= = = PRIME NUMBERS OF THE FORM A * 14^B - 1 = = =

Computed by John Eisenmann

The numbers listed on this page have the form A * 14^B - 1, where A and B are integers such that 0 < A < 14 and B > 0. In base 14, these numbers consist of a digit followed by at least one D. For example: 1D, 2D, 3D, 4D, 4DD, and 4,DDD.

I am looking for values of A and B such that A * 14^B - 1 is prime.

I used a custom-made program to search for these prime numbers. This program is networked across many computational nodes. The program uses the GMP arithmetic library and the Miller-Rabin primality test. These primes are marked as "Industrial Primes".

In addition, I have performed deterministic tests on some of the industrial primes. For numbers where B <= 1912, I used the Ellipsa Primo application which Dana kindly introduced to me. For numbers where B > 1912, I used my own implementation of the N + 1 algorithm described in a paper by John Brillhart, D. H. Lehmer, and J. L. Selfridge. This implementation uses an algorithm by Victor Shoup to calculate Jacobi symbols, and a formula from mathworld.wolfram.com to calculate Lucas sequences.

All of the numbers I have tested in a deterministic fashion have passed, and have been marked as "Definitely Prime".

Current status of the search program:

Working on candidate 12 * 14^56141 - 1 (base 9885 / 18649)

Currently active computational nodes:

Beryllium2
Beryllium1

Below are all of the A * 14^B - 1 primes I (we?) have found:

#13 * 14^1 - 1 (Definitely Prime)

#26 * 14^1 - 1 (Definitely Prime)

#37 * 14^1 - 1 (Definitely Prime)

#410 * 14^1 - 1 (Definitely Prime)

#512 * 14^1 - 1 (Definitely Prime)

#613 * 14^1 - 1 (Definitely Prime)

#73 * 14^2 - 1 (Definitely Prime)

#88 * 14^2 - 1 (Definitely Prime)

#912 * 14^2 - 1 (Definitely Prime)

#103 * 14^3 - 1 (Definitely Prime)

#117 * 14^3 - 1 (Definitely Prime)

#1213 * 14^3 - 1 (Definitely Prime)

#132 * 14^4 - 1 (Definitely Prime)

#1412 * 14^4 - 1 (Definitely Prime)

#153 * 14^5 - 1 (Definitely Prime)

#167 * 14^5 - 1 (Definitely Prime)

#1713 * 14^5 - 1 (Definitely Prime)

#1812 * 14^6 - 1 (Definitely Prime)

#197 * 14^7 - 1 (Definitely Prime)

#207 * 14^9 - 1 (Definitely Prime)

#212 * 14^10 - 1 (Definitely Prime)

#223 * 14^15 - 1 (Definitely Prime)

#237 * 14^21 - 1 (Definitely Prime)

#2412 * 14^22 - 1 (Definitely Prime)

#2512 * 14^24 - 1 (Definitely Prime)

#2612 * 14^26 - 1 (Definitely Prime)

#276 * 14^27 - 1 (Definitely Prime)

#2813 * 14^27 - 1 (Definitely Prime)

#293 * 14^31 - 1 (Definitely Prime)

#3013 * 14^35 - 1 (Definitely Prime)

#3112 * 14^37 - 1 (Definitely Prime)

#327 * 14^39 - 1 (Definitely Prime)

#3312 * 14^47 - 1 (Definitely Prime)

#343 * 14^53 - 1 (Definitely Prime)

#357 * 14^59 - 1 (Definitely Prime)

#368 * 14^88 - 1 (Definitely Prime)

#377 * 14^99 - 1 (Definitely Prime)

#383 * 14^100 - 1 (Definitely Prime)

#3910 * 14^105 - 1 (Definitely Prime)

#403 * 14^108 - 1 (Definitely Prime)

#412 * 14^124 - 1 (Definitely Prime)

#4213 * 14^165 - 1 (Definitely Prime)

#438 * 14^178 - 1 (Definitely Prime)

#443 * 14^188 - 1 (Definitely Prime)

#4510 * 14^209 - 1 (Definitely Prime)

#4613 * 14^209 - 1 (Definitely Prime)

#4712 * 14^257 - 1 (Definitely Prime)

#487 * 14^323 - 1 (Definitely Prime)

#493 * 14^328 - 1 (Definitely Prime)

#507 * 14^331 - 1 (Definitely Prime)

#517 * 14^359 - 1 (Definitely Prime)

#5212 * 14^374 - 1 (Definitely Prime)

#5310 * 14^505 - 1 (Definitely Prime)

#542 * 14^550 - 1 (Definitely Prime)

#558 * 14^562 - 1 (Definitely Prime)

#563 * 14^568 - 1 (Definitely Prime)

#577 * 14^617 - 1 (Definitely Prime)

#5810 * 14^683 - 1 (Definitely Prime)

#593 * 14^816 - 1 (Definitely Prime)

#6010 * 14^879 - 1 (Definitely Prime)

#613 * 14^1013 - 1 (Definitely Prime)

#6210 * 14^1233 - 1 (Definitely Prime)

#636 * 14^1325 - 1 (Definitely Prime)

#6412 * 14^1848 - 1 (Definitely Prime)

#653 * 14^1912 - 1 (Definitely Prime)

#6613 * 14^2351 - 1 (Definitely Prime)

#678 * 14^2392 - 1 (Definitely Prime)

#687 * 14^2529 - 1 (Definitely Prime)

#6912 * 14^3128 - 1 (Definitely Prime)

#7012 * 14^3468 - 1 (Definitely Prime)

#713 * 14^4008 - 1 (Definitely Prime)

#723 * 14^4155 - 1 (Definitely Prime)

#732 * 14^4720 - 1 (Definitely Prime)

#748 * 14^4946 - 1 (Definitely Prime)

#758 * 14^5816 - 1 (Definitely Prime)

#766 * 14^6375 - 1 (Definitely Prime)

#776 * 14^6899 - 1 (Definitely Prime)

#7812 * 14^6943 - 1 (Definitely Prime)

#793 * 14^8686 - 1 (Definitely Prime)

#8012 * 14^9663 - 1 (Definitely Prime)

#813 * 14^10747 - 1 (Definitely Prime)

#8213 * 14^11277 - 1 (Definitely Prime)

#8310 * 14^11647 - 1 (Definitely Prime)

#846 * 14^13445 - 1 (Definitely Prime)

#852 * 14^14644 - 1 (Definitely Prime)

#865 * 14^19698 - 1 (Definitely Prime)

#8713 * 14^21807 - 1 (Definitely Prime)

#882 * 14^23896 - 1 (Definitely Prime)

#8913 * 14^25453 - 1 (Definitely Prime)

#9012 * 14^33936 - 1 (Definitely Prime)

#9112 * 14^36609 - 1 (Definitely Prime)

#923 * 14^38872 - 1 (Industrial Prime; 12912 Bases Tested)

#9312 * 14^39139 - 1 (Industrial Prime; 13001 Bases Tested)

#943 * 14^43261 - 1 (Industrial Prime; 14370 Bases Tested)

#9510 * 14^44393 - 1 (Industrial Prime; 14747 Bases Tested)

#966 * 14^47629 - 1 (Industrial Prime; 15822 Bases Tested)

#973 * 14^49285 - 1 (Industrial Prime; 16372 Bases Tested)

#9813 * 14^52443 - 1 (Industrial Prime; 17421 Bases Tested)