I found that 1248829 | fsq_n for n > 1248827. Searching the OEIS I found that this was already known in 2004, see https://oeis.org/A100289. I agree with your heuristic -- there should be infinitely many fixed prime divisors, and so any appropriately natural class of primes with positive relative density should have infinitely many instances. Charles Greathouse Analyst/Programmer Case Western Reserve University On Fri, Apr 10, 2015 at 10:14 AM, Warren D Smith <warren.wds@gmail.com> wrote:
To make it even worse: I hereby proclaim that I am "heuristically certain" that fsq_n is, for all except a finite number of n, divisible both by an upper, and by a lower, TWIN prime. (If p and p+2 both are prime, than p is the lower and p+2 is the upper twin.) The same sort of proof must exist, but now finding that proof is likely to be so hard that nobody will ever do it!
--CORRECTION: sorry, my brain switched to wrong. I was basing the above on Brun's theorem that the sum of 1/t, where t are twin primes, diverges. But actually Brun says the opposite: this sum is finite.
OK, so to fix this: I am heuristically certain that for all but a finite set of n, fsq_n is divisible by a prime P<n such that P is of the form P=1000000000*k+1. [Pick your favorite number of zeros, I'm still certain.] And a proof of this should be available findable by the technique I mentioned... but that proof will be extremely hard to find.
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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