Hi

I have run:

http://groups.yahoo.com/group/primeform/files/Probable%

20Primes/cubic_mr.c

on a 1GHz Athlon for about ten days; for x = 2 upto 10,000,000,001

and found 527,345,506 probable primes and no "counterexample".

The program tests "f|x^f-x=>5 times Miller-Rabin probable prime where

f=x^3-x-1 and x>1". The probability of a composite f:f|x^f-x getting

through 5 Miller Rabin rounds is less than 4^-5.

Some statistics:-

341 is the smallest base 2 Fermat pseudoprime.

561 is the smallest Carmichael number.

705 is the smallest pseudoprime n for the test "the sum of the nth

powers of the roots of x^2-x-1 is congruent to 1 modulo n".

124 is the smallest x for which the test "f|x^f-x where f=x^2-x-1"

produces a pseudoprime f.

19802 is the smallest x for which the test "f|x^f-x where f=x^2-x-1"

produces a Carmichael number f.

271441 is the smallest Perrin pseudoprime. (The nth term in the

Perrin Sequence is equal to the sum of the nth powers of the roots of

x^3-x-1.)

27664033 is the smallest symmetric pseudoprime relative to the

elementary symmetric functions of the roots of x^3-x-1.

7045248121 is the smallest Perrin pseudoprime that is also a

Carmichael number.

What is the smallest composite f:f|x^f-x where f=x^3-x-1? David

Broadhurst reckons Erdos' Carmichael conjecture implies x ~ 10^13 for

a Carmichael number ;-)

Paul