Sophie-Germain primes as Mersenne exponents

ProximaCentauri · 2014-12-03, 08:36

Suppose, an exponent p (1 mod 4) yields a Mersenne Prime M^p=2^p-1.

If q = 2*p-1 (3 mod 4), will the resulting M_q be composite? If yes, why?

q is a Sophie-Germain prime.

Thanks in advance!

Batalov · 2014-12-03, 12:40

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?

ProximaCentauri · 2014-12-03, 12:47

No, only p is a SGP, not q. Sorry for mixing this up!

Nevertheless, can q yield a Mersenne Prime?

Batalov · 2014-12-03, 12:50

Then q is 2p+1 ?

You cannot have

Quote:

p (1 mod 4) ...
q = 2*p-1 (3 mod 4)

ProximaCentauri · 2014-12-03, 12:52

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?

ProximaCentauri · 2014-12-03, 14:22

To be sure about this, i will be "Lucas-Lehmering" 2^{(2*43112609+1)}-1

VBCurtis · 2014-12-03, 17:13

Quote:

Originally Posted by ProximaCentauri

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?

ProximaCentauri · 2014-12-03, 18:47

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!

Batalov · 2014-12-03, 19:40

No, this ("I see a strong correlation") is properly called apophenia.
Or a clustering illusion, or any of the interrelated cognitive biases.

ProximaCentauri · 2014-12-03, 20:18

I am not suffering "onset schizophrenia" like u wanted to tell me, Serge!!!

Don´t worry and take care of yourself pls!

Batalov · 2014-12-03, 21:24

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.

2014-12-03, 08:36	#1
ProximaCentauri "M49" Dec 2014 Austria 2³×3 Posts	Sophie-Germain primes as Mersenne exponents Suppose, an exponent p (1 mod 4) yields a Mersenne Prime M^p=2^p-1. If q = 2*p-1 (3 mod 4), will the resulting M_q be composite? If yes, why? q is a Sophie-Germain prime. Thanks in advance!

2014-12-03, 12:52	#5
ProximaCentauri "M49" Dec 2014 Austria 24₁₀ Posts	And of course q = 2p+1, not 2p-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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2014-12-03, 12:47	#3
ProximaCentauri "M49" Dec 2014 Austria 18₁₆ Posts	No, only p is a SGP, not q. Sorry for mixing this up! Nevertheless, can q yield a Mersenne Prime?

2014-12-03, 14:22	#6
ProximaCentauri "M49" Dec 2014 Austria 2³·3 Posts	To be sure about this, i will be "Lucas-Lehmering" 2^{(2*43112609+1)}-1

2014-12-03, 18:47	#8
ProximaCentauri "M49" Dec 2014 Austria 11000₂ 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 3⁶·13 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 24₁₀ 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 10010100000101₂ 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.