Dirichlet's Theorem
The infinite arithmetic series: a, a+b, a+2b, a+3b, ... contains infinitely many primes [if hcf (a,b) = 1]
Proof:
Let p0,p1,p2... be the positive primes, including 1, in inceasing order.
Let a = (p(i0)**j0)(p(i1)**j1)...
Let Pn=b*p0p1p2...p(i0-1)p(i0+1)...p(i1-1)p(i1+1)...pn + a
i.e. Pn=b*p0p1p2...p(i0-1)p(i0+1)...p(i1-1)p(i1+1)...pn + (p(i0)**j0)(p(i1)**j1)...
Note that hcf (Pn, p0p1p2p3p4...pn) = 1 [if hcf (a,b) = 1] and that Pn is itself a member of the series
Then Pn is either prime or
Pn is divisible by a prime greater than pn
Let n->99999...
=> pn->99999... [Euclid]
=> Pn is prime
=> There exists an infinite number of primes, Pn, which are members of the series
Copyright 1998 James Wanless