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Old 2021-09-24, 09:36   #45
ryanp
 
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For those who continue to believe that M1061 was factored by ECM: https://eprint.iacr.org/2012/444.pdf
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Old 2021-09-24, 13:26   #46
Viliam Furik
 
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Okay, okay, sorry...
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Old 2021-09-24, 14:10   #47
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Quote:
Originally Posted by tuckerkao View Post
Hopefully, there will be a different type of PRP-like test to run someday in the future, then check whether a specific Mersenne exponent is a semi-prime or not.

M1061 is a certified semi-prime. M1277 will most likely be a semi-prime as well.
"Certified semi-prime?" What the heck is that supposed to mean? AFAIK the only way to "certify" a number N >= M1277 known to be composite, is a P2, is to factor it.

In theory, it is possible, if the smaller factor of a P2 number N is larger than the cube root of N, to prove N is a P2 without factoring it, by doing TF up to the cube root of N. For N >= M1277 this is out of the question. And, of course, it is entirely possible for a P2 number N to have one factor smaller than the cube root of N.

As already pointed out, there is plenty of room left after all effort to date, for M1277 to have more than two prime factors. At this point, the best available way to decide whether M1277 is a P2 is to factor it. And previous posts to this thread have already indicated the kind of effort that would be required to accomplish that.

Or are you merely engaging in wishful thinking about some pie-in-the-sky, computationally cheap "test" that would reveal the number of prime factors of a given number without actually factoring it?

If you actually devise such a method - complete with a proof that it works, of course - by all means get it published. But please, no nattering about nonexistent methods or ''tests."

I heartily endorse Batalov's suggestion: Use the available data on factors of Mp to at least make an informed guess about the likelihood of M1277 having k prime factors for k = 2, 3, 4,...
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Old 2021-09-24, 16:14   #48
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Quote:
Originally Posted by tuckerkao View Post
....It's extremely possible....
You're in a math forum, talking about something for which actual numerical probabilities are not hard to compute- that is Batalov's homework assignment for you, after all. So, you read the assignment, and decide that "extremely possible" is a more accurate thing to say than "most likely"?

Your high school writing instructor should have failed you for putting words like "extremely possible" together. You won't do your math homework, either. Tough to contribute to an academic forum this way, you know?
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Old 2021-09-24, 17:56   #49
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Quote:
Originally Posted by tuckerkao View Post
....It's extremely possible....
I ran some prep work, because that phrase bothered me too.

So - both in game and/or betting world and/or medical genetics field, and in statistics that underlies all them, there is quantification of odds (odds ratio).

Example: you say "extremely possible", and I offer $100 against your $300, is it wise for you to take this bet? "Extremely" usually would rather be like 10:1 in betting world so my offer of 3:1 should be good enough. no? Easy money that you are "extremely possibly" likely to get for free, right?

"Highly likely", "most likely" - what do these words mean? Totally different to different people.
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Old 2021-09-24, 18:31   #50
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Quote:
Originally Posted by petrw1 View Post
So if I may try to summarize briefly (and I have no issue being corrected).

Any more TF is completely out of the question; as is any more P-1.

Completing the current ECM up to 65 digits (which is almost half done) has a few percent chance of finding a factor.

ECM beyond that (i.e. 70 or 75 or 80 digits), which is NOT currently set up in Prime95 but has been attempted a bit by some, would take significantly longer but still with a small chance of finding a factor.

If the smallest factor is actually much larger than 65 digits than the only option is SNFS ... which I know nothing about, but the comments above suggest the world does NOT yet have a computer big enough or fast enough to factor this exponent in our lifetimes.

HOWEVER, in the meantime some members here are still actively running ECM on 1277 because there is still a better than 0.0% chance of it finding a factor.

Feel free to join them.
I experimented with this a couple of years ago. I devised an automated process which had GMP-ECM run it in small increments with many iterations in each step. I believe I started at 2e10. When it passed 2e18 I stopped. From what I read above, I never got close. Someone will eventually solve this. We do not have the necessary hardware to do it now. Anyone taking it on currently would find the process dreadfully slow. However, to those that are, I say, "Keep chewing away."
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Old 2021-09-24, 20:24   #51
petrw1
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Default Question on other related small exponents... for Ryan

For this middle section: ECM on Mersenne numbers with no known factors
Here: https://www.mersenne.org/report_ecm/...cmnof_hi=19999

There are several exponents that are notably lower in the ECM progress.
Example: 4363,4567,4583,4703.

I (or another curious lot) may want to tackle these "low hanging fruit" some day.
However, with the huge amount of ECM Ryan Propper did a few year ago finding dozens of factors; I find it curious that some of these show so much less ECM done.

I think only Ryan can answer this ... and no disrespect intended ...
Is there any chance he already did much more/deeper ECM on these and others in the low end but for whatever reason was unable to upload these NF-ECM results?
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Old 2021-09-24, 22:37   #52
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Quote:
Originally Posted by Batalov View Post
Ok,let's count this for half of the first step. You established the prior.I will even grant you that there are no expected factors below 2^212 (which is 1/6th of the size)
Now what?
Focus specifically on the possible factors between 2^400 to 2^450, the range which the possibility resulting in a semi-prime exceeds 85% if at least 1 hidden factor is indeed there and the remaining co-factor is a prime too.

Last fiddled with by tuckerkao on 2021-09-24 at 22:48
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Old 2021-09-24, 22:48   #53
chalsall
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Quote:
Originally Posted by tuckerkao View Post
Focus specifically on the possible factors between 2^400 to 2^450, the range which the possibility resulting in a semi-prime exceeds 85% if at least 1 hidden factor is indeed there.
Please don't take this the wrong way. But, demonstrably, there's just no "there" there.

Those who know giggle in your general direction...

Deal with it.
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Old 2021-09-24, 23:23   #54
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Quote:
Originally Posted by Batalov View Post
I ran some prep work, because that phrase bothered me too.

So - both in game and/or betting world and/or medical genetics field, and in statistics that underlies all them, there is quantification of odds (odds ratio).

Example: you say "extremely possible", and I offer $100 against your $300, is it wise for you to take this bet? "Extremely" usually would rather be like 10:1 in betting world so my offer of 3:1 should be good enough. no? Easy money that you are "extremely possibly" likely to get for free, right?

"Highly likely", "most likely" - what do these words mean? Totally different to different people.
Lol..
Being literal.."highly" is a directional term and "most" is comparative ..but compared to what. The enumeration of all possible outcomes where only one unique outcome can exist within any unique instant of "time" is a step in the right direction regarding extremes...and so too is the concept of a singularity...but then again..totally different meanings to different modes of comprehension.*

*https://en.wikipedia.org/wiki/Singularity

Last fiddled with by jwaltos on 2021-09-24 at 23:26 Reason: addendum
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Old 2021-09-25, 03:05   #55
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Quote:
Originally Posted by tuckerkao View Post
If the current human technology is able to discover the lower factor of M1061 which is merely 473.9 bits, but unable to do the same for M1277. It's extremely possible that the lower factor of M1277 is larger than that size.
<snip>
Quote:
Originally Posted by tuckerkao View Post
Focus specifically on the possible factors between 2^400 to 2^450
<snip>
You're free to waste your own time and money chasing after your ever-changing will-o'-the-wisps. Don't expect others to waste theirs.
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