The least modulus for which consecutive polynomial values are distinct

JOURNAL OF NUMBER THEORY

卷:160

Article

Sun, Zhi-Wei¹ 

[1] Nanjing Univ, Dept Math, Nanjing 210093, Jiangsu, Peoples R China

关键词: Primes in arithmetic progressions;    Congruences;    Functions taking only prime values;   

DOI  :  10.1016/j.jnt.2015.08.001

来源: Elsevier

【 摘 要 】

Let d >= 4 and c is an element of (-d, d) be relatively prime integers. We show that for any sufficiently large integer n (in particular n > 24 310 suffices for 4 <= d <= 36), the smallest prime p equivalent to c (mod d) with p >= (2dn - c)/(d - 1) is the least positive integer m with 2r(d)k(dk - c) (k = 1,..., n) pairwise distinct modulo m, where r(d) is the radical of d. We also conjecture that for any integer n > 4 the least positive integer m such that vertical bar{k(k - 1)/2 mod m : k = 1,..., n}vertical bar = 1{k(k 1)/2 mod as + 2 : k = 1, = n is the least prime p >= 2n - 1 with p + 2 also prime. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free