20210924, 09:36  #45 
Jun 2012
Boulder, CO
331 Posts 
For those who continue to believe that M1061 was factored by ECM: https://eprint.iacr.org/2012/444.pdf

20210924, 13:26  #46 
"Viliam FurÃk"
Jul 2018
Martin, Slovakia
683 Posts 
Okay, okay, sorry...

20210924, 14:10  #47  
Feb 2017
Nowhere
1001110010110_{2} Posts 
Quote:
In theory, it is possible, if the smaller factor of a P_{2} number N is larger than the cube root of N, to prove N is a P_{2} without factoring it, by doing TF up to the cube root of N. For N >= M_{1277} this is out of the question. And, of course, it is entirely possible for a P_{2} 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 M_{1277} to have more than two prime factors. At this point, the best available way to decide whether M_{1277} is a P_{2} 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 pieinthesky, 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 M_{p} to at least make an informed guess about the likelihood of M_{1277} having k prime factors for k = 2, 3, 4,... 

20210924, 16:14  #48 
"Curtis"
Feb 2005
Riverside, CA
1001110010110_{2} Posts 
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? 
20210924, 17:56  #49 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
5^{2}×383 Posts 
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. 
20210924, 18:31  #50  
Random Account
Aug 2009
11110111111_{2} Posts 
Quote:


20210924, 20:24  #51 
1976 Toyota Corona years forever!
"Wayne"
Nov 2006
Saskatchewan, Canada
4777_{10} Posts 
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 NFECM results? 
20210924, 22:37  #52 
"Tucker Kao"
Jan 2020
Head Base M168202123
2×251 Posts 
Focus specifically on the possible factors between 2^400 to 2^450, the range which the possibility resulting in a semiprime exceeds 85% if at least 1 hidden factor is indeed there and the remaining cofactor is a prime too.
Last fiddled with by tuckerkao on 20210924 at 22:48 
20210924, 22:48  #53  
If I May
"Chris Halsall"
Sep 2002
Barbados
13^{2}·59 Posts 
Quote:
Those who know giggle in your general direction... Deal with it. 

20210924, 23:23  #54  
Apr 2012
Gracie on alert.
401 Posts 
Quote:
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 20210924 at 23:26 Reason: addendum 

20210925, 03:05  #55  
Feb 2017
Nowhere
2×23×109 Posts 
Quote:


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