Factorize a 129-digit number

[mersenneNoob]

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.

[mersenneNoob]

Maybe it has 45 digits or so in the first factor

[frmky]

It looks like you are perhaps trying to factor this number?

33765749444723236975938383549571694531728722961303548619829414070181324100^2+1

[mersenneNoob]

yes I am and I accidentally stumbled on this crap

[charybdis]

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.

[CRGreathouse]

Let s = 33765749444723236975938383549571694531728722961303548619829414070181324100 so your number is s^2 + 1.

Is there some special significance to s or s^2 + 1?

[retina]

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

[a1call]

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:

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

[a1call]

Originally Posted by a1call

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:

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.

[charybdis]

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

166447885293942400205064657834447096885456111524808377
1005632514199652040245190989819182186076324308468303514078693528739687994577

(I didn't find these myself, someone put them in factordb.)

Both factors are 1 mod 8.

[mersenneNoob]

Thanks for the new 54-digit factor!