20201017, 12:09  #1 
Dec 2012
The Netherlands
5AE_{16} Posts 
Peter Cameron question
Prof. Peter Cameron has put a quick question in his blog:
for which positive integers n are \(2^n1\) and \(2^n+1\) both products of at most 2 distinct primes? Link: https://cameroncounts.wordpress.com/...andmersenne/ 
20201017, 14:27  #2 
Einyen
Dec 2003
Denmark
2,969 Posts 
87 Mersenne semi primes known so far.
The last 23 from M86371 to M10443557 are "semi probable primes" or "probable semi primes" (biggest factors are PRP). https://www.mersenne.ca/prp.php Code:
M11 M23 M37 M41 M59 M67 M83 M97 M101 M103 M109 M131 M137 M139 M149 M167 M197 M199 M227 M241 M269 M271 M281 M293 M347 M373 M379 M421 M457 M487 M523 M727 M809 M881 M971 M983 M997 M1061 M1063 M1427 M1487 M1637 M1657 M2357 M2927 M3079 M3259 M3359 M4111 M4243 M4729 M5689 M6043 M6679 M7331 M7757 M10169 M14561 M17029 M26903 M28759 M28771 M58199 M63703 M86371 M106391 M130439 M136883 M151013 M173867 M221509 M271211 M271549 M406583 M432457 M611999 M684127 M1010623 M1168183 M1304983 M1629469 M2327417 M3464473 M4187251 M5240707 M7313983 M10443557 
20201017, 16:30  #3 
"Rashid Naimi"
Oct 2015
Remote to Here/There
789_{16} Posts 
We can only make an unbroken and minimal list of integers exclusively less than 1277, since we don't know if M1277 is a semiprime or not.
ETA As far as the Mersenne number candidates are concerned any nonsemiprine exponent is sieved out. ETA ll Or is it nonprime exponents? Last fiddled with by a1call on 20201017 at 16:44 
20201017, 17:02  #4 
Jun 2003
2^{2}·3^{2}·131 Posts 
We are looking at n prime. Since 2^n+1 has a forced factor of 3, we're looking for Wagstaff primes where the corresponding Mersenne number is a prime or semiprime. There should be infinitely many Wagstaff primes. My guess is infinitely many cases where corresponding Mersenne number is a semiprime.

20201017, 17:26  #5 
"Curtis"
Feb 2005
Riverside, CA
2·3·733 Posts 
Is 2^1277+1 a semiprime? You don't need the status of both 1 and +1 forms if one side is ruled out, the other side doesn't matter.

20201017, 17:50  #6 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3·643 Posts 

20201017, 18:23  #7 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3·643 Posts 
If I read the merssene.org correctly, M5807 is a (Probable) semiprime and so is F5807.
If I read the merssene.org correctly, M14479 is a (Probable) semiprime and so is F14479. If I read the merssene.org correctly, M42737 is a (Probable) semiprime and so is F42737. So no practical upper bound for a continuous list below them. Last fiddled with by a1call on 20201017 at 19:14 
20201017, 20:24  #8  
"Rashid Naimi"
Oct 2015
Remote to Here/There
3·643 Posts 
Quote:


20201017, 21:11  #9 
"Curtis"
Feb 2005
Riverside, CA
2·3·733 Posts 
I appreciate your work to list these, and I think I'll have a go at some ECM on M117239 'cause this is fun.

20201017, 22:32  #10 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3611_{8} Posts 
Glad to have been of service if any.
Otherwise I'm just glad to have something to do with my new build. 
20201017, 23:30  #11  
Jul 2018
Martin, Slovakia
223 Posts 
Quote:


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