[quote=bhelmes;544844]if I regard only the primes p>=5 with p=x²+1 (x>1) and
the primes p with [b]p | (x²+1 ) with p > x[/b]


can i derive any suggestion about the factorisation of p-1 in advance
(except divisible by 4 :-) )[/quote]

[quote=bhelmes;544969]if p=x²+1 then x | p-1


if p | (x²+1) ???


Can I calculate q=x²+1 with q=r*p [b]and x and r known[/b]; then f | p-1 ; f ???[/quote]

[quote=bhelmes;546023]I have the factorisation of
f(n)=n²+1=r*p where r element of N and p is prime[/quote]

It might help if you wouldn't keep trying to change the question.

In the first above-quoted post, you're asking whether knowing the x < p for which p divides x[sup]2[/sup] + 1 is of any help in factoring p-1. The answer is "No."

In the second above-quoted post, your hypothesis "x [i]and[/i] r known" is nonsensical.

In the third above-quoted post, are assuming that [i]n[/i] is given (this variable was named x in previous posts; so in addition to changing the question, you are also gratuitously changing your notation). And, apparently, you are [i]now[/i] asking whether, knowing n and a prime factor p of n[sup]2[/sup] + 1, helps factor p - 1.

Again, the answer is "No." What you [i]do[/i] know is that n (mod p) is one of the square roots of -1 (mod p); -n (mod p) is the other square root of -1 (mod p).

Knowing the square roots of -1 (mod p) can help find a and b such that p = a[sup]2[/sup] + b[sup]2[/sup]. You could then check whether the condition I mentioned in [url=https://www.mersenneforum.org/showpost.php?p=544877&postcount=13]this post[/url] is satisfied; and, [i]if[/i] you're [i]very[/i] lucky and it [i]is[/i] satisfied, obtain a (hopefully non-trivial) factorization of p-1. But as I pointed out, the primes p for which the condition is satisfied are thin on the ground. And unfortunately, they appear to be [i]thinner[/i] on the ground than primes p == 1 (mod 4) for which (p-1)/4 is actually [i]prime[/i]. I'm guessing that p == 1 (mod 4), p <= X for which (p-1)/4 is prime have an asymptotic of order X/log[sup]2[/sup](X). The ones satisfying the condition I mentioned before, I have no idea, except numerical data indicate a smaller asymptotic.

Perhaps someone who knows the subject better than I could indicate what is known for the proportion of primes p == 1 (mod 4) for which the largest prime factor of p-1 is greater than p[sup]c[/sup] where 0 < c < 1.

A peaceful evening for you,

perhaps my mathematical skill is not the best in explaining,
but i am sure that the math behind it is right.

Therefore I try again to explain it and will give an example
which deals although for 20 digit numbers :cool::

Let f(n)=2n²-1

f(n0)=f(2)=7

Substitution with n=7k+2

f(7k+2)=2(7k+2)²-1= 98k²+56k+7 | :7
f(7k+2)/7 = 14k²+8k+1 | -1 because I am looking for the factorization of f(n)-1

f(7k+2)/7-1 = 14k²+8k = 2k*(7k+4)

Therefore I can predict the factorization of f(7k+2)/7-1

k=3 f(7*3+2)/7 - 1 = 150 3|150
k=5 f(7*5+2)/7 - 1 = 390 5|390
k=7 f(7*7+2)/7 - 1 = 742 7|742
and so on.

That is not a factorization but a prediction which is helpful.

I think this explication is mathematical o.k.:
making a substitution for x0, subtraction one and finishing.

@LaurV I think you will understand why this is a progress, or :brian-e:

Enjoy the evening. :cmd:
Bernhard

[QUOTE=bhelmes;551121]
@LaurV I think you will understand why this is a progress, or :brian-e:
[/QUOTE]
Well, honestly, I have a bit of an issue understanding how you pick your substitution. I think that sieving was better :smile: With the current concept it seems to me that you moved the dead cat from factoring to picking the substitution (which is kinda random. Or :shock:)

[QUOTE=LaurV;551156]Well, honestly, I have a bit of an issue understanding how you pick your substitution. [/QUOTE]

In general :

let f(n)=2n²-1

f(n0)=r

then the substitution is n=f(n0)*k+n0 with the following division by f(n0)


I think I will combine the sieving algorithm with the prediction by adding a value for the k for every prime.

Well... come up with one previously-unknown ~80 bits factor and all naysayers will keep quite for a while... :razz:
But for that, you will need n to be as large as 40 to 45 bits (unless extremely lucky), which may take like few months or a year so (wild ass guess here), even slower than a PRP test.

A peaceful day for you, LaurV



[QUOTE=LaurV;551417]Well... come up with one previously-unknown ~80 bits factor and all naysayers will keep quite for a while... :razz:
But for that, you will need n to be as large as 40 to 45 bits (unless extremely lucky), which may take like few months or a year so (wild ass guess here), even slower than a PRP test.[/QUOTE]


I think I will use the function f(n)=2n²-1 from n=1 up to 2^40,
will calculating the factors / primes f(m) with f>m and will storing the m for each f, will make a jump to n=2^46 and a sieve up to 2^46+2^40.
You can use the chinese remainder lemma for using the prediction.
(Any idea what I try to achieve ?)



All in all, will finish work before Christmas,
today it is hot in Germany and not the best time for programming.



What about a mathemtical prediction about the density /

distribution of mersenne factors concerning f(n)=2n²-1 ?


Greetings
Bernhard

[QUOTE=bhelmes;553033]
What about a mathemtical prediction about the density /
distribution of mersenne factors concerning f(n)=2n²-1 ?

[/QUOTE]


I have some datas up to 2^40 for the primesieving for the polynomial

f(n)=2n²-1;
[url]http://devalco.de/quadr_Sieb_2x%5E2-1.php#4a[/url]


I think the density of "non reducible primes" (p=f(n))
is an upper limit for the density of Mersenne primes.


Hence a complete factorisation for f(n) from 1 up to 2^40 should give 6,8439 % of "non coverage"



I am not very skilled in these questions.


May be someone has a better approximation.


Greetings :cmd: :gah: :petrw1:

Bernhard

A peaceful and pleasant night for you,


I have primes p ~ 120 bits big,

I want to factorize p-1,
I do a f=gcd (8*9*25*7*11*13*17*19*23*29*31*37*41, p-1) and
check wether 2^[(p-1)/f]=1 mod p
If this is true I check if g=(p-1)/f is prime,
otherwise I try to make a factorisation of g with ecm


I tried with 5 steps:
[CODE]
switch (count_try)
{
case 0 : g1=ecm_factor (res, input, 3000, 0); break;
case 1 : g1=ecm_factor (res, input, 4000, 0); break;
case 2 : g1=ecm_factor (res, input, 5000, 0); break;
case 3 : g1=ecm_factor (res, input, 6000, 0); break;
case 4 : g1=ecm_factor (res, input, 7000, 0); break;
}
[/CODE]Can I improve the B1 limits ?

(Actual I get 1/25 non factorized candidates)

The program is nearly ready for a longer search.

Thanks if you spend me some lines :loco::brian-e: :truck:

Bernhard

A peaceful day,


this is the end of a wonderful programming episode:
Running of the program was only one day,
I used 59 GByte Ram for storing the sieving primes,
used ecm-library and a parallisation on 12 cores
and sieved a segment for n=2^62 for the function f(n)=2n²-1


The wonderful results can be found here:
M* is the Mersenne number, factor of Mp, n, ratio
[URL]http://devalco.de/results_62.html[/URL]


The ratio = (factor-1) / Mp is unfortunately low. :cry: :cmd:



If someone knows a good sentence in order to close the thread,
be free and transmit it. :tom: :tom: :tom:

Those are all [U][B]trivial[/B][/U] factors, mostly SG factors (p is 4k+3 and 2p+1 is prime) which can be found with no effort**, or they have a very small k (like q=2kp+1, with k=1, 3, 4, 5), and none of the mersenne numbers attached to them are in the GIMPS 1G range of interest (not even under 2^32, or under 10G range of interest for mersenne.ca).

--------
if(p%4=3 and isprime(p) and isprime(q=2*p+1), print(p,q))