Share N+/-1 Primality Proofs

[sweety439]

The generalized repunit probable prime, R2731(685), N-1 has 31.345% factored, all algebraic factors are already entered.

Factoring this 167 digit number (a factor of Phi(390,685), i.e. a factor of 685,195+) will enable N-1 proof for R2731(685), since this will make N-1 have >33.333% factored.

[MDaniello]

2684720974...01   N+/-1  350 digits
5548042917...01   N+/-1   497 digits
(86^294+2)/6          N-1   598 digits
(19607^353+1)/19608   N-1   1511 digits
4687274111...01      N-1   1381 digits
(13088^373-1)/13087   N-1   1532 digits
(17200^457+1)/17201   N-1   1932 digits
(5183^521+1)/5184      N-1   1932 digits
(17195^457-1)/17194   N-1   1932 digits

In addition to these, several Carol-Kynea primes that lacked any kind of primality proof.
Some other primes of this kind were missing from the db at all.

[MDaniello]

(65540^101-1)/65539   N-1   482 digits
(65656^109-1)/65655   N-1   521 digits
(65560^113-1)/65559   N-1   540 digits
(65638^113-1)/65637   N-1   540 digits
(65528^113-1)/65527   N-1   540 digits
(20589^127-1)/20588   N-1   544 digits
(20881^127-1)/20880   N-1   545 digits
(20902^127-1)/20901   N-1   545 digits
(65592^127-1)/65591   N-1   607 digits
80513^544-2   N+1   2669 digits
(1858^919+1)/1859   N-1   3001 digits

[sweety439]

Many of these numbers (not including numbers of the form k*b^n+1 and k*b^n-1, since N-1 or N+1 are trivially 100% factored) are of these forms:

(b^n-1)/(b-1) (use N-1) (n must be prime)
(b^n+1)/(b+1) (use N-1) (n must be prime)
(b^(2^n)+1)/2 (use N-1) (b is odd)
b^n+2 (use N-1)
b^n-2 (use N+1)
b^n+(b-1) (use N+1)
b^n-(b-1) (use N-1)
b^n+(b+1) (use N-1)
b^n-(b+1) (use N+1)
((b-2)*b^n+1)/(b-1) (use N-1)

For these numbers, factor N-1 or N+1 is equivalent to factor the Cunningham number b^n+-1 (or b^(n-1)+-1), and the smallest primes of these forms are minimal "prime > base" in base b (this is an interesting problem, there are 77 such primes in decimal (base 10), and they are 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027)

Originally Posted by MDaniello

In addition to these, several Carol-Kynea primes that lacked any kind of primality proof.
Some other primes of this kind were missing from the db at all.

For the Carol primes (b^n-1)^2-2 and Kynea primes (b^n+1)^2-2, N-1 is trivially 50% factored

[chris2be8]

Is anyone on mersenneforum generating Primo certs for numbers around 964 digits? I'm running a script to prove small PRPs, working from 300 digits upwards, and I've noticed a lot of numbers where someone else proved them just before my script was about to prove them. So we could easily waste effort by working on the same numbers.

I've seen a few by kotenok2000 around 763 digits. But the latest batch were anonymous.

Moving to 980 digits would avoid collisions for a few weeks. Or better keep an eye on http://factordb.com/stat_1.php?prp and avoid the first few hundred numbers it shows.

I've got another system working from 1000 digits upwards, So please also avoid the first few hundred numbers over 1000 digits.