20211208, 01:39  #12  
Apr 2020
2^{2}×3×7^{2} Posts 
Quote:
The rational number 1/3 is the thing that makes 1 when you multiply it by 3. Similarly (3 mod M(p))^{1} is the thing that makes 1 mod M(p) when you multiply it by 3 mod M(p). So "2/3" is just the value that makes 2 when multiplied by 3 mod M(p). And if we take W(p) and multiply it by 3, we get 2^p+1, which is indeed 2 mod M(p). Last fiddled with by charybdis on 20211208 at 01:40 Reason: field > ring: M(p) isn't necessarily prime! 

20211208, 01:55  #13 
Bemusing Prompter
"Danny"
Dec 2002
California
97F_{16} Posts 
I see, thanks.
The Wikipedia article isn't very clear and doesn't mention rings until much later. I should update it when I have time. 
20211208, 11:51  #14  
Feb 2004
France
3^{2}·103 Posts 
Quote:
I see that you got your answer. What I could add is about how to compute 2/3 mod Mq by hand and find an integer between 0 and Mq1. Since Mq = 0 mod Mq, x*Mq/3 is still 0 mod Mq. So you can write : (x*Mq+2)/3 = 0 mod Mq . Now, forget mod Mq and try to find an integer x*Mq+2 which is divisible by 3. Let's try with x= 1.... . First, x=1 : (Mq+2)/3=(2^q1+2)/3 = (2^q+1)/3 which is a Wagstaff number, thus an integer, when q is prime. Moreover, does 1/3 mod Mq always exist ? even when Mq is not a prime ? Yes, since (2*Mq+1)/3 is always an integer. 1/3 mod 31 = 21 1/3 mod 127 = 85 1/3 mod 2047 = 1365 Try : Pari/gp : forprime(q=3,1000,Mq=2^q1;print(q," ",((2*Mq+1)/3)lift(Mod(1/3, Mq)))) 

20211210, 10:08  #15  
Mar 2021
France
2·3^{2} Posts 
Quote:


20211210, 13:16  #16 
Feb 2004
France
927_{10} Posts 

20211211, 15:58  #17 
Mar 2021
France
2×3^{2} Posts 
I don't know if it fits with this topic but maybe I found a probable primality test for the number of the form (3^p1)/2. I don't if it's new at all :
Let N = (3^p−1)/2 when p is a prime number p>3 Let S(i) =S(i−1)^3−3*S(i−1) with S0=52 . Then N is prime if and only if S(p−1) ≡ S0 (modN). I choose 52 because this is one of the seeds for the test of LucasLehmer and it seems it works with this seed (see here : https://oeis.org/A018844 ) I checked until p=2000 and I didn't find counterexample. Last fiddled with by kijinSeija on 20211211 at 16:13 
20211211, 16:06  #18 
Sep 2002
Database er0rr
3·1,327 Posts 

20211211, 16:12  #19 
Mar 2021
France
2·3^{2} Posts 
Sure sorry I will edit.
I don't know if "probable primality test" exists by the way :) Nevermind it exists. So when a prime is found by some primality test with no proof for the primality test it's a PRP right ? Last fiddled with by kijinSeija on 20211211 at 16:30 
20211211, 16:35  #20  
Sep 2002
Database er0rr
3×1,327 Posts 
Quote:
When a probable prime (PRP) is found by some PRP test then it may or may not be prime. A deterministic test such as BLS/KP/CGH/ECCP or others is needed to be 100% sure. 

20211211, 16:40  #21 
Mar 2021
France
2·3^{2} Posts 
Ok thank you for that information.
I only know ECCP in your list 
20211211, 16:58  #22  
Sep 2002
Database er0rr
3·1,327 Posts 
Quote:
KP: https://math.dartmouth.edu/~carlp/PDF/110.pdf 30% CHG: https://primes.utm.edu/bios/page.php?id=797 (could not find a paper) >25% The last two algorithms can be found somewhere in the depths of this forum. https://mersenneforum.org/showthread.php?t=24853 Last fiddled with by paulunderwood on 20211211 at 17:15 

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