Re: [math-fun] Factorizations of 1 + n + n^2

On 11/18/2012 12:05 PM, Allan Wechsler wrote:

At least for the first few positive n, the factorization of 1 + n + n^2 contains a prime larger than n. The first counterexample is at n=16; here 1 + n + n^2 = 273 = 3*7*13.

I've only explored the question by hand, but so far I've found no counterexamples for the special case when n is prime. Can somebody with a few free minutes find one?

67^2 + 67 + 1 = 3 * 7^2 * 31; other primes below 200 that work are 79, 137, 149, 163, 181 and 191. Oh, this turns out to be A091490. [And to answer your question from this morning, it's fine with me if you do the OEIS honors for the 3.3.4.3.4 lattice.] -- Fred W. Helenius fredh@ix.netcom.com