20171230, 12:29  #1 
Feb 2012
Prague, Czech Republ
13^{2} Posts 
Can Mersenne composites share "shape"?
A Mersenne composite can be written as , for some .
Let's call the pair the "shape" (or one of the "shapes" if there are more than two factors) of . I wonder if a "shape" is unique, ie. if it can or cannot occur more than once over composite Mersenne numbers? Does anybody know more about this? (My uneducated guess is it cannot, FTR.) 
20171230, 13:09  #2 
"Rashid Naimi"
Oct 2015
Remote to Here/There
7×283 Posts 
That's pretty cool even if it is not unique.
Is that your own discovery, or was it common knowledge before? 
20171230, 13:12  #3  
Feb 2012
Prague, Czech Republ
169_{10} Posts 
Quote:
using a well known fact. Can you please clarify? Thank you, 

20171230, 13:16  #4 
"Rashid Naimi"
Oct 2015
Remote to Here/There
7·283 Posts 
Is it a well known fact that
2^111 = (11k+1)(11l+1) ? 
20171230, 13:23  #5 
Feb 2012
Prague, Czech Republ
13^{2} Posts 
True, but I have no idea whatsoever what it has to do with my question. Every composite
Mersene number with a prime exponent has some k, l that fits the formula in the OP. I was not looking for an example of k and l. My question is if the particular pair (k, l) can occur only in one Mersenne composite with prime exponent or if if can occur in another Mersenne composite number with a different prime exponent. No offense, but please reread the OP. It seems you have misunderstood it. 
20171230, 13:25  #6  
Banned
"Luigi"
Aug 2002
Team Italia
2^{3}×599 Posts 
Quote:


20171230, 13:28  #7 
"Rashid Naimi"
Oct 2015
Remote to Here/There
7×283 Posts 
I am not trying to be sarcastic. I did not know that fact and suspect not many did either.
I understood the OP question and no hijacking was intended. It was a genuine related question. 
20171230, 13:32  #8  
Feb 2012
Prague, Czech Republ
169_{10} Posts 
Quote:
added to my confusion, sorry for that. 

20171230, 13:35  #9 
"Rashid Naimi"
Oct 2015
Remote to Here/There
11110111101_{2} Posts 
Please don't be sorry.

20171230, 15:11  #10 
Feb 2017
Nowhere
2^{4}·3^{2}·29 Posts 
Hmm. The occurrence of 2*p + 1 as a factor ( k = 1) is well known: If p = 4*n + 3 and q = 8*n + 7 are both prime, then q = 2*1*p + 1 divides 2^{p}  1. (The primes p and q are "Sophie Germain primes." It is not known whether there are infinitely many such, but it is widely believed that there are.)
I do not alas know of any corresponding result for any k greater than 1. 
20171230, 15:28  #11 
"Forget I exist"
Jul 2009
Dumbassville
10000011000000_{2} Posts 
Yes all factors of 2^p1 have form 2jp+1 for p a prime.https://primes.utm.edu/mersenne/
Last fiddled with by science_man_88 on 20171230 at 15:31 
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