mersenneforum.org  

Go Back   mersenneforum.org > Extra Stuff > Miscellaneous Math

Reply
 
Thread Tools
Old 2011-07-04, 16:28   #1
firejuggler
 
firejuggler's Avatar
 
Apr 2010
Over the rainbow

97616 Posts
Default Mersenne, another question

First of all, sorry, i don't know how-to Latex.

Let p1 and p2 be random prime
Let M(p1) and M(p2) be composite mersenne
Let C be a number, factor of M(p1) or M(p2)>0
As we know, factor of M(P) are in the form of (2*k*P)+1
Does a couple (M(p1),M(p2)) exist where the factor of p2 is aqual to (((2*k*p1)*p2) +1)*C

Last fiddled with by firejuggler on 2011-07-04 at 16:29
firejuggler is online now   Reply With Quote
Old 2011-07-04, 16:43   #2
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

26×131 Posts
Default

Quote:
Originally Posted by firejuggler View Post
First of all, sorry, i don't know how-to Latex.

Let p1 and p2 be random prime
Let M(p1) and M(p2) be composite mersenne
Let C be a number, factor of M(p1) or M(p2)>0
As we know, factor of M(P) are in the form of (2*k*P)+1
Does a couple (M(p1),M(p2)) exist where the factor of p2 is aqual to (((2*k*p1)*p2) +1)*C
ctan.org helps and so does the tex tags.

Let p1 and p2 be random prime
Let Mp1 and Mp2 be composite mersenne
Let C be a number, factor of Mp1 or Mp2>0
As we know, factor of MP are in the form of (2*k*P)+1
Does a couple (Mp1,Mp2) exist where the factor of p2 is equal to (((2*k*p1)*p2) +1)*C and this didn't need anything but sub tags. and a underline.
science_man_88 is offline   Reply With Quote
Old 2011-07-04, 16:51   #3
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

26×131 Posts
Default

Quote:
Originally Posted by science_man_88 View Post
ctan.org helps and so does the tex tags.

Let p1 and p2 be random prime
Let Mp1 and Mp2 be composite mersenne
Let C be a number, factor of Mp1 or Mp2>0
As we know, factor of MP are in the form of (2*k*P)+1
Does a couple (Mp1,Mp2) exist where the factor of p2 is equal to (((2*k*p1)*p2) +1)*C and this didn't need anything but sub tags. and a underline.
now to answer the question to be of the form 2*t*p2+1 t = k*p1 okay so this then becomes a matter of the mod 3 remainder of t to determine what p2 could be. because for p2=6n-1 t = 1 or 0 mod 3 will work to give a 6n+1 Mersenne number, 6n+1 t= 2 or 0 mod 3 works. oddly enough I was looking at these relations last night by hand for other reasons.

Last fiddled with by science_man_88 on 2011-07-04 at 16:59
science_man_88 is offline   Reply With Quote
Old 2011-07-04, 17:07   #4
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts
Default

Quote:
Originally Posted by science_man_88 View Post
now to answer the question to be of the form 2*t*p2+1 t = k*p1 okay so this then becomes a matter of the mod 3 remainder of t to determine what p2 could be. because for p2=6n-1 t = 1 or 0 mod 3 will work to give a 6n+1 Mersenne number, 6n+1 t= 2 or 0 mod 3 works. oddly enough I was looking at these relations last night by hand for other reasons.
okay that wasn't quite an answer I know , but I'm not a number theory person, R.D. Silverman likely has a clue or more extra to help.
science_man_88 is offline   Reply With Quote
Old 2011-07-04, 18:08   #5
Christenson
 
Christenson's Avatar
 
Dec 2010
Monticello

5·359 Posts
Default

I expect (warning: naively) such a number to exist....you might want to look among all the factored Mersenne numbers to see if there is a couple like that.
Christenson is offline   Reply With Quote
Old 2011-07-04, 19:26   #6
Mini-Geek
Account Deleted
 
Mini-Geek's Avatar
 
"Tim Sorbera"
Aug 2006
San Antonio, TX USA

17×251 Posts
Default

Quote:
Originally Posted by firejuggler View Post
First of all, sorry, i don't know how-to Latex.

Let p1 and p2 be random prime
Let M(p1) and M(p2) be composite mersenne
Let C be a number, factor of M(p1) or M(p2)>0
As we know, factor of M(P) are in the form of (2*k*P)+1
Does a couple (M(p1),M(p2)) exist where the factor of p2 is aqual to (((2*k*p1)*p2) +1)*C
Not sure what you're trying to get at with (((2*k*p1)*p2) +1)*C. Where C is a factor of M(px), let P=px. That final expression is equivalent to:

(2k*p1*p2+1)*(2kP+1)
=2Pk*2k*p1*p2+2kP+2k*p1*p2+1
=4Pk2*p1*p2+2k*p1*p2+2kP+1
=(2*p1*p2)(2Pk2+k)+2kP+1

I don't think this can be a factor of p2 unless P=p2, since otherwise it wouldn't necessarily be of the form 2kp2+1. That means that C is a factor of p2, not p1, and so p1 hardly enters into the equation.

