Infinitely many primes of the form p=6k-1.

Subject:	Infinitely many primes of the form p=6k-1.
Date:	Sat, 5 Dec 2009 20:29:27 -0800 (PST)

Task:Prove infinitely many primes of the form p=3D6k-1.

Argument:

a) Suppose there are just a finite number of these primes, say
p1, ..., pn. Let a=3Dp<1>...p.

b) Rationally, a =3D (=961)modulo 6.

c) If n is even, let b =3D a + 4.
Otherwise let b =3D a + 6.
Then b =3D 5 modulo 6 =3D =961 modulo 6.

d) Any prime q dividing both a and b will divide either 4 or 6, hence
such a q equals 2 or 3.

e) None of the p will divide b.

f) Write b as a product of primes. Since b is odd, every prime factor
q of b can be written as q =3D 6r + 1, q =3D 6r + 3 or q =3D 6r + 5.

g) The last type (=961 modulo 6) is dismissed: none of p divides b.

h) The second type only permits q =3D 3. Therefore b, being the product
of all its prime factors, is either 1 or 3 modulo 6 (since 3 =3D 3
modulo 6): a contradiction=3D=3Dcompletes the proof.

Cheers,
M. Michael Musatov
Founder, http://www.meami.org