On prime chains

TL;DR

This paper investigates properties of prime chains defined by a specific recursive relation involving primes with primitive root 2, providing conditions under which certain divisibility or equality properties hold.

Contribution

It establishes new conditions for prime chains with recursive relations, linking prime properties with primitive roots and divisibility criteria.

Findings

If q divides p(0)+b, then certain divisibility conditions hold.

Either p_0 or p_1 equals q under specified conditions.

The recursive prime sequence is constrained by divisibility and primitive root properties.

Abstract

Let b be an odd integer such that b=+/-1 (mod 8) and let q be a prime with primitive root 2 such that q does not divide b. We show that if (p(k)) is a sequence of odd primes, with 0<=k<=q-2 such that p(k)=2p(k-1)+b for all 1<=k<=q-2, then either (a) q divides p(0)+b, (b) p_0=q or (c) p_1=q.