On Cunningham Chains

Prime clusters and Cunningham chains

We discuss the methods and results of a search for certain types of prime clusters. In particular, we report specific examples of prime 16-tuplets and Cunningham chains of length 14.

The largest known Cunningham chain of length 3 of the first kind

Cunningham chains of length n of the first kind are n long sequences of prime numbers p1, p2, . . . , pn so that pi+1 = 2pi + 1 (for 1 ≤ i < n). In [3] we have devised a plan to find large Cunningham chains of the first kind of length 3 where the primes are of the form pi+1 = (h0 + cx) · 2 − 1 for some integer x with h0 = 5 775, c = 30 030 and e = 34 944. The project was executed on the non-uni...

Sieving for Large Cunningham Chains of Length 3 of the First Kind

In this paper we study the details of sieving for Cunningham chains of the first kind of length 3. To find such prime triplets larger than the ones already known, we have to investigate the primality of 2 numbers, each in the magnitude of 2 (more than 10 500 decimal digits). This would not be feasible if it weren’t for the sieving process which reduces the estimated time of completion to only a...

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 ...

A Note on Shanks's Chains of Primes

For integers a and b we deene the Shanks chain p 1 ; p 2 ; : : : ; p k of length k to be a sequence of k primes such that p i+1 = ap i 2 ? b for i = 1; 2; : : : ; k ? 1. While for Cunningham chains it is conjectured that innnitely long chains exist, this is, in general, not true for Shanks chains. In fact, with s = ab we show that for all but 56 values of s 1000 any corresponding Shanks chain m...

Heavy Chain US11 Interacts with Conventional Kinesin Herpes Simplex Virus Tegument Protein