mersenneforum.org  

Go Back   mersenneforum.org > New To GIMPS? Start Here! > Homework Help

Reply
 
Thread Tools
Old 2022-05-19, 09:39   #1
Tomazio
 
May 2022

18 Posts
Default 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.
Tomazio is offline   Reply With Quote
Old 2022-05-24, 23:11   #2
Dobri
 
"Καλός"
May 2018

17×19 Posts
Default

Quote:
Originally Posted by Tomazio View Post
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>.
Dobri is offline   Reply With Quote
Old 2022-05-25, 05:33   #3
sweety439
 
sweety439's Avatar
 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

5×13×53 Posts
Default

Quote:
Originally Posted by Tomazio View Post
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.
sweety439 is offline   Reply With Quote
Old 2022-05-25, 09:17   #4
Dobri
 
"Καλός"
May 2018

17·19 Posts
Default

Quote:
Originally Posted by Tomazio View Post
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 is offline   Reply With Quote
Old 2022-05-30, 22:52   #5
Dobri
 
"Καλός"
May 2018

17·19 Posts
Default

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 2022-05-30 at 23:03
Dobri is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Sophie Germain Primes, Mersenne numbers and Wagstaff numbers Connection JCoveiro Number Theory Discussion Group 2 2022-02-08 14:43
Mersenne prime exponents that are also Sophie Germain primes carpetpool Miscellaneous Math 4 2021-11-12 00:10
Sophie Germain Twins Trejack Twin Prime Search 10 2016-06-23 15:10
Sophie-Germain primes as Mersenne exponents ProximaCentauri Miscellaneous Math 15 2014-12-25 14:26
Sophie-Germain sieve firejuggler Software 4 2014-01-10 00:09

All times are UTC. The time now is 05:10.


Sat Jun 25 05:10:44 UTC 2022 up 72 days, 3:12, 0 users, load averages: 1.04, 1.21, 1.22

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2022, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.

≠ ± ∓ ÷ × · − √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °
∠ ∟ ° ≅ ~ ‖ ⟂ ⫛
≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘ ∏ ∐ ∑ ∧ ∨ ∩ ∪ ⨀ ⊕ ⊗ 𝖕 𝖖 𝖗 ⊲ ⊳
∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟
¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣ … ⋯ ⋮ ⋰ ⋱
∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ
𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎𝜍 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