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Old 2017-12-30, 12:29   #1
jnml
 
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Default 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.)
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Old 2017-12-30, 13:09   #2
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That's pretty cool even if it is not unique.
Is that your own discovery, or was it common knowledge before?
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Old 2017-12-30, 13:12   #3
jnml
 
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Quote:
Originally Posted by a1call View Post
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,
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Old 2017-12-30, 13:16   #4
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Is it a well known fact that
2^11-1 = (11k+1)(11l+1)
?
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Old 2017-12-30, 13:23   #5
jnml
 
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Quote:
Originally Posted by a1call View Post
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.
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Old 2017-12-30, 13:25   #6
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Quote:
Originally Posted by jnml View Post
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.
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Old 2017-12-30, 13:28   #7
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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.
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Old 2017-12-30, 13:32   #8
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Quote:
Originally Posted by a1call View Post
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.
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Old 2017-12-30, 13:35   #9
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Please don't be sorry.
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Old 2017-12-30, 15:11   #10
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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.
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Old 2017-12-30, 15:28   #11
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Quote:
Originally Posted by a1call View Post
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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