Predict the number of digits from within the factor for M1277

sweety439 · 2019-10-06, 00:52

With the smaller factor of M1061 found, let's guess that of M1277 (predict the number of digits)

a1call · 2019-10-06, 03:49

What is the Mersenne number with a prime exponent which has the highest known number of prime factors?

Batalov · 2019-10-06, 05:20

Quote:

Originally Posted by a1call

What is the Mersenne number with a prime exponent which has the highest known number of prime factors?

https://www.mersenne.ca/manyfactors.php

a1call · 2019-10-06, 19:41

Fantastic query.
Thank you.

LaurV · 2019-10-09, 08:16

We had a similar guess thread for M1061 where we (royal we) hit the "3 factors" cdot. We were far away in the woods, and at the time we voted we knew that we work against the probability, but it was "cute" to have a 3-way split, and not many people voted it, so that is why. With how much ECM was done here, and how many prime candidates for the factor (i.e. there are more "large" primes than "small" primes, fighting for the "largest factor" honours), the best chances is still a split which is as much uneven as possible. That would mean the smallest factor somewhere at 120 digits or so. But we didn't vote yet. Still thinking about...

On the other hand, chances to have less than 100 digits are almost non-existent, as well as the chances for a 4-way split (it would mean that the smallest factor is under 96 digits, most probably under 75 or so, considering that we won't have a 4-brilliant).

Edit: voted... but OTOH, we are thinking to edit the poll to make the voter's name visible once you voted (as we had other polls in the past). Otherwise, how can I prove that I was right and you were wrong?

a1call · 2019-10-14, 19:01

Probably obvious to some but hopefully not to all here are some pointers which are absolutely of no use for factoring M1277.

* As with any Mersenne number Mq with a prime exponent, M1277 has at least one prime factor p where valuation(p-1,2) = 1

* As with any composite Mersenne numberMq with a prime exponent, M1277 has at least one factor a = Mq/p where valuation(a-1,2) > 1

* As with any composite Mersenne number Mq with a prime exponent, M1277 has at least one prime factor p and a factor a = Mq/p coprime to p where valuation(p^2-1,2) = valuation(a^2-1,2) > 2

* As with any composite Mersenne number Mq with a prime exponent, M1277 has at least one prime factor p and a factor a = Mq/p coprime to p where valuation((p.a)^2-1,2) = q+1 = 1278

Batalov · 2019-10-14, 21:42

Quote:

Originally Posted by LaurV

On the other hand, chances to have less than 100 digits are almost non-existent, as well as the chances for a 4-way split ...

We had a 4-way split just recently in the Fibonacci subproject
φ(1441)/φ(131)φ(11) primitive cofactor = p64 * p64 * p66 * p80
Not really a miss. Nearly a 4-brilliant :-)

There is a first time for anything.

Zhangrc · 2022-06-11, 11:04

I guess 160 digits (about 2^530). Might be larger than the factor of M1061.
(Seems that we haven't found a Mersenne prime factor larger than 2^500 yet)

View Poll Results: Predict the number of digits from within the factor for M1277
<=90 digits	1	5.00%
91-105 digits	0	0%
106-120 digits	0	0%
121-135 digits	7	35.00%
136-150 digits	2	10.00%
151-165 digits	2	10.00%
166-180 digits	3	15.00%
181-193 digits (since M1277 has 385 digits, it cannot be more than 193 digits)	1	5.00%
has 3 prime factors	3	15.00%
has >=4 prime factors	1	5.00%
Voters: 20. You may not vote on this poll

2019-10-06, 00:52	#1
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 110111111001₂ Posts	Predict the number of digits from within the factor for M1277 With the smaller factor of M1061 found, let's guess that of M1277 (predict the number of digits)

2019-10-09, 08:16	#5
LaurV Romulan Interpreter "name field" Jun 2011 Thailand 5×2,003 Posts	We had a similar guess thread for M1061 where we (royal we) hit the "3 factors" cdot. We were far away in the woods, and at the time we voted we knew that we work against the probability, but it was "cute" to have a 3-way split, and not many people voted it, so that is why. With how much ECM was done here, and how many prime candidates for the factor (i.e. there are more "large" primes than "small" primes, fighting for the "largest factor" honours), the best chances is still a split which is as much uneven as possible. That would mean the smallest factor somewhere at 120 digits or so. But we didn't vote yet. Still thinking about... On the other hand, chances to have less than 100 digits are almost non-existent, as well as the chances for a 4-way split (it would mean that the smallest factor is under 96 digits, most probably under 75 or so, considering that we won't have a 4-brilliant). Edit: voted... but OTOH, we are thinking to edit the poll to make the voter's name visible once you voted (as we had other polls in the past). Otherwise, how can I prove that I was right and you were wrong? Last fiddled with by LaurV on 2019-10-09 at 08:25

2019-10-14, 19:01	#6
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 100011100001₂ Posts	Probably obvious to some but hopefully not to all here are some pointers which are absolutely of no use for factoring M1277. * As with any Mersenne number Mq with a prime exponent, M1277 has at least one prime factor p where valuation(p-1,2) = 1 * As with any composite Mersenne numberMq with a prime exponent, M1277 has at least one factor a = Mq/p where valuation(a-1,2) > 1 * As with any composite Mersenne number Mq with a prime exponent, M1277 has at least one prime factor p and a factor a = Mq/p coprime to p where valuation(p^2-1,2) = valuation(a^2-1,2) > 2 * As with any composite Mersenne number Mq with a prime exponent, M1277 has at least one prime factor p and a factor a = Mq/p coprime to p where valuation((p.a)^2-1,2) = q+1 = 1278 Last fiddled with by a1call on 2019-10-14 at 19:26 Reason: mQ

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2019-10-06, 03:49	#2
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 2,273 Posts	What is the Mersenne number with a prime exponent which has the highest known number of prime factors?

2019-10-06, 19:41	#4
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 2,273 Posts	Fantastic query. Thank you.

2022-06-11, 11:04	#8
Zhangrc "University student" May 2021 Beijing, China 2·7·19 Posts	I guess 160 digits (about 2^530). Might be larger than the factor of M1061. (Seems that we haven't found a Mersenne prime factor larger than 2^500 yet)