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 Nick 2020-10-17 12:09

Peter Cameron question

Prof. Peter Cameron has put a quick question in his blog:
for which positive integers n are \(2^n-1\) and \(2^n+1\) both products of at most 2 distinct primes?
Link: [URL]https://cameroncounts.wordpress.com/2020/10/07/between-fermat-and-mersenne/[/URL]

 ATH 2020-10-17 14:27

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).

[url]https://www.mersenne.ca/prp.php[/url]

[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
[/CODE]

 a1call 2020-10-17 16:30

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 non-semiprine exponent is sieved out.
ETA ll Or is it non-prime exponents?

 axn 2020-10-17 17:02

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.

 VBCurtis 2020-10-17 17:26

[QUOTE=a1call;560151]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.[/QUOTE]

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.

 a1call 2020-10-17 17:50

[QUOTE=VBCurtis;560156]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.[/QUOTE]

I should stop posting in my sleep. Please disregard my post.
(3, 888793, 3432577, ... ) | F1277

 a1call 2020-10-17 18:23

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.:smile:

 a1call 2020-10-17 20:24

[QUOTE=a1call;560160]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.:smile:[/QUOTE]

If I read the merssene.org correctly, M117239 has no known factors and F117239 is a (Probable) semiprime. So any complete list will have to be exclusively less than 117239.:smile:

 VBCurtis 2020-10-17 21:11

[QUOTE=a1call;560161]If I read the merssene.org correctly, M117239 has no known factors and F117239 is a (Probable) semiprime. So any complete list will have to be exclusively less than 117239.:smile:[/QUOTE]

I appreciate your work to list these, and I think I'll have a go at some ECM on M117239 'cause this is fun.

 a1call 2020-10-17 22:32

Glad to have been of service if any.
Otherwise I'm just glad to have something to do with my new build.:smile:

 Viliam Furik 2020-10-17 23:30

[QUOTE=a1call;560160]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.:smile:[/QUOTE]

No, all those Mersenne numbers you listed have one factor and a (definitely) composite cofactor, so at least three factors.

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