Cantor Primes as Prime-Valued Cyclotomic Polynomials

Details Cantor Primes as Prime-Valued Cyclotomic Polynomials Cantor Primes as Prime-Valued Cyclotomic Polynomials

2012-03-18

arxiv.org/abs/1203.3969v1

Mathematics

Number Theory

arXiv admin note: text overlap with arXiv:0906.0465

Scientific paper

Cantor primes are primes p such that 1/p belongs to the middle-third Cantor set. One way to look at them is as 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.

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