- Guys,

Here is a new Carol Prime....(2^175749 - 1)^2 - 2. This number has 105812 digits. This is a new record. To date, it is the largets Carol/Kynea prime found. It is the first 100K Carol or Kynea Prime. It took PFGW 30 minutes to test it 2-PRP and 2 hours 23 minutes to run an N+1 test. If only I could prove my conjecture, it would have saved me 2 hours.

Steven, I've checked Carol from 150,001 - 175,928 with only this new prime in that range. This number is also the first 100K+ digits Carol or Kynea prime. This is one of my goals. The next goals is for anyone or us to find a Carol/Kynea prime with 1M+ digits and then 10M+ digits.

I think that a 100K digit Kynea is near. The Kynea was at 110K and the search is up to 220K with no other Kynea Prime in sight. If we keep looking, I think that we will find one soon. I hope to submit a Carol/Kynea paper t a journal soon. I just have to find the time to do so.....

---Cletus Emmanuel

P.S.: I tried to subimt this number to Chris Caldwell's page and it prompt me to enter a password to access the DB, but I don't know that I have access to the site via that route, muchless a password. Can someone help me submit?......

---Cletus Emmanuel

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[Non-text portions of this message have been removed] - At 01:40 PM 10/8/2004, Cletus Emmanuel wrote:
>Here is a new Carol Prime....(2^175749 - 1)^2 - 2. This number has 105812 digits. This is a new record. To date, it is the largets Carol/Kynea prime found. It is the first 100K Carol or Kynea Prime. It took PFGW 30 minutes to test it 2-PRP and 2 hours 23 minutes to run an N+1 test. If only I could prove my conjecture, it would have saved me 2 hours.

Go to your page: http://primes.utm.edu/bios/page.php?id=374 and use the links at the bottom. There is one to submit the prime, another to change your password, and if you have lost yours, just click on edit entry--when you fail to enter the right password it will offer a link to send you a new one. (I should make this less hidden... but I haven't).

>

>P.S.: I tried to subimt this number to Chris Caldwell's page and it prompt me to enter a password to access the DB, but I don't know that I have access to the site via that route, muchless a password. Can someone help me submit?......

"Carol Prime" is not an archivable comment, so don't bother adding it.

Chris - --- In primenumbers@yahoogroups.com, Cletus Emmanuel <cemmanu@y...> wrote:
> Guys,

105812 digits. This is a new record. To date, it is the largets

> Here is a new Carol Prime....(2^175749 - 1)^2 - 2. This number has

Carol/Kynea prime found. It is the first 100K Carol or Kynea Prime.

It took PFGW 30 minutes to test it 2-PRP and 2 hours 23 minutes to run

an N+1 test. If only I could prove my conjecture, it would have saved

me 2 hours.

Congratulations on your find!

I have tested n^2-2 is 2-PRP => n^2-2 is ( 5 miller rounds probable )

prime for all odd n < 47,088,000,003 ( 1,851,251,401 PrP's )

Paul - Paul,

did you find any of those PRP's to be composite?.....

Paul Underwood <paulunderwood@...> wrote:

--- In primenumbers@yahoogroups.com, Cletus Emmanuel <cemmanu@y...> wrote:

> Guys,

> Here is a new Carol Prime....(2^175749 - 1)^2 - 2. This number has

105812 digits. This is a new record. To date, it is the largets

Carol/Kynea prime found. It is the first 100K Carol or Kynea Prime.

It took PFGW 30 minutes to test it 2-PRP and 2 hours 23 minutes to run

an N+1 test. If only I could prove my conjecture, it would have saved

me 2 hours.

Congratulations on your find!

I have tested n^2-2 is 2-PRP => n^2-2 is ( 5 miller rounds probable )

prime for all odd n < 47,088,000,003 ( 1,851,251,401 PrP's )

Paul

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[Non-text portions of this message have been removed] - Cletus,

no, I did not find any of those 2-PRP's composite by using 5 rounds of

the Miller-Rabin test on each one found.

The process is slow ~ 120 MHz

Here is the GNU Multiple Precision program:

//outputs x:x^f-x is NOT 5 times Miller Rabin where f=x^2-2

is 2-PRP

//(PU/22/03/04)

#include <stdio.h>

#include <gmp.h>

int main( int argc, char *argv[] ) {

mpz_t x; //inputted minimum

mpz_t f; //=x^2-2

mpz_t k; //prp counter

mpz_t t; //temporary variable

int i, j; //loop counter

FILE *fp; //for x-PRP.out

if ( argc != 3 ) {

printf( "usage : quadratic minimum_x prp_count\n" );

exit( 1 ); }

else printf( "Working...\n" );

mpz_init_set_str( x, *++argv, 10 );

mpz_init_set_str( k, *++argv, 10 );

mpz_init( f );

mpz_init( t );

while ( 1 ) {

for( i=0; i<1000000; i++ ) {

mpz_add_ui( x, x, 2 );

mpz_pow_ui( f, x, 2 );

mpz_sub_ui( f, f, 2 );

mpz_set_ui( t, 2 );

mpz_powm( t, t, f, f );

if ( mpz_cmp_ui( t, 2 ) == 0 ) {

if( mpz_probab_prime_p( f, 5 ) ) mpz_add_ui( k, k, 1 );

else {

printf( "x-PRP:x=" ); mpz_out_str( NULL, 10, x );

printf( "\n" );

fp = fopen( "x-PRP.out", "a" );

fprintf( fp, "x-PRP:x=" ); mpz_out_str( fp, 10, x );

fprintf( fp, "\n" );

fclose( fp ); } } }

printf( "status: x=" ); mpz_out_str( NULL, 10, x );

printf( " x-prp_count=" ); mpz_out_str( NULL, 10, k );

printf( "\n" );

fp = fopen( "x-PRP.out", "a" );

fprintf( fp, "status: x=" ); mpz_out_str( fp, 10, x );

fprintf( fp, " x-prp_count=" ); mpz_out_str( fp, 10, k );

fprintf( fp, "\n" );

fclose( fp );

}

}

Of course I can unroll my loops by using the method of differences.

And I need only test 2^((n^2-3)/2)=1 mod n^2-2. But speed up in these

tricks are insignificant compared to the modular exponentiation and

any of subsequent Miller Rabin probable prime tests.

Paul

--- In primenumbers@yahoogroups.com, Cletus Emmanuel <cemmanu@y...> wrote:

> Paul,

> did you find any of those PRP's to be composite?.....

>

> Paul Underwood <paulunderwood@m...> wrote:

>

> --- In primenumbers@yahoogroups.com, Cletus Emmanuel <cemmanu@y...>

wrote:

> > Guys,

> > Here is a new Carol Prime....(2^175749 - 1)^2 - 2. This number has

> 105812 digits. This is a new record. To date, it is the largets

> Carol/Kynea prime found. It is the first 100K Carol or Kynea Prime.

> It took PFGW 30 minutes to test it 2-PRP and 2 hours 23 minutes to run

> an N+1 test. If only I could prove my conjecture, it would have saved

> me 2 hours.

>

> Congratulations on your find!

>

> I have tested n^2-2 is 2-PRP => n^2-2 is ( 5 miller rounds probable )

> prime for all odd n < 47,088,000,003 ( 1,851,251,401 PrP's )

>

> Paul

>

>