Peter Cameron question

Nick · 2020-10-17, 12:09

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: https://cameroncounts.wordpress.com/...-and-mersenne/

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).

https://www.mersenne.ca/prp.php

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:

Originally Posted by a1call

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.

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:

Originally Posted by VBCurtis

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.

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.

a1call · 2020-10-17, 20:24

Quote:

Originally Posted by a1call

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.

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.

VBCurtis · 2020-10-17, 21:11

Quote:

Originally Posted by a1call

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.

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.

Viliam Furik · 2020-10-17, 23:30

Quote:

Originally Posted by a1call

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.

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

2020-10-17, 12:09	#1
Nick Dec 2012 The Netherlands 2⁴×101 Posts	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: https://cameroncounts.wordpress.com/...-and-mersenne/

2020-10-17, 16:30	#3
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 2⁶·31 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 non-semiprine exponent is sieved out. ETA ll Or is it non-prime exponents? Last fiddled with by a1call on 2020-10-17 at 16:44

2020-10-17, 18:23	#7
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 2⁶·31 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 2020-10-17 at 19:14

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2020-10-17, 17:02	#4
axn Jun 2003 12F8₁₆ 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.

2020-10-17, 22:32	#10
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 2⁶·31 Posts	Glad to have been of service if any. Otherwise I'm just glad to have something to do with my new build.