Let 1666c89a81" title="Click to view the MathML source">d⩾4 and c∈(−d,d) be relatively prime integers. We show that for any sufficiently large integer n (in particular suffices for 4⩽d⩽36), the smallest prime 16" width="101" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X15002693-si174.gif"> 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 is the least prime p⩾2n−1 with p+2 also prime.