[seqfan] Re: Conjectures relating to twin primes and Lucas numbers
Andrew Weimholt
andrew.weimholt at gmail.com
Wed Dec 30 23:45:08 CET 2009
On Wed, Dec 30, 2009 at 2:01 PM, Creighton Kenneth Dement
<creighton.k.dement at mail.uni-oldenburg.de> wrote:
>
> I have two more variations involving Lucas numbers.
>
> Conjecture II:
> Let p be an odd prime.
> p, p+2 are twin primes if and only if
> p+2 divides Lucas(p+2) - 1 = A000032(p+2) - 1
>
> Conjecture III:
> Let n be any integer > 2.
> Then n, n+2 are twin primes if and only if
>
> n divides Lucas(n) - 1 = A000032(n) - 1
> and
> n+2 divides Lucas(n+2) - 1 = A000032(n+2) - 1
>
For any prime p, L(p) == 1 mod p,
however the converse is not necessarily true.
L(n) == 1 mod n does not necessarily mean n is prime.
Composite numbers for which L(n)==1 mod n are called Lucas
Pseudoprimes (A005845)
Your conjecture II is equivalent to saying
"if p is prime, p+2 is not a Lucas Pseudoprime"
Your conjecture III is equivalent to saying
"there are no Lucas Pseudoprimes which differ by 2 from either a prime
or another Lucas Pseudoprime"
Andrew
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