The least modulus for which consecutive polynomial values are distinct

الأصل · EN

Let d≥4 and c∈(-d,d) be relatively prime integers. We show that for any sufficiently large integer n (in particular n>24310 suffices for 4≤ d≤ 36), the smallest prime p≡ c 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 |{k(k-1)/2modm:k=1,,n}|= |{k(k-1)/2modm+2:k=1,,n}|=n is the least prime p≥ 2n-1 with p+2 also prime.