What's known about Mersenne prime exponent composites?

[bhelmes]

A peaceful and pleasant night for you

Mersenne numbers and Mersenne prime numbers are "function values" of
the quadratic irreducible polynomial f(n)=2n²-1 whereby n is a potence of 2.

An easy elementary proof can be found under http://devalco.de/quadr_Sieb_2x%5E2-1.php#1

The distribution of Mp can be explained by a sieving procedure similar to the sieve of Eratosthenes
concerning this polynomial, which is the only known for me and really beautiful and a theoretical explication
for the distribution of Mps, but unfortunately too slow for practical reasons.

A composite Mersenne number has at least one factor which "appears" earlier in the sieving procedure.
More infos ca be found under
http://devalco.de/quadr_Sieb_2x^2-1.php or if you prefer a pdf
http://devalco.de/quadratic_prime_sieves.pdf

There was 3 years ago a destructive and violating statement concerning this topic from R.D. Silvermann
https://www.mersenneforum.org/showpo...1&postcount=14
It would be nice to know what he thinks now about this topic.

Is it mathematical possible to make a substitution for n concerning the polynomial f(n)=2n²-1
and decline a faster growing function (also a quadratic polynomial but with better coefficients) ?

[R.D. Silverman]

Originally Posted by bhelmes

A peaceful and pleasant night for you

Mersenne numbers and Mersenne prime numbers are "function values" of
the quadratic irreducible polynomial f(n)=2n²-1 whereby n is a potence of 2.

An easy elementary proof can be found under http://devalco.de/quadr_Sieb_2x%5E2-1.php#1

The distribution of Mp can be explained by a sieving procedure similar to the sieve of Eratosthenes
concerning this polynomial, which is the only known for me and really beautiful and a theoretical explication
for the distribution of Mps, but unfortunately too slow for practical reasons.

A composite Mersenne number has at least one factor which "appears" earlier in the sieving procedure.
More infos ca be found under
http://devalco.de/quadr_Sieb_2x^2-1.php or if you prefer a pdf
http://devalco.de/quadratic_prime_sieves.pdf

There was 3 years ago a destructive and violating statement concerning this topic from R.D. Silvermann
https://www.mersenneforum.org/showpo...1&postcount=14
It would be nice to know what he thinks now about this topic.

Is it mathematical possible to make a substitution for n concerning the polynomial f(n)=2n²-1
and decline a faster growing function (also a quadratic polynomial but with better coefficients) ?

It is still complete crap.

[Andrew Usher]

(This thread's having been moved means that no one important is going to read it, and my work of composing the following reply is probably wasted. There is only one math forum that everyone follows.)

I myself don't think there was anything especially interesting about semiprime Mersennes, but for heuristics, they should be the same as for semiprime integers: there are infinitely many, their limiting density is zero but it falls off more slowly than the primes, and so for their density should exceed that of the primes for all but a small initial interval - and so on for any fixed number of prime factors.

As charybdis points out, though there are already more discovered semiprimes than primes, there are even more undiscovered ones (though none among PRP-tested numbers, obviously) - if I see the infinite series correctly, for a constant factoring limit the ratio of unfactored to factored semiprimes should rise as the square of the exponent.

On the other hand there was nothing wrong with compiling and posting the list of known semiprimes. The accumulation of data, though not the most important part of mathematics, still is a part; it often leads to useful conjectures and insights, and even when it gets nowhere, it is still permanent reference material that avoids needless repetition (much like the Primenet database). It is not a standard meaning of the word to term that 'numerology', even if done without any understanding. So long as the data is not presented as anything more than that, it is not numerology and should not be so dismissed.

It might be more interesting (but a hard problem) to ask what the average number of factors in a Mersenne is; a famous result is that for all integers the limit is log log N, and my guess would be that for Mersennes and other classes of number whose factorisation does not follow from the definition, it also has that limit, or a fixed multiple of it.

My reply to science_man_88 was entirely serious and intended to be helpful - the 'other thread' is of course https://www.mersenneforum.org/showthread.php?t=28604 where it's fairly clear what he's getting at but the fact that it is true of all moduli, and why, don't even seem to occur to him - I could compose a proof (not relying on any previous knowledge of group theory) in a few minutes.

[bhelmes]

Originally Posted by R.D. Silverman

It is still complete crap.

My mum told me not to speak about math with women, if I search a girlfriend,
actually I would really enjoy it to talk only about math and eating ice cream after the sex.

Please, even if it is really hard for you (your history is impressiv),
return to mathematical logic and arguments.

You participate in a wonderful math forum and the aim should be talking about math.

Suffering from pain might influence thinking and conversation with other persons,
escaping into madness is an option, but IMHO not the best choice.

Personally I prefer walking, mostly in the night, juggling some balls or playing table tennis with some friends.

[bhelmes]

Originally Posted by bhelmes

Is it mathematical possible to make a substitution for n concerning the polynomial f(n)=2n²-1
and decline a faster growing function (also a quadratic polynomial but with better coefficients) ?

IMHO yes:
Let f(n)=2n²-1

Substitution for n=(2kp+1)+s
where k is the new variable, p from Mp and s a variable which is chosen later.
leads to a better quadratic irreducible polynomial g(k, s).

With f(n0)=g(k,s0) for k=0
s0=n0-1

Example:
Mp=2047=2¹¹-1, p=11
n0=32, f(32)=2047
s0=32-1
g(k, s0)=2(22k+1+32-1)²-1
=2(22k+2⁵)²-1
=2(22²k²+2*22*32k+2¹⁰)-1
=968k²+2816k+2047

Is it possible to make a sieving construction with that new polynomial ?
(Maybe in two steps, first a presieve and than a second sieve ?)