Autour des plus grands facteurs premiers d’entiers consÉcutifs voisins d’un entier criblé

Abstract

Denote by P+(n) (resp. P−(n)) the largest (resp. the smallest) prime factor of the integer n. In this paper, we prove that there exists a positive proportion of integers n having no small prime factor such that P+(n)<P+(n+2). Especially, we prove that the pattern P+(P3)<P+(P3+2) is realized by a positive proportion of P3 with P−(P3)>x1/3−δ,0<δ≤112, where P3 denote an integer having at most three prime factors taken with multiplicity. We also prove that the pattern P+(p−1)<P+(p+1) holds for a positive proportion of primes under the Elliott-Halberstam conjecture.

ECM and the Elliott–Halberstam conjecture for quadratic fields

“…The second term of the right member is treated by Wang [Wan18]. His proof applies directly in the case of cyclotomic fields K and can be adapted in the other cases:…”

“…The most celebrated example is the precise version of the twin primes conjecture due to Hardy and Littlewood. Two other problems concern P1: the smoothness of A a = {p+a : p prime}, for a constant a; see [Wan18] and [LWX20] for the newest results, both of them are conditional on the Elliott-Halberstam conjecture; P2: the primality of A E = {|E(F p )| : p prime , p ∆(E)} where E is an elliptic curve with rational coefficients and E(F p ) is the reduction of E modulo p; see [Zyw11] for a review of the literature. In this work we address a series of questions related to P3: the smoothness of the set A E above.…”

“…We follow the strategy of Wang [Wan18], so we assume Conjecture 1. Since classical EH conjecture is a strengthening of the statement of the Vinogradov-Bombieri (BV) theorem, the number field version of EH is the same strengthening of the number field BV, which due to Huxley.…”

On a conjecture on shifted primes with large prime factors

“…In [8], I also pointed out that Chen and Chen's conjecture is already in contradiction with the Elliott-Halberstam conjecture according to the works of Pomerance [22], Granville [13], Wang [23] and Wu [24]. In fact, one has…”