On functions taking only prime values

JOURNAL OF NUMBER THEORY

卷:133

Article

Sun, Zhi-Wei

关键词: Primes;    Congruences;    Functions taking only prime values;   

DOI  :  10.1016/j.jnt.2013.02.003

来源: Elsevier

【 摘 要 】

For n = 1, 2, 3, ... define S(n) as the smallest integer m > 1 such that those 2k(k - 1) mod m for k = 1, ... , n are pairwise distinct; we show that S(n) is the least prime greater than 2n - 2 and hence the value set of the function S(n) is exactly the set of all prime numbers. For every n = 4, 5, ... , we prove that the least prime p > 3n with p equivalent to 1 (mod 3) is just the least positive integer m such that 18k(3k - 1) (k = 1, ... , n) are pairwise distinct modulo m. For d is an element of {4, 6, 12} and n = 3, 4, ... , we show that the least prime p >= 2n - 1 with p equivalent to -1 (mod d) is the smallest integer m such that those (2k - 1)(d) for k = 1, ... , n are pairwise distinct modulo m. We also pose several challenging conjectures on primes. For example, we find a surprising recurrence for primes, namely, for every n = 10, 11, ... the (n + 1)-th prime p(n+1) is just the least positive integer m such that 2s(k)(2) (k = 1, ..., n) are pairwise distinct modulo m where s(k) = Sigma(k)(j=1) (-1)(k-j) p(j). We also conjecture that for any positive integer m there are consecutive primes p(k), ... , p(n) (k < n) not exceeding 2m + 2.2 root m such that m = p(n) - p(n-1) + ... + (-1)(n-k) p(k). (C) 2013 Elsevier Inc. All rights reserved.

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