mersenneforum.org Factorize a 129-digit number
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 2021-04-09, 07:26 #1 mersenneNoob   "Nigel" Apr 2021 178 Posts Factorize a 129-digit number Code: 167385405371362584782846408982573842467658758597134437038484622185119073107121016763971211022426995195588062521818021967340171529 is a 129 digit number and I have tried to factor it using ECM efforts, but it wouldn't bulge. If you use the calculator at alpertron.com it won't produce the factors quickly, because I am already at the curve 2150 and no factors were produced. May someone help factor this. Last fiddled with by Dr Sardonicus on 2021-04-23 at 13:54
 2021-04-09, 07:47 #2 mersenneNoob   "Nigel" Apr 2021 1510 Posts First factor estimate Maybe it has 45 digits or so in the first factor
 2021-04-09, 07:50 #3 frmky     Jul 2003 So Cal 32·233 Posts It looks like you are perhaps trying to factor this number? Code: 33765749444723236975938383549571694531728722961303548619829414070181324100^2+1
 2021-04-09, 07:52 #4 mersenneNoob   "Nigel" Apr 2021 F16 Posts yes I am and I accidentally stumbled on this crap
 2021-04-09, 12:55 #5 charybdis     Apr 2020 2×7×19 Posts Normally when I stumble across some crap I want to clean it up, not find its factors. But if you desperately want to know the factors of this number, the best way is to find them yourself. There's some excellent advice in this thread, particularly post #8, assuming you're running Windows. A 129-digit number should take less than a day to factor using yafu+ggnfs on a modern quad-core.
 2021-04-09, 19:35 #6 CRGreathouse     Aug 2006 32×5×7×19 Posts Let s = 33765749444723236975938383549571694531728722961303548619829414070181324100 so your number is s^2 + 1. Is there some special significance to s or s^2 + 1?
2021-04-09, 20:23   #7
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

6,143 Posts

Quote:
 Originally Posted by CRGreathouse Let s = 33765749444723236975938383549571694531728722961303548619829414070181324100 so your number is s^2 + 1.
Hmmm?

Your number ends in ...00
Squaring +1 gives ...0001
The number in #1 ends in ...1529

 2021-04-09, 21:44 #8 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 23·11·23 Posts Code: 33765749444723236975938383549571694531728722961303548619829414070181324100^2+1 = 6811381392746490569 x 167385405371362584782846408982573842467658758597134437038484622185119073107121016763971211022426995195588062521818021967340171529 I have no idea how frmky could have figured that out or what the significance of any of the numbers are. ETA: PARI-GP code: Code: factor(6811381392746490569 ) %8 = [ 109 1] [ 23917 1] [2612774911073 1] ETA II One thing you can be sure of is that none of the prime factors p of OP-Number will be of the form 2n+1 where n is an odd integer, so you can rule out the infinite set 3, 7, 11,19, ....as candidates Last fiddled with by a1call on 2021-04-09 at 22:13
2021-04-09, 22:50   #9
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

23·11·23 Posts

Quote:
 Originally Posted by a1call Code: 33765749444723236975938383549571694531728722961303548619829414070181324100^2+1 = 6811381392746490569 x 167385405371362584782846408982573842467658758597134437038484622185119073107121016763971211022426995195588062521818021967340171529 I have no idea how frmky could have figured that out or what the significance of any of the numbers are. ETA: PARI-GP code: Code: factor(6811381392746490569 ) %8 = [ 109 1] [ 23917 1] [2612774911073 1] ETA II One thing you can be sure of is that none of the prime factors p of OP-Number will be of the form 2n+1 where n is an odd integer, so you can rule out the infinite set 3, 7, 11,19, ....as candidates
ETA III You can also be sure that at least one of the prime factors of OP-Number will be of the form 4*n+1 where n is an odd integer.

2021-04-09, 23:14   #10
charybdis

Apr 2020

2·7·19 Posts

Quote:
 Originally Posted by a1call ETA III You can also be sure that at least one of the prime factors of OP-Number will be of the form 4*n+1 where n is an odd integer.
Wrong.

The factors of OP's number are
Code:
166447885293942400205064657834447096885456111524808377
1005632514199652040245190989819182186076324308468303514078693528739687994577
(I didn't find these myself, someone put them in factordb.)

Both factors are 1 mod 8.

 2021-04-10, 01:44 #11 mersenneNoob   "Nigel" Apr 2021 3·5 Posts Thanks Thanks for the new 54-digit factor!

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