## fact or fiction

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• 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
Message 1 of 4 , Feb 1, 2010
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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. :)
Message 2 of 4 , Feb 1, 2010
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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
>
>
>
>
>
> ------------------------------------
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://www.primepages.org/
>
>
>
>
>
>
• ... It takes less than 2 seconds to prove that there are at least 124880 counterexamples with N
Message 3 of 4 , Feb 1, 2010
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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.

It takes less than 2 seconds to prove that there are
at least 124880 counterexamples with N < 10^15. See:

It is likely that there are *precisely*
124880 counterexamples with N < 10^15

David
• ... In less than 20 seconds, I found 993305 counterexamples in tables from William Galway and Richard Pinch. There are 319226 numbers N
Message 4 of 4 , Feb 2, 2010
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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