The least modulus for which consecutive polynomial values are distinct

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• 作者：Zhi-Wei Sun¹ ; ^{zwsun@nju.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 11A41 ; 11B25 ; secondary ; 11A07 ; 11B75 ; 11N13 ; 11Y11
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：160
• 期：Complete
• 页码：108-116
• 全文大小：257 K

文摘

Let d⩾4$d⩾4$ and c∈(−d,d)$c\in (-d,d)$ be relatively prime integers. We show that for any sufficiently large integer n   (in particular $n>24310$ suffices for 4⩽d⩽36$4⩽d⩽36$), the smallest prime 43aee4">$p\equiv c(\mathrm{mod}d)$ with p⩾(2dn−c)/(d−1)$p⩾(2dn-c)/(d-1)$ is the least positive integer m   with 2r(d)k(dk−c)$2r(d)k(dk-c)$ (k=1,…,n$k=1,\dots ,n$) pairwise distinct modulo m  , where r(d)$r(d)$ is the radical of d  . We also conjecture that for any integer n>4$n>4$ the least positive integer m   such that $|\{k(k-1)/2\mathrm{mod}m:k=1,\dots ,n\}|=|\{k(k-1)/2\mathrm{mod}m+2:k=1,\dots ,n\}|=n$ is the least prime p⩾2n−1$p⩾2n-1$ with p+2$p+2$ also prime.