[seqfan] Goldilocks primes

Joshua Searle jprsearle at gmail.com
Sun Dec 24 13:53:45 CET 2023

I was reading this page on goldilocks primes, or more specifically, a slight change to the original definition.
Originally p = x^2 - x - 1, this looked at ones of the form p = x^2 - x + 1.

For reasons explained on the page, it is desirable for x to be of the form 2^k but there are very few examples:
k = {1, 2, 4, 32} and no others <4000 (it appears the author missed k = 2?).

The link in question: (no WorldWideWeb) 2π (dot) com/22/goldilocks/

I have now exhaustively checked for primality up to k = 10200, along with the specific values 2^(14, 15, 16, 17) on the off chance that it must itself be a power of two (or even two to the power of a fermat prime, hence up to 2^17). Alas none passed the probable prime test, with k = 2^17 being nearly 80,000 digits. If it had been prime we could have had a rare example of a meaningful triple exponential function!

Could we get another term?


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