Some heuristics and results for small cycles of the discrete logarithm

Abstract

Abstract. Brizolis asked the question: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? The first author previously extended this question to ask about not only fixed points but also two-cycles, and gave heuristics (building on work of Zhang, Cobeli, Zaharescu, Campbell, and Pomerance) for estimating the number of such pairs given certain conditions on g and h. In this paper we extend these heuristics and prove results for some of them, building again on the aforementioned work. We also make some new conjectures and prove some average versions of the results.

Mapping the discrete logarithm

“…In [Holden 2002;Holden and Moree 2004;2006], heuristics and observed values for the number of small cycles (fixed points and two-cycles) in discrete exponentiation graphs are given. Our methods build on this to generate experimental data for the parameters described by the theoretical predictions in Section 3.…”

COUNTING FIXED POINTS, TWO-CYCLES, AND COLLISIONS OF THE DISCRETE EXPONENTIAL FUNCTION USING p-ADIC METHODS

“…considered (with e = 1) in [15] as closely related to (3). Another problem that should be tractable using our methods is finding solutions of (11) g x−1+c ≡ x mod p e for c fixed.…”

Artin's Primitive Root Conjecture – A Survey

“…A related conjecture by Holden and Moree [236] states that one should have F (p) = (1 + o (1))p. Bourgain et al [44] showed that F (p) = p + O(p 4/5+ǫ ) for a set of primes p of relative density 1 and in a later paper, [45],…”

“…Let F (p) denote the number of pairs (g, h) satisfying g h ≡ h(mod p) with 1 ≤ g, h ≤ p − 1. Let G(p) denote the number of pairs (g, h) satisfying g h ≡ h(mod p) with 1 ≤ g, h ≤ p − 1, where in addition we require g to be a primitive root modulo p. Holden and Moree [236,Conjecture 8.3] …”