Re: a conjecture based on Euler pol. and the Little Fermat Theorem

simon plouffe wrote :

I had this idea one night a while ago,

if k = n*n - n + 41 and if 2^(k-1) = 1 mod k then
k is prime.

It seems to work all the time, I verified for n up to
500,000,000.

In other words : n*n - n + 41 is never a 2-psp, a
pseudo-prime
in base 2 (that is the conjecture).

what if k equals a polynomial that does not have coefficients larger than 1 ?

***
tomic polynomial:

all coefficients are E [-1,0,1]

none of its zero's is a root of unity / [-1,1]

zero is not a root of the polynomial
***

are there tomic polynomials that satisfy :

if k = tomic(n) and if 2^(k-1) = 1 mod k then
k is prime.

??

regards

tommy1729


There is another one too : if k = n*n + 3 and if
2^(k-1) = 1 mod k
then k is prime.

I tried to construct an argument about the
factorization of
n*n - n + 41 and 2-psp's but failed to see any
convincing details.

I have other values that seems to work also like
k = n*n + 163 (only 1 fail), another counter-example
is

103*n*n - 3945*n + 34831 when n = 2400371 it does
satisfy the
little fermat theorem but the number is composite :
5315987*111635707.

and that polynomial (103*n*n..) is known to produce a
lot of primes.

Based on this I launched a series of computations but
(as you may
know),
it all comes back to the fact that the little fermat
theorem produces
primes very often and if we combine it with a simple
g.f. like the
Euler polynomial, chances are that it will produce
many primes.
Even by knowing this, the best probable prime I could
find is about
35000 digits.

1) I could not find any reference about this.
2) So far n*n + 3 and the Euler pol. always produces
primes and
avoid the 2-psp's elegantly.
3) Maybe there is a link between class numbers and
the little fermat
theorem but I can't see it.

My question is do you see any connections ?

Simon Plouffe

.

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