--- In

primenumbers@yahoogroups.com,

Bill Bouris <leavemsg1@...> wrote:

> if 'N' passes the 2-PRP test, then...

> if 3, 5, or 7 divides 'N', then 'N' is composite...

> (or) if either ALL or NONE of them divide 'N-1',

> then a simple 3-PRP will conclude that 'N' is composite,

> without a doubt.

In less than 20 seconds, I found 993305 counterexamples

in tables from William Galway and Richard Pinch.

There are 319226 numbers N < 10^15 that are 2-PSP and 3-PSP.

Of these, 287672 are coprime to 3*5*7.

Of these, 119450 have N-1 divisible by 3*5*7

and another 5430 have N-1 coprime to 3*5*7

giving us 124880 counterexamples below 10^15.

There are 1296432 Carmichael numbers C between 10^15 and 10^18.

Of these, 1131205 are coprime to 3*5*7.

Of these, 868340 have C-1 divisible by 3*5*7

and another 85 have C-1 coprime to 3*5*7

giving us 868425 additional counterexamples.

Tally and timing from Pari-GP:

124880 + 868425 = 993305 counterexamples in 19320 milliseconds

David