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Topic: six strong pseudoprimes, one degree 4096 polynomial
Replies: 10   Last Post: Jun 21, 2013 12:46 PM

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Phil Carmody

Posts: 2,205
Registered: 12/7/04
Re: six strong pseudoprimes, one degree 4096 polynomial
Posted: Jun 11, 2013 9:10 AM
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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 pseudo-prime,
> passing the Rabin-Miller 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 psudo-prime and
> x <= 14845.
>
> P(14845) has about 17000 digits, and
> 17000-digit 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.



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