mersenneforum.org  

Go Back   mersenneforum.org > Great Internet Mersenne Prime Search > Math > Number Theory Discussion Group

Reply
 
Thread Tools
Old 2022-02-06, 19:22   #1
JCoveiro
 
"Jorge Coveiro"
Nov 2006
Moura, Portugal

24×3 Posts
Default Sophie Germain Primes, Mersenne numbers and Wagstaff numbers Connection

Hi all.

I think I've found a "Sophie Germain Primes" connection with Mersenne numbers and Wagstaff numbers.

Code:
k=1;forprime(x=1,1000,if(isprime(2*x+1),print1("SophieGermain("k")*2+1 = ("x"*2+1) = "x*2+1" --> "x*2+1" | ");if(x%4==3,print1("Mersenne(");print1(x") --> M("x") Mod ("x"*2+1) = ");print1((2^x-1)%(2*x+1)),print1("Wagstaff(");print1(x") --> W("x") Mod ("x"*2+1) = ");print1((2^x+1)/3%(2*x+1)););print();k++);)                                                                                                                                                            

SophieGermain(1)*2+1 = (2*2+1) = 5 --> 5 | Wagstaff(2) --> W(2) Mod (2*2+1) = 0                                                                                                                                                          
SophieGermain(2)*2+1 = (3*2+1) = 7 --> 7 | Mersenne(3) --> M(3) Mod (3*2+1) = 0
SophieGermain(3)*2+1 = (5*2+1) = 11 --> 11 | Wagstaff(5) --> W(5) Mod (5*2+1) = 0
SophieGermain(4)*2+1 = (11*2+1) = 23 --> 23 | Mersenne(11) --> M(11) Mod (11*2+1) = 0
SophieGermain(5)*2+1 = (23*2+1) = 47 --> 47 | Mersenne(23) --> M(23) Mod (23*2+1) = 0
SophieGermain(6)*2+1 = (29*2+1) = 59 --> 59 | Wagstaff(29) --> W(29) Mod (29*2+1) = 0
SophieGermain(7)*2+1 = (41*2+1) = 83 --> 83 | Wagstaff(41) --> W(41) Mod (41*2+1) = 0
SophieGermain(8)*2+1 = (53*2+1) = 107 --> 107 | Wagstaff(53) --> W(53) Mod (53*2+1) = 0
SophieGermain(9)*2+1 = (83*2+1) = 167 --> 167 | Mersenne(83) --> M(83) Mod (83*2+1) = 0
SophieGermain(10)*2+1 = (89*2+1) = 179 --> 179 | Wagstaff(89) --> W(89) Mod (89*2+1) = 0
SophieGermain(11)*2+1 = (113*2+1) = 227 --> 227 | Wagstaff(113) --> W(113) Mod (113*2+1) = 0
SophieGermain(12)*2+1 = (131*2+1) = 263 --> 263 | Mersenne(131) --> M(131) Mod (131*2+1) = 0
SophieGermain(13)*2+1 = (173*2+1) = 347 --> 347 | Wagstaff(173) --> W(173) Mod (173*2+1) = 0
