Sophie Germain prime and Mersenne number 2^p-1

[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.

[Dobri]

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>.

[sweety439]

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.

[Dobri]

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.

[Dobri]

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>.