Mersenne Prime Exponent Distribution

[PawnProver44]

This question has never been considered before, so is there a way to determine the number of Mersenne prime Expoenents (p such that 2^p-1 is prime) less than n. A second similar problem is what is the chance 2^p-1 is prime for a random prime p. For example, is 2^71324207525210468041-1 prime. If not, what is the nearest prime p such that 2^p-1 is prime for. (I don't really expect specific answers to these examples, but this is just to show the types of problems no one has really encountered before.)

[science_man_88]

Originally Posted by PawnProver44

This question has never been considered before, so is there a way to determine the number of Mersenne prime Expoenents (p such that 2^p-1 is prime) less than n. A second similar problem is what is the chance 2^p-1 is prime for a random prime p. For example, is 2^71324207525210468041-1 prime. If not, what is the nearest prime p such that 2^p-1 is prime for. (I don't really expect specific answers to these examples, but this is just to show the types of problems no one has really encountered before.)

it has to be less than the number of primes by at least the number of Sophie Germain primes that are 3 mod 4. As already stated if p is a 3 mod 4 Sophie Germain prime 2p+1 divides Mp. if we could answer such questions completely, especially if in simple form GIMPS would not be needed as we could predict what the next exponent would be and therefore just calculate the exponents upto as high as possible.

[VBCurtis]

Originally Posted by PawnProver44

This question has never been considered before,

How would you know? You can't even say it hasn't been considered on this forum before.

To find the number of mersenne primes less than n, count them. There's a list readily available of known mersenne primes, so for any n below the double-check line your question is trivial.

[PawnProver44]

Originally Posted by science_man_88

it has to be less than the number of primes by at least the number of Sophie Germain primes that are 3 mod 4. As already stated if p is a 3 mod 4 Sophie Germain prime 2p+1 divides Mp. if we could answer such questions completely, especially if in simple form GIMPS would not be needed as we could predict what the next exponent would be and therefore just calculate the exponents upto as high as possible.

71324207525210468041 is already congruent to 1 (mod 4), and no cofactors known either, so there still may be a chance that 2^71324207525210468041-1 is prime. This is just an example, that I am taking into account though.

[science_man_88]

Originally Posted by PawnProver44

71324207525210468041 is already congruent to 1 (mod 4), and no cofactors known either, so there still may be a chance that 2^71324207525210468041-1 is prime. This is just an example, that I am taking into account though.

right but you asked about the odds and I'm just saying that there's one case where it could be 0.

[PawnProver44]

Originally Posted by science_man_88

right but you asked about the odds and I'm just saying that there's one case where it could be 0.

Not all odds (primes p) congruent to 3 mod 4 are not possible choices for 2^p-1 to be primes, just those such that 2p+1 is also prime.

[Uncwilly]

Originally Posted by PawnProver44

there still may be a chance that 2^71324207525210468041-1 is prime. This is just an example, that I am taking into account though.

How much factoring work have you done on this exponent?

[ATH]

Originally Posted by PawnProver44

This question has never been considered before, so is there a way to determine the number of Mersenne prime Expoenents (p such that 2^p-1 is prime) less than n. A second similar problem is what is the chance 2^p-1 is prime for a random prime p. For example, is 2^71324207525210468041-1 prime. If not, what is the nearest prime p such that 2^p-1 is prime for. (I don't really expect specific answers to these examples, but this is just to show the types of problems no one has really encountered before.)

Considering how many smart people have thought about math for thousands of years, there is very few things you or I can think of, that have not been considered and encountered before, so don't expect many of your ideas to be original.

If we knew exactly where and how many Mersenne primes there are then where would be no point of GIMPS, but there are of course conjectures:

https://primes.utm.edu/mersenne/heuristic.html
https://primes.utm.edu/notes/faq/NextMersenne.html

[PawnProver44]

Using that information, there are probably 2 undiscovered primes p such that 2^p-1 is prime between 74,000,000 and 600,000,000, roughly speaking.

[science_man_88]

Originally Posted by PawnProver44

Using that information, there are probably 2 undiscovered primes p such that 2^p-1 is prime between 74,000,000 and 600,000,000, roughly speaking.

really I get 5 potentially. 600/74> 3 <1.5^3>1.47...^3 okay technically I changed one thing late.point still stands.

[PawnProver44]

Originally Posted by science_man_88

really I get 5 potentially. 600/74> 3 >1.5^3>1.47...^3

Except for 1, there is always a prime p such that 2^p-1 is prime between 10^n and 1.5*10^n.