You said: 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. Me: but would it make a good sequence? What are the first few such primes? If it is not in the OEIS please submit it! Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Wed, Dec 16, 2015 at 9:14 PM, Dan Asimov <asimov@msri.org> wrote:
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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun