Form of mersenne divisors???
 

The mersenne divisors have the form of 2kp+1 and this fact was discovered about 3 decades ago. Do we have an improved form or is GIMPS still using this form for trial division?
My next question is the special number field sieve using this form to find factors of a mersenne number.
Tank you...

Read this: [url]http://www.mersenne.org/various/math.php#trial_factoring[/url]

Also, "this fact was discovered [STRIKE]about 3 decades ago[/STRIKE]" >240 years ago.

[QUOTE=kurtulmehtap;382104]The mersenne divisors have the form of 2kp+1 and this fact was discovered about 3 decades ago. Do we have an improved form or is GIMPS still using this form for trial division?
My next question is the special number field sieve using this form to find factors of a mersenne number.
Tank you...[/QUOTE]

The fact was discovered about three centuries ago, I think - Euler knew it.

There's also the requirement that the factors of Mersenne numbers are +-1 modulo 8 (by quadratic reciprocity, because we know that 2 has a square root modulo them, namely 2^((p+1)/2)), which was discovered at about the same time.

I believe there are no other modular constraints.

However nobody knows whether it's possible to take advantage of the form for SNFS.

[QUOTE=kurtulmehtap;382104]The mersenne divisors have the form of 2kp+1 and this fact was discovered about 3 decades ago. Do we have an improved form or is GIMPS still using this form for trial division?
My next question is the special number field sieve using this form to find factors of a mersenne number.
Tank you...[/QUOTE]

See the previous follow-on, then repeat after me: Google is my friend.
There are many elementary articles discussing how NFS works. READ ONE.

Further, you need to explain what you mean by "improved form".
Define it, please.

[QUOTE=fivemack;382106]The fact was discovered about three centuries ago, I think - Euler knew it.

There's also the requirement that the factors of Mersenne numbers are +-1 modulo 8 (by quadratic reciprocity, because we know that 2 has a square root modulo them, namely 2^((p+1)/2)), which was discovered at about the same time.

I believe there are no other modular constraints.

However nobody knows whether it's possible to take advantage of the form for SNFS.[/QUOTE]

Yes, we do know.

[QUOTE=R.D. Silverman;382107]See the previous follow-on, then repeat after me: Google is my friend.
There are many elementary articles discussing how NFS works. READ ONE.

Further, you need to explain what you mean by "improved form".
Define it, please.[/QUOTE]

I Meant by "improved form" a better way than filtering the possible values of "k" other than the modified sieve of Erasthotenes or a deterministic way of finding "k" for a given prime exponent.

Time to move to Misc.Math.

Reminder: in Misc.Math, posters are protected from abuse. Let sm_88 chip some thoughts and maybe someone else... In Mike Myers' words, "talk amongst yourselves. I'll give you a topic: <<insert OP here>>".

[QUOTE=kurtulmehtap;382109]I Meant by "improved form" a better way than filtering the possible values of "k" other than the modified sieve of Erasthotenes or a deterministic way of finding "k" for a given prime exponent.[/QUOTE]

May I suggest that you read how trial division algorithms work? There is
a very good description in Knuth's TAOCP, Vol II.

Further, we do have a deterministic way of determining k: search all
possible values until we find one that works. This is definitely deterministic.
There is no randomness involved.

Or do you have some other definition of "deterministic" in mind??

[QUOTE=R.D. Silverman;382111]Or do you have some other definition of "deterministic" in mind??[/QUOTE]


I think the OP means something along the lines of instead of trial division with k that are in certain modular classes is there a way to determine exactly which k must yield prime 2kp+1 but the best I can think of is a modified Atkins sieve without the square-free step completely done and with a change to the equations so they could say if 2kp+1 is in this modular class and p this modular class, k must have an odd number of solutions to <altered equation> for 2kp+1 to be prime but the modular classes k would be in to even attempt to be prime would change for every different modular class p could be in. but I know there's better sieves I don't know really anything about.

To quote the late Dorothy Parker: "What fresh hell is this?"

One modification, if it were possible and economical would be to ensure that all tested 2*k*P + 1 are actually prime before actually spending the time to do the division.

The upside is that many candidate factors are skipped to save time. The downside is composite factors are missed, and if the smaller prime factors weren't found in the first place, we would miss the composite factors too.

EDIT: I think this is the kind of thing the OP might have been after.