For integers a and b we define 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 arbitrarily 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 must have bounded length. For this, we study certain properties of functional digraphs of quadratic functions over prime fields, both in theory and practice. We give efficient algorithms to investigate these properties and present a selection of our experimental results.
['Edlyn Teske', 'Hugh C. Williams']
A Note on Shanks's Chains of Primes