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 2017-12-30, 12:29 #1 jnml   Feb 2012 Prague, Czech Republ 132 Posts Can Mersenne composites share "shape"? A Mersenne composite $M_p, p \in \text{primes}$ can be written as $(2kp+1)(2lp+1)$, for some $k, l \in \mathbb{N}$. Let's call the pair $(k, l)$ the "shape" (or one of the "shapes" if there are more than two factors) of $M_p$. 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.)
 2017-12-30, 13:09 #2 a1call     "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?
2017-12-30, 13:12   #3
jnml

Feb 2012
Prague, Czech Republ

16910 Posts

Quote:
 Originally Posted by a1call That's pretty cool even if it is not unique. Is that your own discovery, or was it common knowledge before?
I have no idea what "discovery" you're talking about. The OP contains none such thing, but it does contain a question formulated
using a well known fact. Can you please clarify? Thank you,

 2017-12-30, 13:16 #4 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 7·283 Posts Is it a well known fact that 2^11-1 = (11k+1)(11l+1) ?
2017-12-30, 13:23   #5
jnml

Feb 2012
Prague, Czech Republ

132 Posts

Quote:
 Originally Posted by a1call Is it a well known fact that 2^11-1 = (11k+1)(11l+1) ?
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.

2017-12-30, 13:25   #6
ET_
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"Luigi"
Aug 2002
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Quote:
 Originally Posted by jnml 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.
The answer is no for Fermat numbers, IIRC.

 2017-12-30, 13:28 #7 a1call     "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.
2017-12-30, 13:32   #8
jnml

Feb 2012
Prague, Czech Republ

16910 Posts

Quote:
 Originally Posted by a1call 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.
Then please accept my sincere apologies. I misunderstood your post and the example for
$M_{11}$ added to my confusion, sorry for that.

 2017-12-30, 13:35 #9 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 111101111012 Posts Please don't be sorry.
 2017-12-30, 15:11 #10 Dr Sardonicus     Feb 2017 Nowhere 24·32·29 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 2p - 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.
2017-12-30, 15:28   #11
science_man_88

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Quote:
 Originally Posted by a1call Is it a well known fact that 2^11-1 = (11k+1)(11l+1) ?
Yes all factors of 2^p-1 have form 2jp+1 for p a prime.https://primes.utm.edu/mersenne/

Last fiddled with by science_man_88 on 2017-12-30 at 15:31

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