## Lucas-Lehmer like test

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• Hello, Group. Let n= 1, 2, 3, etc. and Nn= 3^n -2. Does the sequence 1, 7, 25, 79, 241, 727, 1185, etc. have a test for verifying the primality of Nn similar
Message 1 of 2 , Mar 1 12:49 PM
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Hello, Group.

Let n= 1, 2, 3, etc. and Nn= 3^n -2.

Does the sequence 1, 7, 25, 79, 241, 727, 1185, etc. have a test for
verifying the primality of Nn similar to the LL test for Mersenne Mp=
2^p -1?

I tried for several weeks to find one and couldn't.

Bill
• ... No. Note N+1 == 3^n-2+1 == 3^n-1 and N-1 == 3^n-2-1 == 3*(3^(n-1)-1 So given enough factors of N+-1 the combined classical tests can be used:
Message 2 of 2 , Mar 1 5:54 PM
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>
> Let n= 1, 2, 3, etc. and Nn= 3^n -2.
>
> Does the sequence 1, 7, 25, 79, 241, 727, 1185, etc. have a test for
> verifying the primality of Nn similar to the LL test for Mersenne Mp=
> 2^p -1?
>

No.

Note N+1 == 3^n-2+1 == 3^n-1
and N-1 == 3^n-2-1 == 3*(3^(n-1)-1

So given enough factors of N+-1 the combined "classical tests" can be
used:
http://primes.utm.edu/prove/index.html

These have been implemented in PFGW:
http://groups.yahoo.com/group/primeform/

At the cutting edge, a less than 33.33 % factored percentage could
lead to a proof: "Konyagin-Pomerance" (KP) or "Coppersmith
Howgrave-Graham" (CHG).

If the number is sub-gigantic (<10,000 digits) it can be proven with
Marcel Martin's ECCP implementation, Primo:
http://www.ellipsa.net/

Wojciech Florek has some good pages covering 3^n-2:
http://perta.fizyka.amu.edu.pl/pnq/

> I tried for several weeks to find one and couldn't.
>

You aren't the first to try ;-)

Paul
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