SophieGermain(14)*2+1 = (179*2+1) = 359 --> 359 | Mersenne(179) --> M(179) Mod (179*2+1) = 0
SophieGermain(15)*2+1 = (191*2+1) = 383 --> 383 | Mersenne(191) --> M(191) Mod (191*2+1) = 0
SophieGermain(16)*2+1 = (233*2+1) = 467 --> 467 | Wagstaff(233) --> W(233) Mod (233*2+1) = 0
SophieGermain(17)*2+1 = (239*2+1) = 479 --> 479 | Mersenne(239) --> M(239) Mod (239*2+1) = 0
SophieGermain(18)*2+1 = (251*2+1) = 503 --> 503 | Mersenne(251) --> M(251) Mod (251*2+1) = 0
SophieGermain(19)*2+1 = (281*2+1) = 563 --> 563 | Wagstaff(281) --> W(281) Mod (281*2+1) = 0
SophieGermain(20)*2+1 = (293*2+1) = 587 --> 587 | Wagstaff(293) --> W(293) Mod (293*2+1) = 0
SophieGermain(21)*2+1 = (359*2+1) = 719 --> 719 | Mersenne(359) --> M(359) Mod (359*2+1) = 0
SophieGermain(22)*2+1 = (419*2+1) = 839 --> 839 | Mersenne(419) --> M(419) Mod (419*2+1) = 0
SophieGermain(23)*2+1 = (431*2+1) = 863 --> 863 | Mersenne(431) --> M(431) Mod (431*2+1) = 0
SophieGermain(24)*2+1 = (443*2+1) = 887 --> 887 | Mersenne(443) --> M(443) Mod (443*2+1) = 0
SophieGermain(25)*2+1 = (491*2+1) = 983 --> 983 | Mersenne(491) --> M(491) Mod (491*2+1) = 0
SophieGermain(26)*2+1 = (509*2+1) = 1019 --> 1019 | Wagstaff(509) --> W(509) Mod (509*2+1) = 0
SophieGermain(27)*2+1 = (593*2+1) = 1187 --> 1187 | Wagstaff(593) --> W(593) Mod (593*2+1) = 0
SophieGermain(28)*2+1 = (641*2+1) = 1283 --> 1283 | Wagstaff(641) --> W(641) Mod (641*2+1) = 0
SophieGermain(29)*2+1 = (653*2+1) = 1307 --> 1307 | Wagstaff(653) --> W(653) Mod (653*2+1) = 0
SophieGermain(30)*2+1 = (659*2+1) = 1319 --> 1319 | Mersenne(659) --> M(659) Mod (659*2+1) = 0
SophieGermain(31)*2+1 = (683*2+1) = 1367 --> 1367 | Mersenne(683) --> M(683) Mod (683*2+1) = 0
SophieGermain(32)*2+1 = (719*2+1) = 1439 --> 1439 | Mersenne(719) --> M(719) Mod (719*2+1) = 0
SophieGermain(33)*2+1 = (743*2+1) = 1487 --> 1487 | Mersenne(743) --> M(743) Mod (743*2+1) = 0
SophieGermain(34)*2+1 = (761*2+1) = 1523 --> 1523 | Wagstaff(761) --> W(761) Mod (761*2+1) = 0
SophieGermain(35)*2+1 = (809*2+1) = 1619 --> 1619 | Wagstaff(809) --> W(809) Mod (809*2+1) = 0
SophieGermain(36)*2+1 = (911*2+1) = 1823 --> 1823 | Mersenne(911) --> M(911) Mod (911*2+1) = 0
SophieGermain(37)*2+1 = (953*2+1) = 1907 --> 1907 | Wagstaff(953) --> W(953) Mod (953*2+1) = 0

