Correctly enumerate primes between 4295098369 and 3825123056546413050.

Prior to this commit, primes(6) relied solely on sieving with primes up
to 65537, with the effect that composite numbers which are the product
of two non-16-bit primes would be incorrectly identified as prime.  For
example,
# primes 1099511627800 1099511627820
would output
1099511627803
1099511627807
1099511627813
when in fact only the first of those values is prime.

This commit adds strong pseudoprime tests to validate the candidates
which pass the initial sieving stage, using bases of 2, 3, 5, 7, 11,
13, 17, 19, and 23.  Thanks to papers from C. Pomerance, J.L. Selfridge,
and S.S. Wagstaff, Jr.; G. Jaeschke; and Y. Jiang and Y. Deng, we know
that the smallest value which passes these tests is 3825123056546413051.

At present we do not know how many strong pseudoprime tests are required
to prove primality for values larger than 3825123056546413050, so we
force primes(6) to stop at that point.

Reviewed by:	jmg
Relnotes:	primes(6) now correctly enumerates primes up to
		3825123056546413050
MFC after:	7 days
Sponsored by:	EuroBSDCon devsummit

1 files changed, 7 insertions, 1 deletions

diff --git a/games/factor/factor.6 b/games/factor/factor.6
index a4d35e9..508a98b 100644
--- a/games/factor/factor.6
+++ b/games/factor/factor.6
@@ -90,7 +90,7 @@ value must not be greater than the maximum.
 The default and maximum value of
 .Ar stop
 is 4294967295 on 32-bit architectures
-and 18446744073709551615 on 64-bit ones.
+and 3825123056546413050 on 64-bit ones.
 .Pp
 When the
 .Nm primes
@@ -120,3 +120,9 @@ cannot handle the
 factor list,
 .Nm primes
 will not get you a world record.
+.Pp
+.Nm primes
+is unable to list primes between 3825123056546413050 and 18446744073709551615
+since it relies on strong pseudoprime tests after sieving, and nobody has
+proven how many strong pseudoprime tests are required to prove primality for
+integers larger than 3825123056546413050.