What's known about Mersenne prime exponent composites?

[jnml]

In particular, the two prime factors ones, like \(M_{11}, M_{23}\).

Are there finitely many?

Infinitely many?

We don't know, but we expect some heuristics, like proportional to \(p^r\), for some \(r\)?

Is anything else interesting to note about them?

[kriesel]

Some statistics related to Mersenne primes, composites, and factors are here.

[science_man_88]

Originally Posted by jnml

In particular, the two prime factors ones, like \(M_{11}, M_{23}\).

If \((2kp+1)(2jp+1)=2(2jkp+k+j)p+1\) then \(2jkp+k+j\) must not be expressible in the same form for different variable values. That's all I state.

[Andrew Usher]

The thing is, science_man_88, that as in the other thread your math is correct, but utterly trivial. Presenting it as though it were a real insight seems bizarre.

[R.D. Silverman]

Originally Posted by Andrew Usher

The thing is, science_man_88, that as in the other thread your math is correct, but utterly trivial. Presenting it as though it were a real insight seems bizarre.

Oh, the irony.

[cxc]

According to mersenne.ca/prp.php, there are currently known to be 89 fully-factored Mersenne numbers with two and only two non-trivial factors; here are their exponents: {11,23,37,41,59,67,83,97,101,103, 109,131,137,139,149,167,197,199,227,241, 269,271,281,293,347,373,379,421,457,487, 523,727,809,881,971,983,997,1061,1063,1427, 1487,1637,1657,2357,2927,3079,3259,3359,4111,4243, 4729,5689,6043,6679,7331,7757,10169,14561,17029,26903, 28759,28771,58199,63703,86371,106391,130439,136883,151013,173867, 221509,245107,271211,271549,406583,432457,611999,684127,1010623,1168183, 1304983,1629469,2327417,3464473,4187251,5240707,7313983,10443557,12588091}

In terms of heuristics about their rarity, the range occupied by the 89 2-factor composites contains the first 38 Mersenne primes, so they might be more plentiful than their prime siblings over a larger range. (The wavefront for PRP testing cofactors of partially-factored Mersennes is considerably behind the waves for first-time primality tests and double checks, as testing is almost as expensive as the first-time primality test for the same exponent.)

[kriesel]

12558091 is composite.

[cxc]

I see someone was paying attention :) The correct exponent is, of course, 12588091.

[R.D. Silverman]

Originally Posted by cxc

According to mersenne.ca/prp.php, there are currently known to be 89 fully-factored Mersenne numbers with two and only two non-trivial factors; here are their exponents: {11,23,37,41,59,67,83,97,101,103, 109,131,137,139,149,167,197,199,227,241, 269,271,281,293,347,373,379,421,457,487, 523,727,809,881,971,983,997,1061,1063,1427, 1487,1637,1657,2357,2927,3079,3259,3359,4111,4243, 4729,5689,6043,6679,7331,7757,10169,14561,17029,26903, 28759,28771,58199,63703,86371,106391,130439,136883,151013,173867, 221509,245107,271211,271549,406583,432457,611999,684127,1010623,1168183, 1304983,1629469,2327417,3464473,4187251,5240707,7313983,10443557,12588091}

In terms of heuristics about their rarity, the range occupied by the 89 2-factor composites contains the first 38 Mersenne primes, so they might be more plentiful than their prime siblings over a larger range. (The wavefront for PRP testing cofactors of partially-factored Mersennes is considerably behind the waves for first-time primality tests and double checks, as testing is almost as expensive as the first-time primality test for the same exponent.)

This is supposed to be for math discussion. Please take the pure numerology to another forum.
Merely presenting data is not mathematics.

I will offer a hint: See Hardy & Wright's derivation for the distribution of p2's. There is a very strong reason
why we expect p2 Mersenne's to be more plentiful than the primes themselves. Perhaps you might like to
investigate and get back to us?

[charybdis]

Originally Posted by cxc

According to mersenne.ca/prp.php, there are currently known to be 89 fully-factored Mersenne numbers with two and only two non-trivial factors; here are their exponents: ...

In terms of heuristics about their rarity, the range occupied by the 89 2-factor composites contains the first 38 Mersenne primes, so they might be more plentiful than their prime siblings over a larger range. (The wavefront for PRP testing cofactors of partially-factored Mersennes is considerably behind the waves for first-time primality tests and double checks, as testing is almost as expensive as the first-time primality test for the same exponent.)

The PRP-CF wavefront is irrelevant here. We can only test the cofactor for primality if we know a factor in the first place, and any composite Mersenne without known factors could be a semiprime. There will be a *lot* of undiscovered semiprimes in that range.

[R.D. Silverman]

Originally Posted by charybdis

The PRP-CF wavefront is irrelevant here. We can only test the cofactor for primality if we know a factor in the first place, and any composite Mersenne without known factors could be a semiprime. There will be a *lot* of undiscovered semiprimes in that range.

Thank you.