New Mersenne Conjecture

ATH · 2014-03-30, 04:00

The New Mersenne (Prime) Conjecture is basically saying if a prime p is an exponent for both a Mersenne Prime (2^p-1) and a Wagstaff prime (2^p+1)/3 then p is of the form 2^k +/- 1 or 4^k +/- 3 where k is some natural number. The conjecture says that if 2 of these conditions is true then so is the third.

I tried submitting some updates a few times to the list here:
http://primes.utm.edu/mersenne/NewMe...onjecture.html
but it seems it is not being updated anymore.

So I made a new list myself: NMC.html

The conjecture corresponds to there being only 1 or 3 "yes" in each horizontal line. If 2 yes occurred the conjecture would be false.
The green lines are where the conjecture holds, and the 2 red lines are still unknown.

Here are the updates I made since the old list:
p=11213, p=216091, p=1398269: Added factors of (2^p+1)/3 found on factordb.com

p=65537: Added factor of 2^p-1 found on mersenne.org

p=986191, p=4031399, p=13347311, p=13372531: Added the 4 lines for the new Wagstaff primes with factors of 2^p-1 from mersenne.org

p=20966011, p=30402457, p=32582657, p=37156667, p=42643801, p=43112609, p=57885161:
Added the 7 latest Mersenne Primes. Found factors of 3 of the Wagstaff numbers with LLR, and factored the rest to 69-70 bits. Ran the Vrba-Reix test with LLR on the remaining 4 Wagstaff numbers proving them composite:
(2^20996011+1)/3 is not prime. Vrba-Reix RES64: BA75673D7BAB002F Time : 566688.538 sec.
(2^30402457+1)/3 is not prime. Vrba-Reix RES64: 9AD14D52DFCCB2A6 Time : 887355.947 sec.
(2^32582657+1)/3 is not prime. Vrba-Reix RES64: 99985D4C1756CE0F Time : 946154.777 sec.
(2^42643801+1)/3 is not prime. Vrba-Reix RES64: 90ACCFA9ADA98C0D Time : 1575885.158 sec.

p=268435459, p=1073741827: Added these 2 lines so the list roughly goes to p=1 billion like primenet v5. Found 1 factor of Wagstaff number with LLR and factored the other to 76 bits without success. Factored the 2 Mersenne numbers to 80 and 82 bits with mfaktc without success.

Brian-E · 2014-03-30, 09:42

Is the conjecture merely motivated by observation that it works for small p and the probability that two of the conditions (let alone three) are satisfied very quickly becomes vanishingly small? (And hence the conjecture is expected to hold in rather the same way that the conjecture of no Fermat primes beyond F₄ is expected.)

Or is it in fact conjectured that there are more occurrences of all three conditions holding beyond the last known case of p=127?

R.D. Silverman · 2014-03-30, 12:48

Quote:

Originally Posted by Brian-E

Is the conjecture merely motivated by observation that it works for small p and the probability that two of the conditions (let alone three) are satisfied very quickly becomes vanishingly small?

Yes. Furthermore, John indicated that he was not really serious
when he proposed it.

This "conjecture" is basically a joke.

axn · 2014-03-30, 14:24

I always thought that NMC was inspired by Mersenne's original conjecture list 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, which indeed picks out primes of the form 2^k+/-1 and 4^k+/-3 (except 61, which, while of the form 4^k-3 and in fact gives a Mersenne prime, was left out by Mersenne). Could be a coincidence though.

ewmayer · 2014-03-30, 21:31

Quote:

Originally Posted by axn

I always thought that NMC was inspired by Mersenne's original conjecture list 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, which indeed picks out primes of the form 2^k+/-1 and 4^k+/-3 (except 61, which, while of the form 4^k-3 and in fact gives a Mersenne prime, was left out by Mersenne). Could be a coincidence though.

One good joke deserves another.

science_man_88 · 2014-03-31, 01:15

Quote:

Originally Posted by ewmayer

One good joke deserves another.

Can we test the validity of it in this thread ? There's a pattern in the difference between (2^(2n+1) + 1)/3 and the next Mersenne number. Basically, A080674 with an extra 0 on the front. Could we test the validity of it based on this ? Admittedly, I have a portfolio to complete for school, so I might get around to it myself.

CRGreathouse · 2014-03-31, 13:45

Quote:

Originally Posted by R.D. Silverman

This "conjecture" is basically a joke.

Yes. It seems to be an example of a theorem which is true by coincidence. (I would love to be proved wrong here!)

Kathegetes · 2014-03-31, 21:27

Quote:

Originally Posted by ewmayer

One good joke deserves another.

Some think, weight givens, speak without reaching conclusion missing some step. Given; more can be known than is capable of being penetrated by mere mortal logic at some moment. I may have read Mersenne's list may be uncertain in regards to 61 or 67 ie, reading his hand writing. If you please to confirm the following... your friends may be pleased to wager conjectures entrusting all to TIME ; as she alone has proven herself a noble guardian of all entrusted unto her, whether mortal or divine. On the quantity of members of ( S ) less than a given term of G. Let K be a known prime such that 2^K-1 is prime D, such that D(D+1)/2 is some S, sum of her parts. Let G0= triangular radix of 8 = 3.5311... Zero S less than G0. G1=8, G2=36, G3=666, all terms of G follow by triangulation of 8. The members of S show an increase. Present knowledge of proper order of S being incomplete though perhaps correct up to G28. At G30 we delight in desire to know. K.D.S projects merely 53 as a divine statement. All such being feminine intuitive conjectures divined in TIME. When will 5 be proven in any term? Then 6,7,8...should mortals divine no logic in such trivial beauties?

