20210828, 23:15  #12  
May 2018
C2_{16} Posts 
Quote:
I did not intend to create a separate thread or provide extensive details on a work in progress but rather to post a few hints and see if there are similar attempts for factorization of composite Mersenne numbers. One heuristic scheme I am currently testing and modifying is to use the factors of 2^{p1}1 (which are obviously coprime with Mp) as a factor base in a version of the rational sieve and proceed in a series of iterations in potentially finding factors of Mp. As pointed out in a previous post, the factors of 2^{p+1}1 are also known. This gives me an idea to attempt a hybrid rational sieve using a combined factor base. 

20210828, 23:32  #13  
May 2018
302_{8} Posts 
Quote:
https://mathworld.wolfram.com/GeometricSeries.html https://www.wolframalpha.com/input/?i=x%5E111 

20210829, 00:33  #14 
"Tucker Kao"
Jan 2020
Head Base M168202123
751_{8} Posts 
2^{2^n} + 1 have the higher chance to be a prime.
I always wondering why nobody is testing 3^{n}  2 or 3^{n} + 2. Last fiddled with by tuckerkao on 20210829 at 00:34 
20210829, 00:36  #15 
"Viliam Furík"
Jul 2018
Martin, Slovakia
2^{3}·5·17 Posts 

20210829, 00:40  #16  
"Tucker Kao"
Jan 2020
Head Base M168202123
1E9_{16} Posts 
Quote:
x^{2^(p+1)}1 = Maybe a Prime x^{2^(p1)}1 = Always composite, p > 3 x^{2^p+1}1 = Always composite, p ≥ 3 x^{2^p1}1 = Maybe a Prime Last fiddled with by tuckerkao on 20210829 at 01:10 

20210829, 05:52  #17 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2×3^{2}×349 Posts 

20210829, 13:57  #18 
Feb 2017
Nowhere
1001101111011_{2} Posts 

20210829, 16:10  #19  
Feb 2017
Nowhere
4,987 Posts 
Quote:
Quote:


20210829, 18:29  #20 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3^{2}×239 Posts 
FWIW, for odd positive integers a, b & n = a*b, there are relationships between the factors of n1 and factors of n which can be used to make the solutionspace for factors of n smaller. However this reduction may or may not be marginal. Unfortunately for large composites it often (but not always) is.
For example: GCD(a1,n1) == GCD(b1,n1) always holds true. Example: 65341= 361*181 => gcd(653411, 3611) == gcd(653411, 1811) == 180 => 65341 == (2*180+1) * (1*180+1) PariGP code: Code:
\\EDE100A Relevance of factors of n1 to factors of n by Rashid Naimi 2021/8/29 \\ The following PariGP code shows that for odd positive integers a, b & n= a*b, GCD(a1,n1) == GCD(b1,n1) runningN = 1 runningGcd = 1 theExp = 2 \\determines the upper range for both a & b forstep(a=3,19^theExp ,2,{ forstep(b=3,19^theExp ,2, n=a*b; ag=gcd(a1,n1); bg=gcd(b1,n1); if(ag > runningGcd && !ispower(n), runningN =n; runningGcd =ag; ); print("** ",n," >> ",ag); if(ag != bg, print("**** Counterexample ****"); next(19); \\\\ Halt RUN if a counterexample is found ); ); }) print("***** ",runningN ," >> ",runningGcd ); factor(runningN) Last fiddled with by a1call on 20210829 at 18:32 
20210831, 01:19  #21 
"Tucker Kao"
Jan 2020
Head Base M168202123
489_{10} Posts 

20210831, 01:42  #22  
Feb 2017
Nowhere
137B_{16} Posts 
Quote:
I advise you to hit "Preview Post," and check the preview every time before you post a message. That way, you won't have to bother trying to remember which nested bbcode tags work and which ones don't. If any nested bbcodes are squirrelly, you'll see it in the preview window. You can then edit the message in the Message window (NOT the Preview window). You can check "Preview Message" repeatedly and edit repeatedly until it looks satisfactory, before you hit "Submit Reply." 

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