20050523, 19:51  #1 
May 2005
3C_{16} Posts 
Factorization attempt to a c163  a new Odd Perfect Number roadblock
Currently, I am attempting to prove that an Odd Perfect Number must have at least 9 distinct prime factors (I did 8 in 1979). I need to get just one factor out of a c163 from the L Aurifeillian of 17^2891 or alternatively prove that c163 has at most 4 factors (i.e. to show that c163 has no factor less than c163^(1/5)) by a nonprobabilistic algorithm.
Note: If someone can prove that the c163 has at most 3 factors that would be even better for my work. I just completed the 2614th curve at the 35 digits level. The completed factorization of (17^2891)/16 is Code:
10949 * 1749233 * 2699538733 * 8093 * 202879 * 21366108595042598039019343 * 1268289320653338013663048107658421895784820383933483845536741745292852579037241336594064422676519255617544021091795698651078540915868211167 * 1080976398676005497687700789001811475732289777254501989753747835521426921182991116146472692199375816076769108716604114107484303070529988526405561481325009425864203 (163digit composite). Code:
Factoring 1080976398676005497687700789001811475732289777254501989753747835521426921182991116146472692199375816076769108716604114107484303070529988526405561481325009425864203 (163 digits) Limit (B1=11000000; B2=1100000000) Curve 2614 Digits in factor: >= 15 >= 20 >= 25 >= 30 >= 35 >= 40 Probability: 100% 100% 100% 100% 99% 40% Joseph E.Z. Chein Last fiddled with by akruppa on 20050524 at 20:24 Reason: added CODE tags to avoid horizontal scrolling 
20050523, 20:20  #2  
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
2^{2}·7·383 Posts 
Quote:
I think I'd want to see ECM work done to around the p50 level before recommending it be done by GNFS. Paul 

20050523, 20:51  #3 
"Phil"
Sep 2002
Tracktown, U.S.A.
3·373 Posts 
So did the 139 digit number come from some sort of Aurifeuillean factorization? What are the polynomials of those factors? I don't know, but I'm sure that others will, whether or not SNFS might be of use here.

20050523, 21:16  #4  
Nov 2003
2^{2}·5·373 Posts 
Quote:
It is either A + B or AB with A,B polynomials with coefficients: A : (1, 9, 11, 5, 15, 5, 11, 9, 1) (degree 8 in x = 17^H) B: 17^K(1, 3, 17, 3, 3, 17, 3, 1) (degree 7 in 17^H) with H = 2K1 , H = 17 You should be able to reduce this octic to a quartic in (x+17/x). I don't have Macsyma or Maple to do the manipulation, and I would hate to do it by hand. A C163 via SNFS is quite easy. 

20050523, 22:00  #5 
May 2003
11000001011_{2} Posts 
Joseph,
I took a look at this problem a few summers ago. I'm glad someone is finally tackling it! It is about time. When I looked at it, I was just going to show that 3 and 5 both had to divide an odd perfect number of 8 factors. But a friend of mine already did it (he is a fellow grad student here at Berkeleybut didn't publish that result). We were thinking about eventually trying to do the whole shibang, but never got around to it. I'm currently using my computer to do another SNFS factorization, but once that is done (should stop tomorrow) I'd be willing to take a crack at your number if you'd like (or if nobody else gets to it first). By the way, I'm wondering why you even need to factor that number. It looks like [17^289 1]/16 already has at least 8 factors. Thus any OPN number N with 17  N and [17^2891]/16  N must have at least 9 factors. So why do you need to factor this number? Best, Pace Nielsen 
20050523, 22:03  #6 
Jul 2004
Potsdam, Germany
3·277 Posts 
I did a P1 run with B1=1100000000 (11e8), B2=593887176568 (gmpecm default)  no factor found.
More is not wise with 1 GB RAM... 
20050524, 00:22  #7 
"Phil"
Sep 2002
Tracktown, U.S.A.
3·373 Posts 
I'm not getting the Aurifeuillian factorization to work out. I should be getting A^2B^2 to be equal to (17^2891)/(17^171), right? I get the first 50 or so digits the same, but after that, the agreement ends. I'm not sure if I am misinterpreting something or the coefficients aren't quite right.
A and B are symmetric polynomials, but could the large coefficient 17^9 multiplying B cause problems? 202879 is the only known small factor of this one, so if SNFS is the way to go, it will be a 168digit factorization. 
20050524, 05:35  #8  
"Phil"
Sep 2002
Tracktown, U.S.A.
45F_{16} Posts 
Quote:
B: 17^K(1, 3, 1, 3, 3, 1, 3, 1) (degree 7 in 17^H) with H = 2K1 , H = 17 Now I get A^2  B^2 coming out correctly. I am having trouble reducing the octic to a quartic, though. The octic is A +/ B, and even though A and B are symmetric, because they are of different degrees, the sum or difference is not. Ideas, anyone? 

20050524, 06:01  #9 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
That's exactly where I got stuck last night.
Bob's suggestion was to not do a polynomial in (1+1/x), which would require the coefficients to be symmetric, but in (1+17/x). Unfortunately, I didn't manage to make a quartic this way, either  the coefficients never matched up. The octic would be useless for sieving  the coefficients are quite large (not that much smaller than they'd be for GNFS) and the degree too high... Alex Last fiddled with by akruppa on 20050524 at 06:02 
20050524, 10:05  #10 
Apr 2004
Copenhagen, Denmark
2^{2}·29 Posts 
Can someone with Maple experience please explain the best way to manipulate polynomial expressions like that in Maple. I've been fiddleing with the algsubs command, but it doesn't seem to do what I want.
I've tried playing around with the 12'th degree polynomial you get by giving Maple the command normal(x^13+1)/(x+1); This can be reduced to degree 6. This I know how to do. What I don't know is how to make Maple do it. If I multiply by x^(6) to get the symmetric polynomial I thought of something like Code:
algsubs(1/x+x=y,x^12x^11+x^10x^9+x^8x^7+x^6x^5+x^4x^3+x^2x+1); What do I do?  Cheers, Jes 
20050524, 13:35  #11  
Nov 2003
1110100100100_{2} Posts 
Quote:
computations by hand........ Unfortunately, I don't have any CA software available. Try expressing the octic as a quartic in (ax + b/x) for some a,b. I'm not sure if this will work. Maybe Peter Montgomery has some ideas; he is much more intuitive about such manipulations than I am.... You could also try a quartic in (a * 17^c + b * 17^d) for different c,d. 

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