16 Dec
2015
16 Dec
'15
7:15 p.m.
According to something I've read, Legendre asserted that: ----- If p_n denotes the nth prime, then for any integer K at least one of the odd integers 2K + 1, 2K + 3, . . ., 2K + 2p_(n-1) + 1 is not divisible by any of the primes p_1, p_2, . . ., p_n. ----- It turns out that this is not true. The situation is now completely understood: The set of exactly those p_n for which the statement holds is known. The answer is easy to state but does not seem to be easy to prove, so this is not necessarily good puzzle material. But just in case anyone (who doesn't already know the answer) would like to conjecture the makeup of the set of p_n for which this statement holds, feel free — and I'll post the answer soon. —Dan