The least modulus for which consecutive polynomial values are distinct

Let d⩾4d⩾4 and c∈(−d,d)c∈(−d,d) be relatively prime integers. We show that for any sufficiently large integer n   (in particular n>24310 suffices for 4⩽d⩽364⩽d⩽36), the smallest prime p≡c(modd) 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,…,nk=1,…,n) pairwise distinct modulo m  , where r(d)r(d) is the radical of d  . We also conjecture that for any integer n>4n>4 the least positive integer m   such that |{k(k−1)/2modm:k=1,…,n}|=|{k(k−1)/2modm+2:k=1,…,n}|=n is the least prime p⩾2n−1p⩾2n−1 with p+2p+2 also prime.