20181103, 16:04  #1 
"Mihai Preda"
Apr 2015
2537_{8} Posts 
Smallest exponent for mersenne notfactored
Hi, how can I query, or find, the smallest exponents "p" for which M(p) has no known factors?
I.e. it is known that M(p) is composite (e.g. through an LL or PRP test), but no factor was found through TF or P1. 
20181103, 16:08  #2 
1976 Toyota Corona years forever!
"Wayne"
Nov 2006
Saskatchewan, Canada
2^{2}×3^{5}×5 Posts 
I think you want this

20181103, 20:28  #3  
"Mihai Preda"
Apr 2015
5^{3}×11 Posts 
Quote:
I'm surprised by how small those exponents are, without known factors. The smallest is 1277! I looked at the P1 already done. For 1277: https://www.mersenne.org/report_expo...o=1277&exp_hi= B1= 5 000 000 000 003 B2= 400 000 000 000 241 So somebody did a P1 firststage to 5*10^12, which is impressive! I wonder, is that residue (after the P1 firststage to high B1), the full 1277 bits of it, available somewhere? 

20181103, 20:50  #4  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 
Quote:
also TF of M1277 to 72 bits has as many k values as TF of 74000051 to 87 bits 

20181103, 20:52  #5  
Sep 2003
5·11·47 Posts 
Quote:
The previous low record holder was M1061. It was finally factored by NFS into P143*P177, that is, its smallest factor is 474 bits and P−1 would have required B1=5.87906e+15 and B2=1.13383e+98 (!!) Everyone looks at M1277 and imagines for a moment that they will be the hero. But that's not realistic. Edit: There is surprisingly little overlap between factors found by TF and factors found by P−1. Most factors found by TF have k that is far too unsmooth to be findable by P−1, and most factors found by P−1 have bit size that is far too large to be findable by TF. However, I suspect that there is much more overlap between the factors findable by P−1 and the factors findable with a very sustained and deep ECM effort. In all likelihood, there has been more ECM, to deeper limits, than what has been recorded on our ECM progress page, and that probably already rules out the usefulness of further P−1 testing. Last fiddled with by GP2 on 20181103 at 21:49 

20181103, 21:03  #6  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
17·349 Posts 
Quote:
Last fiddled with by kriesel on 20181103 at 21:06 

20181103, 21:35  #7 
Sep 2003
5·11·47 Posts 
Offtopic
Here's something that we rarely think about: if our sole goal is to find new Mersenne primes, then factoring is actually not that significant at all.
When you do trialfactoring, the difficulty doubles every time you increment the number of bits. Exponentially increasing difficulty, so it's a very rapid transition from ridiculously easy to impossibly hard, with only a narrow transition zone. With P−1 it's a similar story: you can find a lot of factors with ridiculously small B1 and B2, but then there are rapidly diminishing returns. So you have tons of lowhanging fruit (which was picked many years ago) and tons of highhanging fruit (which we will never ever reach), and only a relatively small amount of middlehanging fruit. About 55 to 60% of exponents have lowhanging fruit, about 30 to 35% have only highhanging fruit. That leaves only 10% or so in the middlehanging segment. All of the tremendous TF factoring effort that is being done, not just now but into the indefinite future... will only ever find factors for a singledigit percentage of exponents, even under the most ideal circumstances. Imagine for a moment if there had never been any factoring effort beyond the lowhanging fruit: just very basic TF to low bit depth and very basic P−1 with small limits. How much of a setback would that be for GIMPS, how much further behind would we be? The answer is, maybe only 5% behind, compared to where we actually are in presentday reality. In all likelihood we would already have discovered all of the 50 known Mersenne primes except possibly the most recent one, which might have taken an extra year or so to discover. Yes, factoring is worth it if it saves more effort than two LL tests (though it's not clear that the current overkill deep TF work still meets those criteria), and factors are very cool in their own right, I search for them myself. A factored exponent is more secure knowledge than a primalitytested exponent, since verification can be done in a millionth of a second rather than weeks. But overall, factoring has much less of an impact on the progress of the GIMPS project than we often realize. 
20181103, 23:10  #8  
"Curtis"
Feb 2005
Riverside, CA
5^{2}·7·29 Posts 
Quote:
An SNFS factoring attempt on this number would require quite a lot of memory, so ECM to large bounds is our only hope in the next couple of years. Last fiddled with by VBCurtis on 20181103 at 23:11 

20181103, 23:17  #9 
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 

20181103, 23:21  #10 
Einyen
Dec 2003
Denmark
110010000100_{2} Posts 
Further trial factoring from where we are now might have "little" impact, but in total it is a pretty big impact.
Up to 100M: 3,735,132 of 5,761,455 exponents (64,8%) has at least 1 known factor. Up to 1000M: 29,489,324 of 50,847,534 exponents (58,0%) has at least 1 known factor. I'm not sure what the ratio is of factors found by TF, ECM, P1 is, and some of them was found after 1 or 2 LL test was already done, but it is still significant. But yes there is still at lot of TF, ECM and P1 going on below the double check limit which has zero impact on finding Mersenne Primes. Last fiddled with by ATH on 20181103 at 23:24 
20181104, 00:47  #11  
"GIMFS"
Sep 2002
Oeiras, Portugal
17·89 Posts 
Quote:
They ARE there, so it is sort of an irresistible thing to try and dig them up. 

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