wanless'theorem-twinprimeconjecture

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]