
Re: six strong pseudoprimes, one degree 4096 polynomial
Posted:
Jun 11, 2013 9:10 AM


David Bernier <david250@videotron.ca> writes: > If I start with the polynomial function: > g(x) = x^4  x^2 +1 then > > P(x) := g(g(g(g(g(g(x)))))) is also a polynomial function of x. > > The degree of P in x is: 4*4*4*4*4*4 = 4^6 = 2^12 = 4096. > > Letting 'x' go from 2 to 14,845 > 6 values of 'x' give that P(x) is a strong pseudoprime, > passing the RabinMiller test with 60 random bases in each > case. > > x= 55, 1948, 3269, 3981, 7341, 14845 > are those 6 values. > > Beyond ~= 3269, I was only testing x with exactly > two prime factors, so there could possibly be > more x with P(x) a strong psudoprime and > x <= 14845. > > P(14845) has about 17000 digits, and > 17000digit numbers are prime once in > 2.5 *17000 or one time in 42500. > > With x in {55, 1948, 3269, 3981, 7341, 14845} > each yielding a probable prime for P(x), the > (with "high probability"), the primality > statistics for the P(x), heuristics, seem to > be out of whack. > > So, it could be chance. But it looks "whacked" to me, > at least so far.
Look at divisivility by small factors.
g(x) = Phi(12) Therefore any divisors will be of form 12k+1. So they won't be divisible by 2, 3, 5, 7, or 11. That boosts expected density. Alas, they're 4 times as likely to be divisible by 13 as arbitrary numbers, so you take a hit there, but nothing compared to the boost you get by excluding all of 2, 3, 5, 7, and 11. (And 17, 19, 23, 29, 31, ...)
The sieve is a great leveller.
Phil  "In a world of magnets and miracles"  Insane Clown Posse, Miracles, 2009. Much derided. "Magnets, how do they work"  Pink Floyd, High Hopes, 1994. Lauded as lyrical geniuses.

