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 $\Phi_s(3^{s^j}) \equiv 1$ (mod 4). Significant open problems concern the infinitude of these, making Cantor primes perhaps more interesting than previously realised.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2012
- DOI:
- arXiv:
- arXiv:1203.3969
- Bibcode:
- 2012arXiv1203.3969S
- Keywords:
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- Mathematics - Number Theory;
- 11A41;
- 11R09
- E-Print:
- arXiv admin note: text overlap with arXiv:0906.0465