Prime Chains and Pratt Trees

Abstract

Prime chains are sequences p 1 , . . . , p k of primes for which p j+1 ≡ 1 (mod p j ) for each j. We introduce three new methods for counting long prime chains. The first is used to show that N (is the number of chains with p 1 = p and p k px. The second method is used to show that the number of prime chains ending at p is log p for most p. The third method produces the first nontrivial upper bounds on H(p), the length of the longest chain with p k = p, valid for almost all p. As a consequence, we also settle a conjecture of Erdős, Granville, Pomerance and Spiro from 1990. A probabilistic model of H(p), based on the theory of branching random walks, is introduced and analyzed. The model suggests that for most p x, H(p) stays very close to e log log x.

Common values of the arithmetic functions ϕ and σ

“…Key ingredients in the proof are a very recent bound on counts of prime chains from [14] (see §3 for a definition) and estimates for primes in arithmetic progressions. The possible existence of Siegel zeros (see §2 for a definition) creates a major obstacle for the success of our argument.…”

Two remarks on iterates of Euler’s totient function

“…This tree structure first appears in work of Pratt [13] and was extensively studied in recent work of Ford et al [5]. Our second theorem is a lower bound for the number of distinct primes in the Pratt tree.…”

Artin's Primitive Root Conjecture – A Survey

“…, p k be a chain of primes such that for 2 ≤ j ≤ k we have p j = 2p j−1 + 1; that is we have Conditonally one can do far better than this. Assuming that P(2)(x) ∼ Aπ(x), which by Hooley's result is true under GRH, Ford et al [160] have shown that…”