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 2314X15002693&_mathId=si1.gif&_user=111111111&_pii=S0022314X15002693&_rdoc=1&_issn=0022314X&md5=fc889f17be9a246708bb291666c89a81" title="Click to view the MathML source">d⩾4$d⩾4$ and 2314X15002693&_mathId=si2.gif&_user=111111111&_pii=S0022314X15002693&_rdoc=1&_issn=0022314X&md5=fdeafde7a2634ccc954030e4db3993a4" title="Click to view the MathML source">c∈(−d,d)$c\in (-d,d)$ be relatively prime integers. We show that for any sufficiently large integer n   (in particular 2314X15002693&_mathId=si3.gif&_user=111111111&_pii=S0022314X15002693&_rdoc=1&_issn=0022314X&md5=eb1fb30221a2423ade4f9379ecf8d41e">2314X15002693-si3.gif">$n>24310$ suffices for 2314X15002693&_mathId=si37.gif&_user=111111111&_pii=S0022314X15002693&_rdoc=1&_issn=0022314X&md5=ddce0c37cb0ebfeb1f9fcc7e98bcf901" title="Click to view the MathML source">4⩽d⩽36$4⩽d⩽36$), the smallest prime 2314X15002693&_mathId=si174.gif&_user=111111111&_pii=S0022314X15002693&_rdoc=1&_issn=0022314X&md5=361894d08ffbd6a141e212677443aee4">2314X15002693-si174.gif">$p\equiv c(\mathrm{mod}d)$ with 2314X15002693&_mathId=si36.gif&_user=111111111&_pii=S0022314X15002693&_rdoc=1&_issn=0022314X&md5=c21d86dd71a21c9980ec72dd47fdb639" title="Click to view the MathML source">p⩾(2dn−c)/(d−1)$p⩾(2dn-c)/(d-1)$ is the least positive integer m   with 2314X15002693&_mathId=si7.gif&_user=111111111&_pii=S0022314X15002693&_rdoc=1&_issn=0022314X&md5=74adceecbf33db58b0c2ad1cfee3c26e" title="Click to view the MathML source">2r(d)k(dk−c)$2r(d)k(dk-c)$ (2314X15002693&_mathId=si34.gif&_user=111111111&_pii=S0022314X15002693&_rdoc=1&_issn=0022314X&md5=b1dca492f2d755a61a19fd353cf3f075" title="Click to view the MathML source">k=1,…,n$k=1,\dots ,n$) pairwise distinct modulo m  , where 2314X15002693&_mathId=si30.gif&_user=111111111&_pii=S0022314X15002693&_rdoc=1&_issn=0022314X&md5=b5255bb592876e79b5571f91832b1d0b" title="Click to view the MathML source">r(d)$r(d)$ is the radical of d  . We also conjecture that for any integer 2314X15002693&_mathId=si10.gif&_user=111111111&_pii=S0022314X15002693&_rdoc=1&_issn=0022314X&md5=5387df988d675b325253c24a36f33a65" title="Click to view the MathML source">n>4$n>4$ the least positive integer m   such that 2314X15002693&_mathId=si11.gif&_user=111111111&_pii=S0022314X15002693&_rdoc=1&_issn=0022314X&md5=7a51545f9b2ad594a974fabf76c020ae">2314X15002693-si11.gif">$|\{k(k-1)/2\mathrm{mod}m:k=1,\dots ,n\}|=|\{k(k-1)/2\mathrm{mod}m+2:k=1,\dots ,n\}|=n$ is the least prime 2314X15002693&_mathId=si12.gif&_user=111111111&_pii=S0022314X15002693&_rdoc=1&_issn=0022314X&md5=f100401ff3b6c1d0c77361f069883a61" title="Click to view the MathML source">p⩾2n−1$p⩾2n-1$ with 2314X15002693&_mathId=si13.gif&_user=111111111&_pii=S0022314X15002693&_rdoc=1&_issn=0022314X&md5=aa804f0b32a15837430eaa30f8371b20" title="Click to view the MathML source">p+2$p+2$ also prime.