k=1;forprime(x=1,1000,if(isprime(2*x+1),if(x%4==3,print1("Mersenne(SophieGermain("k")) Mod (SophieGermain("k")*2+1) == ");print1((2^x-1)%(2*x+1)),print1("Wagstaff(SophieGermain("k")) Mod (SophieGermain("k")*2+1) == ");print1((2^x+1)/3%(2*x+1)););print();k++);)       
                                                                                                                                                                                                     
Wagstaff(SophieGermain(1)) Mod (SophieGermain(1)*2+1) == 0                                                                                                                                                                               
Mersenne(SophieGermain(2)) Mod (SophieGermain(2)*2+1) == 0
Wagstaff(SophieGermain(3)) Mod (SophieGermain(3)*2+1) == 0
Mersenne(SophieGermain(4)) Mod (SophieGermain(4)*2+1) == 0
Mersenne(SophieGermain(5)) Mod (SophieGermain(5)*2+1) == 0
Wagstaff(SophieGermain(6)) Mod (SophieGermain(6)*2+1) == 0
Wagstaff(SophieGermain(7)) Mod (SophieGermain(7)*2+1) == 0
Wagstaff(SophieGermain(8)) Mod (SophieGermain(8)*2+1) == 0
Mersenne(SophieGermain(9)) Mod (SophieGermain(9)*2+1) == 0
Wagstaff(SophieGermain(10)) Mod (SophieGermain(10)*2+1) == 0
Wagstaff(SophieGermain(11)) Mod (SophieGermain(11)*2+1) == 0
Mersenne(SophieGermain(12)) Mod (SophieGermain(12)*2+1) == 0
Wagstaff(SophieGermain(13)) Mod (SophieGermain(13)*2+1) == 0
Mersenne(SophieGermain(14)) Mod (SophieGermain(14)*2+1) == 0
Mersenne(SophieGermain(15)) Mod (SophieGermain(15)*2+1) == 0
Wagstaff(SophieGermain(16)) Mod (SophieGermain(16)*2+1) == 0
Mersenne(SophieGermain(17)) Mod (SophieGermain(17)*2+1) == 0
Mersenne(SophieGermain(18)) Mod (SophieGermain(18)*2+1) == 0
Wagstaff(SophieGermain(19)) Mod (SophieGermain(19)*2+1) == 0
Wagstaff(SophieGermain(20)) Mod (SophieGermain(20)*2+1) == 0
Mersenne(SophieGermain(21)) Mod (SophieGermain(21)*2+1) == 0
Mersenne(SophieGermain(22)) Mod (SophieGermain(22)*2+1) == 0
Mersenne(SophieGermain(23)) Mod (SophieGermain(23)*2+1) == 0
Mersenne(SophieGermain(24)) Mod (SophieGermain(24)*2+1) == 0
Mersenne(SophieGermain(25)) Mod (SophieGermain(25)*2+1) == 0
Wagstaff(SophieGermain(26)) Mod (SophieGermain(26)*2+1) == 0
Wagstaff(SophieGermain(27)) Mod (SophieGermain(27)*2+1) == 0
Wagstaff(SophieGermain(28)) Mod (SophieGermain(28)*2+1) == 0
Wagstaff(SophieGermain(29)) Mod (SophieGermain(29)*2+1) == 0
Mersenne(SophieGermain(30)) Mod (SophieGermain(30)*2+1) == 0
Mersenne(SophieGermain(31)) Mod (SophieGermain(31)*2+1) == 0
Mersenne(SophieGermain(32)) Mod (SophieGermain(32)*2+1) == 0
Mersenne(SophieGermain(33)) Mod (SophieGermain(33)*2+1) == 0
Wagstaff(SophieGermain(34)) Mod (SophieGermain(34)*2+1) == 0
Wagstaff(SophieGermain(35)) Mod (SophieGermain(35)*2+1) == 0
Mersenne(SophieGermain(36)) Mod (SophieGermain(36)*2+1) == 0
Wagstaff(SophieGermain(37)) Mod (SophieGermain(37)*2+1) == 0
So:
Every: (4^(SophieGermain(x))-1)/3 Mod (SophieGermain(x)*2+1) == 0
So that: (Mersenne(SophieGermain(x))*Wagstaff(SophieGermain(x))) Mod (SophieGermain(x)*2+1) == 0

In conclusion:
If isprime(2*p+1)&&(p%4==3) then (2*p+1) | Mersenne(p)
If isprime(2*p+1)&&(p%4==1) then (2*p+1) | Wagstaff(p)

Do you think that this can be related with the "new Mersenne Prime Conjecture" too?

Last fiddled with by JCoveiro on 2022-02-06 at 19:54
JCoveiro is online now   Reply With Quote
Old 2022-02-08, 14:13   #2
JCoveiro
 
"Jorge Coveiro"
Nov 2006
Moura, Portugal

24×3 Posts
Default also

Also:

If (2*p+1) | Mersenne(p) then: Wagstaff(p) might be a prime.

Code:
3,11,23,191,3539,10691,83339,4031399,13347311
and

If (2*p+1) | Wagstaff(p) then: Mersenne(p) might be a prime.

Code:
5,89,9689,21701,859433,43112609

Last fiddled with by JCoveiro on 2022-02-08 at 14:18
JCoveiro is online now   Reply With Quote
Old 2022-02-08, 14:43   #3
JCoveiro
 
"Jorge Coveiro"
Nov 2006
Moura, Portugal

608 Posts
Default also2

Numbers:
Code:
3,11,23,191,3539,10691,83339,4031399,13347311
can be generated by: isprime(2*p+1)&&(p%4==3)

And numbers:
Code:
5,89,9689,21701,859433,43112609
can be generated by: isprime(2*p+1)&&(p%4==1)
JCoveiro is online now   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Universal seeds of the LLT for Mersenne and Wagstaff numbers T.Rex Math 22 2021-12-11 18:05
Mersenne prime exponents that are also Sophie Germain primes carpetpool Miscellaneous Math 4 2021-11-12 00:10
Status of Wagstaff testing? and testing Mersenne primes for Wagstaff-ness GP2 Wagstaff PRP Search 414 2020-12-27 08:11
Sophie-Germain primes as Mersenne exponents ProximaCentauri Miscellaneous Math 15 2014-12-25 14:26
Sophie Germains, multiple n-ranges, future of TPS MooooMoo Twin Prime Search 8 2008-11-05 15:03

All times are UTC. The time now is 02:37.


Mon Sep 26 02:37:52 UTC 2022 up 39 days, 6 mins, 0 users, load averages: 1.07, 0.99, 1.05

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