R.D. Silverman · 2014-03-31, 22:49

Quote:

Originally Posted by Kathegetes

Some think, weight givens, speak without reaching conclusion missing some step. Given; more can be known than is capable of being penetrated by mere mortal logic at some moment. I may have read Mersenne's list may be uncertain in regards to 61 or 67 ie, reading his hand writing. If you please to confirm the following... your friends may be pleased to wager conjectures entrusting all to TIME ; as she alone has proven herself a noble guardian of all entrusted unto her, whether mortal or divine. On the quantity of members of ( S ) less than a given term of G. Let K be a known prime such that 2^K-1 is prime D, such that D(D+1)/2 is some S, sum of her parts. Let G0= triangular radix of 8 = 3.5311... Zero S less than G0. G1=8, G2=36, G3=666, all terms of G follow by triangulation of 8. The members of S show an increase. Present knowledge of proper order of S being incomplete though perhaps correct up to G28. At G30 we delight in desire to know. K.D.S projects merely 53 as a divine statement. All such being feminine intuitive conjectures divined in TIME. When will 5 be proven in any term? Then 6,7,8...should mortals divine no logic in such trivial beauties?

What kind of drugs are you taking?

NBtarheel_33 · 2014-04-01, 00:46

Quote:

Originally Posted by R.D. Silverman

What kind of drugs are you taking?

Or perhaps (of the psychotropic variety) *not* taking?

TheMawn · 2014-04-01, 04:19

Quote:

Originally Posted by NBtarheel_33

Or perhaps (of the psychotropic variety) *not* taking?

I must be not taking the same drugs. I couldn't even follow it.

2014-03-30, 04:00	#1
ATH Einyen Dec 2003 Denmark 19×181 Posts	New Mersenne Conjecture The New Mersenne (Prime) Conjecture is basically saying if a prime p is an exponent for both a Mersenne Prime (2^p-1) and a Wagstaff prime (2^p+1)/3 then p is of the form 2^k +/- 1 or 4^k +/- 3 where k is some natural number. The conjecture says that if 2 of these conditions is true then so is the third. I tried submitting some updates a few times to the list here: http://primes.utm.edu/mersenne/NewMe...onjecture.html but it seems it is not being updated anymore. So I made a new list myself: NMC.html The conjecture corresponds to there being only 1 or 3 "yes" in each horizontal line. If 2 yes occurred the conjecture would be false. The green lines are where the conjecture holds, and the 2 red lines are still unknown. Here are the updates I made since the old list: p=11213, p=216091, p=1398269: Added factors of (2^p+1)/3 found on factordb.com p=65537: Added factor of 2^p-1 found on mersenne.org p=986191, p=4031399, p=13347311, p=13372531: Added the 4 lines for the new Wagstaff primes with factors of 2^p-1 from mersenne.org p=20966011, p=30402457, p=32582657, p=37156667, p=42643801, p=43112609, p=57885161: Added the 7 latest Mersenne Primes. Found factors of 3 of the Wagstaff numbers with LLR, and factored the rest to 69-70 bits. Ran the Vrba-Reix test with LLR on the remaining 4 Wagstaff numbers proving them composite: (2^20996011+1)/3 is not prime. Vrba-Reix RES64: BA75673D7BAB002F Time : 566688.538 sec. (2^30402457+1)/3 is not prime. Vrba-Reix RES64: 9AD14D52DFCCB2A6 Time : 887355.947 sec. (2^32582657+1)/3 is not prime. Vrba-Reix RES64: 99985D4C1756CE0F Time : 946154.777 sec. (2^42643801+1)/3 is not prime. Vrba-Reix RES64: 90ACCFA9ADA98C0D Time : 1575885.158 sec. p=268435459, p=1073741827: Added these 2 lines so the list roughly goes to p=1 billion like primenet v5. Found 1 factor of Wagstaff number with LLR and factored the other to 76 bits without success. Factored the 2 Mersenne numbers to 80 and 82 bits with mfaktc without success. Last fiddled with by ATH on 2014-03-30 at 04:03

2014-03-30, 09:42	#2
Brian-E "Brian" Jul 2007 The Netherlands 2·11·149 Posts	Is the conjecture merely motivated by observation that it works for small p and the probability that two of the conditions (let alone three) are satisfied very quickly becomes vanishingly small? (And hence the conjecture is expected to hold in rather the same way that the conjecture of no Fermat primes beyond F₄ is expected.) Or is it in fact conjectured that there are more occurrences of all three conditions holding beyond the last known case of p=127? Last fiddled with by Brian-E on 2014-03-30 at 09:43

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2014-03-30, 14:24	#4
axn Jun 2003 2×2,719 Posts	I always thought that NMC was inspired by Mersenne's original conjecture list 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, which indeed picks out primes of the form 2^k+/-1 and 4^k+/-3 (except 61, which, while of the form 4^k-3 and in fact gives a Mersenne prime, was left out by Mersenne). Could be a coincidence though.