20220519, 09:39  #1 
May 2022
1_{10} Posts 
Sophie Germain prime and Mersenne number 2^p1
How can I prove that if p=3 (mod 4) is a Sophie Germain prime then the Mersenne number 2^p1 is composite?
Thanks in advance. 
20220524, 23:11  #2  
"Καλός"
May 2018
2^{2}×5×17 Posts 
Quote:
Édouard Lucas, Théorie des Fonctions Numériques Simplement Périodiques, American Journal of Mathematics, Vol. 1, No. 4 (1878), pp. 184240 and 289321 (in French). Available: <http://edouardlucas.free.fr/oeuvres/...eriodiques.pdf>. 

20220525, 05:33  #3  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
7×509 Posts 
Quote:
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^p1)/11 is composite. 

20220525, 09:17  #4  
"Καλός"
May 2018
2^{2}×5×17 Posts 
Quote:
Joseph Louis de Lagrange, Recherches d'arithmétique (1775), pp. 695795 (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. 

20220530, 22:52  #5 
"Καλός"
May 2018
340_{10} Posts 
See also the web page on "Euler and Lagrange on Mersenne Divisors" by Chris K. Caldwell at <https://primes.utm.edu/notes/proofs/MerDiv2.html>.
Last fiddled with by Dobri on 20220530 at 23:03 
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