2019-07-15, 13:12 | #1 |
Mar 2018
17·31 Posts |
pg primes of the form 41s+r
Pg numbers are numbers of the form
(2^k-1)*10^d+2^(k-1)-1 where d is the number of decimal digits of 2^(k-1)-1 pg(215), pg(51456), pg(69660), pg(92020) and pg(541456) are probable primes where k is congruent to 10^m mod 41, for m some nonnegative integer. 215 for example is 10 mod 41. now when k is congruent to 10^m mod 41 and pg(k) is probable prime, then 10*pg(k) is congruent to 10^s mod 37 or mod 307 where s is a nonnegative integer 215*10 is 1 mod 307 51456*10 is 1 mod 37 69660*10 is 1 mod 37 92020*10 is 10 mod 37 541456*10 is 1 mod 307 37 and 307 are primes with the same first and last digit (3 and 7) so it seems that if pg(k) is prime and k is 10^m mod 41, then: or pg(k) is 10^s mod 37 (cases 51456, 69660, 92020) or if pg(k) is not 10^s mod 37 then 10*pg(k) is 10^s mod 307 (cases 215 and 541456) Last fiddled with by enzocreti on 2019-07-15 at 14:37 |
Thread Tools | |
Similar Threads | ||||
Thread | Thread Starter | Forum | Replies | Last Post |
Primes of the form (b+-1)*b^n+-1 and b^n+-(b+-1) | sweety439 | sweety439 | 162 | 2020-05-15 18:33 |
Primes of the form 4*10^n+1 and primes of the form 16*100^n+1 | enzocreti | enzocreti | 7 | 2019-05-05 13:19 |
Primes of the Form Mod(p,q) = Mod(x,q) | a1call | Miscellaneous Math | 6 | 2018-12-11 03:34 |
Primes of the form n+-phi(n) | carpetpool | carpetpool | 3 | 2017-01-26 01:29 |
Primes of the form 2.3^n+1 | Dougy | Math | 8 | 2009-09-03 02:44 |