Wanless' Theorem - Twin Prime Conjecture
There exist an infinite number of pairs of primes, P and Q, s.t. P-Q=2N, for all N
(Corollaries include affirmation of the Twin Prime Conjecture)
Proof:
Let p0,p1,p2... be the positive primes, including 1, in inceasing order.
Let Pn=p0p1p2...p(i0-1)p(i0+1)...p(i1-1)p(i1+1)...pn + (p(i0)**j0)(p(i1)**j1)... and
let Qn=p0p1p2...p(i0-1)p(i0+1)...p(i1-1)p(i1+1)...pn - (p(i0)**j0)(p(i1)**j1)... [any i, j]
Note that hcf (Pn, p0p1p2p3p4...pn) = 1 and hcf (Qn, p0p1p2p3p4...pn) = 1 [Euclid]
Then Pn and Qn are either both prime or
Pn is divisible by a prime greater than pn or
Qn is divisible by a prime greater than pn
Let n->99999...
=> pn->99999... [Euclid]
=> Pn and Qn are both prime
=> There exist an infinite number of pairs of primes, Pn and Qn, s.t. Pn-Qn=2N [any N>=0] (1)
=> There exist an infinite number of pairs of primes, Qn and Pn, s.t. Qn-Pn=2N [any N<=0] (2)
(1) and (2) => There exist an infinite number of pairs of primes, P and Q, s.t. P-Q=2N [any N] (3)
Corollary (Twin Prime Conjecture):
(3) [N=1]
Copyright 1997 James Wanless