dirichlet'stheorem

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