20101011, 17:21  #12 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
2^{2}·11·97 Posts 
Even faster with B1=34631 and no stage 2. This 78 bit prime factor can be found in ~95 seconds on one core of a modern CPU. Pretty incredible.

20101011, 17:58  #13  
Sep 2010
Scandinavia
3·5·41 Posts 
Quote:
I'm doing a large number of P1 in that exponent area with B1~ double the above, and no stage 2. Might add on stage 2 later. But as low as the limits are on some exponents in the area I think this might be a fairly effective approach. On the other end we have M234473 with its 2kp+1=750020570278149061867959407 k= 41*234,473*161,503,549*241,537,413,779 Good luck finding that with P1! Would be nice to hear other more or less extreme examples, maybe that has/deserves a thread of its own. Last fiddled with by lorgix on 20101011 at 18:00 Reason: typo 

20101012, 06:49  #14 
Sep 2010
Scandinavia
3×5×41 Posts 
My smallest factor yet;
M2428781 has a factor: 5404906126626057367 k=3*7*89*7583*78509 Evaded previous TF by 1.2bits. Another one; M2428661 has a factor: 1082894272153874474993 k=2*2*2*13553*22307*92177 
20101013, 07:05  #15 
Sep 2010
Scandinavia
3·5·41 Posts 
More cases where the largest prime factor of k is relatively small:
M2430457 has a factor: 113034265381620576607487  77bit k= 17*541*4987*14303*35447 M4538137 has a factor: 326333958725131675555063  79bit k= 3*7*109*15271*20509*50153 Is my intuition right, or are cases like these actually quite common? 
20101013, 08:57  #16 
Aug 2002
Termonfeckin, IE
2^{2}×691 Posts 
<silverman> Read my paper with Wagstaff.</silverman>.

20101016, 12:47  #17  
"Oliver"
Mar 2005
Germany
11×101 Posts 
Hi,
Quote:
M49915309 has a factor: 2085962683046854861393 (70.82 Bits; k = 20895019231944 = 2 * 2 * 2 * 3 * 11 * 13 * 37 * 227 * 347 * 2089) M50739071 has a factor: 10474816683392115991831 (73.14 Bits; k = 103222393285365 = 3 * 3 * 5 * 19 * 181 * 227 * 769 * 3821) M51027377 has a factor: 77850684812475802805663 (76.04 Bits; k = 762832516479103 = 7 * 17 * 139 * 191 * 239 * 257 * 3931) M51679921 has a factor: 451068222670482355938121 (78.57 Bits; k = 4364056812997860 = 2 * 2 * 3 * 5 * 11 * 307 * 953 * 2957 * 7643) M53196851 has a factor: 129375114189794147350126111 (86.74 Bits; k = 1216003501690298805 = 3 * 5 * 11 * 17 * 71 * 79 * 2113 * 3571 * 10243) M51007903 has a factor: 416373044176966390884620641 (88.42 Bits; k = 4081456202747311440 = 2 * 2 * 2 * 2 * 3 * 3 * 3 * 5 * 13 * 59 * 89 * 563 * 4441 * 11071) M51094921 has a factor: 620135167283167713317314151 (89.00 Bits; k = 6068461944418778075 = 5 * 5 * 271 * 2851 * 3371 * 8429 * 11057) M51139447 has a factor: 1873562419055481575542379948831 (100.56 Bits; k = 18318172457510946251945 = 5 * 7 * 19 * 23 * 73 * 359 * 18457 * 35597 * 69557) Oliver 

20101016, 13:35  #18 
Aug 2002
Buenos Aires, Argentina
1380_{10} Posts 
But notice the latest factors my computer found using ECM:
M400087 has a factor: 286218557414155282359049 (k = 2 ^ 2 x 3 x 41737 x 714185251333) M400277 has a factor: 2081610233687632912124807 (k = 11 x 107 x 2209186189633807) These factors could not have been discovered using P1. Last fiddled with by alpertron on 20101016 at 13:36 
20101016, 13:55  #19  
Sep 2010
Scandinavia
3·5·41 Posts 
Quote:
Quote:
Yes, both extremes exist. But I'm curious about how common cases like these are. A 3D (exponent size, number of factors of k, sizes of factors of k) graph or a distribution table would be extremely nice. But I doubt that's at hand. Any thought or info/ideas? 

20101016, 14:17  #20 
Apr 2010
Over the rainbow
2×3^{3}×7^{2} Posts 
not found by me but still
M52000043 has a factor : 104000087 (k=2) thats an extreme... 
20101016, 14:50  #21  
Sep 2010
Scandinavia
1001100111_{2} Posts 
Quote:
A little sample; these should be the known ones in the range. M33623 has 67247 M34283 has 68567 M34319 has 68639 M34439 has 68879 M34631 has 69263 M34883 has 69767 M35099 has 70199 M35111 has 70223 M35291 has 70583 M35831 has 71663 M35999 has 71999 And it goes on.... I think it's safe to say that k=2 is common for small p. But just how does the distribution look? I kinda feel like this info should be available somewhere. P.S. Smallest p for which k=2 11, 23, 83, 131, 179, 191, 239, 251, 359, 419, 431, 443 After looking those up manually I realize that this is A002515. 

20101016, 18:12  #22 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
1000010101100_{2} Posts 
These "k=2" you are talking about are really k=1. Factors are of the form 2kp+1. In other words, they're mp+1, with m always even. With these factors, m=2 and k=1, since the factor is equal to 2*p+1.
I'd bet that the factors of the k's break down, on average, like the factors of any natural number of about their size. And that the chance of any given k producing a factor is related to the equation given at http://www.mersenne.org/various/math.php: "(how_far_factored1) / (exponent times Euler's constant (0.577...))". I don't know what that means precisely as far as how many k's will be smooth to suchandsuch bounds, but the GIMPS Math page says "The chance of finding a factor and the factoring cost both vary with different B1 and B2 values. Dickman's function (see Knuth's Art of Computer Programming Vol. 2) is used to determine the probability of finding a factor, that is k is B1smooth or B1smooth with just one factor between B1 and B2. The program tries many values of B1 and if there is sufficient available memory several values of B2, selecting the B1 and B2 values that maximize the formula above." Last fiddled with by MiniGeek on 20101016 at 18:18 
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