Cantor Primes as Prime-Valued Cyclotomic Polynomials

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 s(3 s j