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 2023-04-28, 19:47 #12 bhelmes     Mar 2016 431 Posts 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) ?
2023-04-28, 20:58   #13
R.D. Silverman

"Bob Silverman"
Nov 2003
North of Boston

22·1,889 Posts

Quote:
 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.

2023-04-30, 00:31   #15
bhelmes

Mar 2016

43110 Posts

Quote:
 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),

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.

2023-05-20, 02:09   #16
bhelmes

Mar 2016

431 Posts

Quote:
 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 ?)

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