On the divisibility of Fermat quotients Yurii Shteynikov. 3rd Workshop on Analysis, Geometry and Probability. September 28 - October 2, 2015 Ulm University, Germany 1 Denitions: p - suciently large prime number Fermat's Little Theorem: For a prime p and for a with gcd(a, p) = 1 ap−1 ≡ 1 (mod p). Denitions For prime p and for a with gcd(a, p) = 1 Fermat quotient qa(p) is dened as ap−1 − 1 qa(p) = ; p lp will denote the smallest a ∈ N, a > 1 that p does not divide qa(p). Divisibility of Fermat quotients has some number-theoretic applications. 1 We consider the task of estimating lp for 1) all primes p; 2) all p with the exception of primes from a set with relative zero density. Another formulation of this task. Let Gp multiplicative subgroup of (Z/p2Z)∗ of order p − 1: Gp = {ap : 1 ≤ a ≤ p − 1} (mod p2). We want to estimate the rst positive integer a: a ∈ N, a > 1 that a does not belong to Gp for: 1) all primes p; 2) all p, with the exception of primes from a set with relative zero density. 2 Previous rezultsEstimates for all p. Theorem (H.W. Lenstra, 1979): p > 3 ⇒ lp ≤ 4(log p)2; p → ∞ ⇒ lp ≤ (4e−2 + o(1))(log p)2. Theorem (J. Bourgain, K. Ford, S. Konyagin, I. Shparlinski, 2010): We have the estimate lp ≤ (log p) 463 252 +o(1) Argument of the proof. For a multiplicative subgroup G ⊂ Z∗n of order t and for positive integer k we dene J(n, G, k) = |({1, ..., k} \ G)|. We need to have good upper T estimates on J(n, G, k) = |({1, ..., k} G)|. The following estimate takes place. 3 Theorem (J.Bourgain, S. Konyagin, I. Shparlinski, 2009) For any ε > 0 the following holds kt Cεk X J(n, G, k) ≤ + Mn(ω, Z, G)|S(w, G)|, n tn ω∈Zn where Z := min(n1+εk −1, n/2), Mn(ω, Z, G) is the number of solutions of the congruence ω ≡ zu (mod n), 1 ≤ |z| ≤ Z, u ∈ G and S(ω, G) := X e2πigω/n g∈G We observe that for integer k X |S(ω, G)|2k = nTk (G), ω∈Zn where Tk (G) := {(x1, . . . , x2k ) ∈ G(2k) : x1+. . .+xk ≡ xk+1 + . . . + x2k (mod n)}. Task: nd upper bounds for Tk (G). 4 The task of divisibility of Fermat quotient corresponds to the case G := Gp. Let t := |G| Theorem(S. Konyagin,D.R. HeathBrown, 2000) Let G ⊆ Z∗p , t < p1/2. We have the bound Tk (G) ≤ C(k)t2k−2+1/2 k−1 , p → ∞. Theorem(I. Shkredov, 2014) We have the bound T2(Gp) ≤ p 32 13 +o(1) , p → ∞. Corrolary(I. Shkredov, 2014) We have the bound lp ≤ (log p) 7829 4284 +o(1) , p → ∞. Theorem(Y.S, 2015) Let G ⊆ Z∗p , t = |G|, t < p1/2. We have the bound 3 +o(1) 4 14 T3(G) ≤ t 5 , p → ∞. II Estimates for lp for almost all primes p. Theorem (J. Bourgain, K. Ford, S. Konyagin, I. Shparlinski, 2009): For every ε > 0 there is δ > 0 such that for all bur O(Q1−δ ) primes p < Q we have lp ≤ (log p) 5 +ε 3 . Theorem (Y.S, 2012): For every ε > 0 there is δ > 0 such that for all bur O(Q1−δ ) primes p < Q we have 3 lp ≤ (log p) 2 +ε. Arguments in the proof: For a multiplicative subgroups G1 ⊂ Z∗p2 and G2 ⊂ Z∗p2 and 1 for positive integer N we dene J(G1, G2, N ) = |({1, ..., N } \ G1 2 \ G2)|. It is not diicult to show that estimate for lp for almost al p can be obtained with estimates on average for J(G1, G2, N ) over dierent pairs G1, G2. 6 How to estimate J(G1, G2, N ) = |({1, ..., N } \ G1 \ G2)|? Let primes p1, p2 be the "same order". Theorem(Y.S., 2015) We have the following estimate J(G1, G2, N ) (N1N2)1/2 where quantities N1, N2 are dened to be the number of solutions to the equations x1−x2 ≡ kp22 (mod p21), x1, x2 ∈ G1, 0 ≤ k ≤ N p21 (mod p22), x1, x2 ∈ G2, 0 ≤ k ≤ N p22 and x1−x2 ≡ kp21 respectively. A little bit other statement was proved in the work of Bourgain, Ford, Konyagin, Shparlinski. 7 Futher task for research 1) Consider the task of divisibility of Fermat quotients on the powers of p: p2, p3... 2) Find non-trivial estimate on the Tk (G), when G is a subgroup of Zp∗r , r ≥ 3. 3) Find estimates on J(G1, G2, G3N ) where J(G1, G2, G3, N ) = |{1, ..., N } 8 \ G1 \ G2 \ G3|