, and I show that in general all Cantor primes > 3 satisfy a closely related equation of the form 2pK + 1 = 3^q, with the base-3 repunits being the special case K = 1. I use this to prove that the Cantor primes > 3 are exactly the prime-valued cyclotomic polynomials of the form $\Phi_s(3^{s^j}) \equiv 1$ (mod 4). Significant open problems concern the infinitude of these, making Cantor primes perhaps more interesting than previously realised.