But I think you're really asking this:
Is there a number of the form 2*k*p1*p2+1 that divides both M(p1) and M(p2)?
I'm with Christenson: my gut says yes, and I don't know of anything preventing the possibility, but I don't know of any specific examples. I'm going to try some brute force searching real quick to see if I can find any examples...
Mini-Geek is online now   Reply With Quote
Old 2011-07-04, 20:25   #7
Mini-Geek
Account Deleted
 
Mini-Geek's Avatar
 
"Tim Sorbera"
Aug 2006
San Antonio, TX USA

17×251 Posts
Default

Quote:
Originally Posted by Mini-Geek View Post
I'm with Christenson: my gut says yes, and I don't know of anything preventing the possibility, but I don't know of any specific examples. I'm going to try some brute force searching real quick to see if I can find any examples...
After searching all combinations where p1 < p2 < 500, k < 10000, and up to p < 1000 with k < 1000, and finding no results, I'm no longer so sure this happens. If it does, it would seem to be rather rare. Of course, it's possible that I've done something wrong with the code. The speed (or lack thereof) makes me not want to go farther with this particular implementation (very simple Java - doesn't even check if the potential factor is +-1 mod 8 or prime, just does a BigInteger.mod on everything). It's finding enough factors that I don't have much reason to doubt it, but you never know.
Edit: Will Edgington's list of Mersenne factors contains no prime twice. I find it unlikely this is a coincidence, as the list starts out quite dense: it includes 143 of the 168 primes below 1000.
Edit 2: Just found on http://www.garlic.com/~wedgingt/mersenne.html, "Lemma 1: A prime number divides at most one prime-exponent Mersenne" (with proof)
So the answer to "Is there a number of the form 2*k*p1*p2+1 that divides both M(p1) and M(p2)?" is no.
I never noticed this fact before, though I have visited W.E.'s pages in the past.

Last fiddled with by Mini-Geek on 2011-07-04 at 21:00
Mini-Geek is online now   Reply With Quote
Old 2011-07-04, 20:27   #8
firejuggler
 
firejuggler's Avatar
 
Apr 2010
Over the rainbow

242210 Posts
Default

what i was meaning is that
M(p1) has a factor of (2*k*p1)+1 and
M(p2) has a factor of (2*k*p1*p2)+1
I was using P as a 'general case
the C i used is a non-specific cofactor
firejuggler is online now   Reply With Quote
Old 2011-07-04, 21:59   #9
R.D. Silverman
 
R.D. Silverman's Avatar
 
Nov 2003

22×5×373 Posts
Default

Quote:
Originally Posted by Mini-Geek View Post

But I think you're really asking this:
Is there a number of the form 2*k*p1*p2+1 that divides both M(p1) and M(p2)?
I'm with Christenson: my gut says yes, and I don't know of anything preventing the possibility, ...
I do. It is an elementary exercize that might be given as a homework
problem in a first year number theory class.
R.D. Silverman is offline   Reply With Quote
Old 2011-07-04, 22:09   #10
R.D. Silverman
 
R.D. Silverman's Avatar
 
Nov 2003

22·5·373 Posts
Default

Quote:
Originally Posted by R.D. Silverman View Post
I do. It is an elementary exercize that might be given as a homework
problem in a first year number theory class.
I will even give a hint:

Ask: what is GCD(2^a-1, 2^b-1)????

When are you people going to stop prattling and making meaningless
"conjectures" and pick up and READ some number theory books????

The question under discussion is well within the scope of any 1st year
number theory book.
R.D. Silverman is offline   Reply With Quote
Old 2011-07-04, 22:30   #11
firejuggler
 
firejuggler's Avatar
 
Apr 2010
Over the rainbow

2×7×173 Posts
Default

why do you think i put it under 'Miscellaneous Math Threads' ? i knew i was risking your anger would it be under ' Math'
firejuggler is online now   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Question about Mersenne divisors paulunderwood Miscellaneous Math 1 2016-01-24 01:41
Mersenne theorems question ShiningArcanine Math 21 2012-04-27 01:38
ECM question for mersenne numbers LaurV Math 11 2012-03-16 12:10
Question about Mersenne Numbers rong123 Marin's Mersenne-aries 7 2007-11-09 00:34
odd divisors of Mersenne-like, question stpascu Factoring 1 2006-10-16 16:31

All times are UTC. The time now is 22:55.

Wed Sep 30 22:55:18 UTC 2020 up 20 days, 20:06, 0 users, load averages: 1.54, 1.69, 1.64

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.