mersenneforum.org Sophie-Germain primes as Mersenne exponents
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 2014-12-03, 08:36 #1 ProximaCentauri     "M49" Dec 2014 Austria 2410 Posts Sophie-Germain primes as Mersenne exponents Suppose, an exponent p (1 mod 4) yields a Mersenne Prime Mp=2p-1. If q = 2*p-1 (3 mod 4), will the resulting Mq be composite? If yes, why? q is a Sophie-Germain prime. Thanks in advance!
2014-12-03, 12:40   #2
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

3·52·131 Posts

Quote:
 Originally Posted by ProximaCentauri ... If q = 2*p-1 (3 mod 4), ... q is a Sophie-Germain prime.
How is q a Sophie-Germain prime? Do you mean that additionally to what you wrote 2q+1 is also prime?

 2014-12-03, 12:47 #3 ProximaCentauri     "M49" Dec 2014 Austria 23×3 Posts No, only p is a SGP, not q. Sorry for mixing this up! Nevertheless, can q yield a Mersenne Prime?
2014-12-03, 12:50   #4
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

982510 Posts

Then q is 2p+1 ?

You cannot have
Quote:
 p (1 mod 4) ... q = 2*p-1 (3 mod 4)

 2014-12-03, 12:52 #5 ProximaCentauri     "M49" Dec 2014 Austria 23·3 Posts And of course q = 2*p+1, not 2*p-1, as I posted before! So q is a safe prime! There are 8 known Mersenne Prime exponents so far, which are Sophie-Germain Primes as well, namely 2 3 5 89 9689 21701 859433 43112609 2 of them have associated safe primes, which also result in a Mersenne prime. M5 and M7 My question is: Are there any more possible of this special kind? Last fiddled with by ProximaCentauri on 2014-12-03 at 13:51
 2014-12-03, 14:22 #6 ProximaCentauri     "M49" Dec 2014 Austria 1816 Posts To be sure about this, i will be "Lucas-Lehmering" 2(2*43112609+1)-1
2014-12-03, 17:13   #7
VBCurtis

"Curtis"
Feb 2005
Riverside, CA

5,279 Posts

Quote:
What does having one more data point, affirmative or negative, make you any more "sure" of?

 2014-12-03, 18:47 #8 ProximaCentauri     "M49" Dec 2014 Austria 23×3 Posts Maybe sure was the wrong word. But I see a strong correlation between Mersenne prime exponents, Sophie-Germain Primes and Twin primes. By now 16 of the 48 Mersenne prime exponents are also part of a TWIN-PRIME, this is 1/3 of all known. 2 3 5 7 13 17 19 31 61 107 521 1279 4423 110503 132049 20996011 21 of 48 Mersenne prime exponents either share the property of being a SGP (Sophie-Germain Prime) and/or being a part of a TWIN-Prime. 2, 3 and 5 have both properties! This is nearly 50% of all, 43,75% to be exact! So I will concentrate on these exponents with above mentioned attributes in the future. Just a strong feeling, no proof at all!
 2014-12-03, 19:40 #9 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3·52·131 Posts No, this ("I see a strong correlation") is properly called apophenia. Or a clustering illusion, or any of the interrelated cognitive biases.
 2014-12-03, 20:18 #10 ProximaCentauri     "M49" Dec 2014 Austria 23·3 Posts I am not suffering "onset schizophrenia" like u wanted to tell me, Serge!!! Don´t worry and take care of yourself pls!
 2014-12-03, 21:24 #11 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3×52×131 Posts Now, you see? You just saw another pattern that you wanted to see -- where there was none. Everyone sees them; not everyone knows proper statistics.

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