On Prime Chains

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 (pk) q−2 k=0 is a sequence of odd primes such that pk = 2pk−1 + b for all 1 ≤ k ≤ q − 2, then either (a) q divides p0 + b, (b) p0 = q or (c) p1 = q. For integers a, b with a ≥ 1, a sequence of primes (pk) λ−1 k=0 such that pk = apk−1+b for all 1 ≤ k ≤ λ − 1 is called a prime chain of length λ based on the pair (a, b). This follows the terminology of Lehmer [7]. The value of pk is given by (1) pk = a p0 + b (a − 1) (a− 1) for all 0 ≤ k ≤ λ− 1. For prime chains based on the pair (2,+1), Cunningham [2, p. 241] listed three prime chains of length 6 and identified some congruences satisfied by the primes within prime chains of length at least 4. Prime chains based on the pair (2,+1) are now called Cunningham chains of the first kind, which we will call C+1 chains, for short. Prime chains based on the pair (2,−1) are called Cunningham chains of the second kind, which we will call C−1 chains. We begin with the following theorem which has ramifications on the maximum length of a prime chain; it is a simple corollary of Fermat’s Little Theorem. A proof is also given by Löh [9]. Moser [10] once posed Theorem 1, with a, b, p0 ≥ 1, as an exercise, for which he received fourteen supposedly correct proofs. Theorem 1. Let (pk)k≥0 be an infinite sequence for which pk = apk−1 + b for all k ≥ 1. Then the set {pk}k≥0 is either finite or contains a composite number. There are some choices of (p0, a, b) that are uninteresting. For example, if b = −(a− 1)p0, then the prime chain is (p0, p0, . . . ). In fact, if pi = pj for any distinct i, j then {pk} ∞ k=0 will be periodic, with period dividing |pi−pj|. Also if gcd(a, b) > 1 then the sequence could only possibly be of length 1, since gcd(a, b) divides ap0+ b. In this article we will therefore assume that (pk) ∞ k=0 is a strictly increasing sequence. Theorem 1 implies that no choice of (p0, a, b) will give rise to a prime chain of infinite length. However, this raises the question, how long can a prime chain be? Green and Tao [5] proved that, for all λ ≥ 1, there exists a prime chain of length λ based on the pair (1, b) for some b. Lehmer [7] remarked that Dickson’s Conjecture [3], should it be true, would imply that there are infinitely many prime chains of length λ based on the pair (a, b), with the exception of some inappropriate pairs (a, b). 2000 Mathematics Subject Classification. Primary 11N13.