[seqfan] Conjectures relating to twin primes and Lucas numbers

[Creighton Kenneth Dement]

Dear Seqfans,

I mentioned the conjecture:
a(n+4) = A001359(n+2) for all n

a while back.

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

The third conjecture looks a bit strange to me... I apologize if I've
overlooked something obvious.

Happy New Year,
Creighton