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)