primes p and p||1

[enzocreti]

(2^487-1)/4871 is prime where 4871 is a prime.
are there other primes p such that (2^p-1)/(p||1) is prime where p||1 is prime? p||1 simply means the concatenation in base ten of p and 1.

[DukeBG]

p||1 = 10*p + 1

since factors of (2^p-1) are of form 2*k*p+1 the question is equal to

"are there M(p) that have exactly two divisors, one of which has k=5".

I've found about 43,000 Mp's with k=5 up to M4,290,333,139 using mersenne.ca. However, the "has exactly two divisors" is a much stricter condition with far less known examples – there's only 330 fully factored ones (excluding mersenne primes themselves) out of which 85 have exactly two factors. Easy to manually look through that list. M487 is the only one suitable on it.

However, I do believe that the amount of numbers like that would be infinite.

Maybe, there are other numbers in primenet with only one k=5 factor found (in the 43,000 list) and the cofactor not PRP-tested.

[CRGreathouse]

Originally Posted by DukeBG

However, the "has exactly two divisors" is a much stricter condition with far less known examples – there's only 330 fully factored ones (excluding mersenne primes themselves) out of which 85 have exactly two factors.

How do you find these? Is there a way to list just the fully factored numbers? (I don't know the mersenne.ca interface. )

[CRGreathouse]

BTW, OEIS has
https://oeis.org/A135978

[DukeBG]

Originally Posted by CRGreathouse

How do you find these? Is there a way to list just the fully factored numbers? (I don't know the mersenne.ca interface. )

This page: https://www.mersenne.ca/prp.php