Statistical Evidence for Small Generating Sets

Abstract

Abstract.For an integer n , let G(n) denote the smallest x such that the primes < x generate the multiplicative group modulo n . We offer heuristic arguments and numerical data supporting the idea that G(n) < (log2)_1 lognloglog« asymptotically. We believe that the coefficient 1 / log 2 is optimal. Finally, we show the average value of G(n) for n < N is at least(1 + o( 1 )) log log N log log log N, and give a heuristic argument that this is also an upper bound. This work gives additional evidence, independent of the ERH, that primality testing can be done in deterministic polynomial time; if our bound on G(n) is correct, there is a deterministic primality test using O(logn)2 multiplications modulo n.

Artin's Primitive Root Conjecture – A Survey

“…Let G(n) be the smallest integer k such that the primes p ≤ k generate the multiplicative group modulo n. Prove or disprove the conjectures formulated by Bach and Huelsbergen [24], see §9.7.55.…”

“…Narkiewicz [384] sharpened this estimate to O(x 1−A/(log 2)+ǫ ). Ram Murty and Srinivasan [379] considered a sum akin to (24). They showed that if…”

“…Assuming GRH, Montgomery [328] showed that G(n) = O(log 2 n), which was made effective by Bach [21] who showed that G(n) ≤ 3 log 2 n for n ≥ 2. As to unconditional results: Bach and Huelsbergen [24] showed that G(n) = O( √ n(log n) log log n), which was sharpened by Burthe [59] to G(n) ≪ ǫ n α+ǫ , where α = 1/(3 √ e) = 0.20217 . .…”

“…. .. Bach and Huelsbergen [24] conjectured that lim sup n→∞ G(n) log n log log n = 1 log 2 and 1 x n≤x G(n) ∼ (log log x) log log log x.…”

Doubly focused enumeration of locally square polynomial values

“…A proof of an explicit bound such as r/log r ≤ 2 lg x would imply, among other things, that there is a deterministic primality-proving algorithm taking essentially cubic time. See [8], [2], [14, Another way to phrase the same result: Every non-square positive integer below 24 · 2 64 is locally non-square at some prime in {2, 3, . .…”