Joint IAS/Princeton University Number Theory Seminar

A sequence of primes p_1, ..., p_k is called a prime chain if p_j | (p_{j+1}-1) for each j ; e.g. 3, 7, 29, 59. We will discuss problems about counting prime chains with certain properties, and about the existence of prime chains with various properties. The Pratt tree for a prime p is the tree with root node p and below p are the Pratt trees of the odd prime factors of p-1. Example: 79 / \ 3 13 \ 3 We are concerned with the normal and extremal behavior of the depth of such trees.