- Here's how I want to define the magnum 357 test!

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

if 3, 5, or 7 divides 'N', then 'N' is composite... and we are done.

(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.

(or) if either one or two of them divide 'N-1', then a test can be

devised to determine whether 'N' is prime or composite, no doubts.

say only 3 doesn't, then either 3^((N-1)/5) mod N or 3^((N-1)/7) mod

N will prove that N is prime or composite when followed by a second

test of either (result)^5 mod N or (result)^7 mod N, respectively.

I believe that the {(result)^(power) mod N} <> abs((power)-3) will

confirm that 'N' is composite, otherwise 'N' will be prime.

There's probably not enough theory to deny or support this conclusion.

Either I'm right or wrong... I can accept it... non-spammingly! I'm also

watching for the Troll-watcher(DBr)... lol

Bill - Try N = 25326001.

Try N = 3215031751.

Then fix the test to handle those correctly. :)

Bill Bouris wrote:> Here's how I want to define the magnum 357 test!

>

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

>

> if 3, 5, or 7 divides 'N', then 'N' is composite... and we are done.

>

> (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.

>

> (or) if either one or two of them divide 'N-1', then a test can be

> devised to determine whether 'N' is prime or composite, no doubts.

>

> say only 3 doesn't, then either 3^((N-1)/5) mod N or 3^((N-1)/7) mod

> N will prove that N is prime or composite when followed by a second

> test of either (result)^5 mod N or (result)^7 mod N, respectively.

>

> I believe that the {(result)^(power) mod N} <> abs((power)-3) will

> confirm that 'N' is composite, otherwise 'N' will be prime.

>

> There's probably not enough theory to deny or support this conclusion.

>

> Either I'm right or wrong... I can accept it... non-spammingly! I'm also

> watching for the Troll-watcher(DBr)... lol

>

> Bill

>

>

>

>

>

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>

> - --- In primenumbers@yahoogroups.com,

Bill Bouris <leavemsg1@...> wrote:

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

It takes less than 2 seconds to prove that there are

> 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.

at least 124880 counterexamples with N < 10^15. See:

http://physics.open.ac.uk/~dbroadhu/cert/ct124880.zip

It is likely that there are *precisely*

124880 counterexamples with N < 10^15

David - --- In primenumbers@yahoogroups.com,

Bill Bouris <leavemsg1@...> wrote:

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

In less than 20 seconds, I found 993305 counterexamples

> 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 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