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Old 2014-12-03, 08:36   #1
ProximaCentauri
 
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Default 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!
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Old 2014-12-03, 12:40   #2
Batalov
 
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Quote:
Originally Posted by ProximaCentauri View Post
...
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?
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Old 2014-12-03, 12:47   #3
ProximaCentauri
 
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No, only p is a SGP, not q. Sorry for mixing this up!

Nevertheless, can q yield a Mersenne Prime?
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Old 2014-12-03, 12:50   #4
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Then q is 2p+1 ?

You cannot have
Quote:
p (1 mod 4) ...
q = 2*p-1 (3 mod 4)
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Old 2014-12-03, 12:52   #5
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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
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Old 2014-12-03, 14:22   #6
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To be sure about this, i will be "Lucas-Lehmering" 2(2*43112609+1)-1
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Old 2014-12-03, 17:13   #7
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Quote:
Originally Posted by ProximaCentauri View Post
To be sure about this, i will be "Lucas-Lehmering" 2(2*43112609+1)-1
What does having one more data point, affirmative or negative, make you any more "sure" of?
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Old 2014-12-03, 18:47   #8
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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!
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Old 2014-12-03, 19:40   #9
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No, this ("I see a strong correlation") is properly called apophenia.
Or a clustering illusion, or any of the interrelated cognitive biases.
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Old 2014-12-03, 20:18   #10
ProximaCentauri
 
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I am not suffering "onset schizophrenia" like u wanted to tell me, Serge!!!

DonĀ“t worry and take care of yourself pls!
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Old 2014-12-03, 21:24   #11
Batalov
 
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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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