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 2022-05-19, 09:39 #1 Tomazio   May 2022 1 Posts Sophie Germain prime and Mersenne number 2^p-1 How can I prove that if p=3 (mod 4) is a Sophie Germain prime then the Mersenne number 2^p-1 is composite? Thanks in advance.
2022-05-24, 23:11   #2
Dobri

"Καλός"
May 2018

17×19 Posts

Quote:
 Originally Posted by Tomazio How can I prove that if p=3 (mod 4) is a Sophie Germain prime then the Mersenne number 2^p-1 is composite? Thanks in advance.
See the following manuscript:
Édouard Lucas, Théorie des Fonctions Numériques Simplement Périodiques, American Journal of Mathematics, Vol. 1, No. 4 (1878), pp. 184-240 and 289-321 (in French).
Available: <http://edouardlucas.free.fr/oeuvres/...eriodiques.pdf>.

2022-05-25, 05:33   #3
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

2·1,723 Posts

Quote:
 Originally Posted by Tomazio How can I prove that if p=3 (mod 4) is a Sophie Germain prime then the Mersenne number 2^p-1 is composite? Thanks in advance.
Because for all such primes p, 2^p-1 is divisible by 2*p+1, since 2*p+1 is == 7 mod 8, thus 2 is a quadratic residue mod 2*p+1, and znorder(Mod(2,2*p+1)) must divide ((2*p+1)-1)/2 = p, thus znorder(Mod(2,2*p+1)) is either 1 or p, but cannot be 1 since there is no prime q such that znorder(Mod(2,q)) = 1 (or q must divide 2-1 = 1, which is impossible), thus znorder(Mod(2,2*p+1)) can only be p, and hence 2*p+1 divides 2^p-1, thus 2^p-1 is composite.

Similarly, if p == 1 mod 4 is a Sophie Germain prime and p > 5, then the Wagstaff number (2^p+1)/3 is composite.

Here is an exercise for you: prove that if p is a Sophie Germain prime other than 2, 3, and 5, then the dozenal repunit (12^p-1)/11 is composite.

2022-05-25, 09:17   #4
Dobri

"Καλός"
May 2018

5038 Posts

Quote:
 Originally Posted by Tomazio How can I prove that if p=3 (mod 4) is a Sophie Germain prime then the Mersenne number 2^p-1 is composite? Thanks in advance.
For the first known proof, see the following manuscript:
Joseph Louis de Lagrange, Recherches d'arithmétique (1775), pp. 695-795 (in French).
Available: <https://gallica.bnf.fr/ark:/12148/bpt6k229222d/f696#>.

See Lemme III in page 778 and also 49. Scolie I in page 794.

 2022-05-30, 22:52 #5 Dobri   "Καλός" May 2018 1010000112 Posts See also the web page on "Euler and Lagrange on Mersenne Divisors" by Chris K. Caldwell at . Last fiddled with by Dobri on 2022-05-30 at 23:03

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