# Thread: Infintely many primes of the form:

1. ## Infintely many primes of the form:

Hi,

Prove that there exists infinetly many primes of the form:

a) 4q + 1 for some q in N U {0}
b) 4q + 3 for some q in N U {0}

Any help would be greatly appreciated.

2. a) consider the polynomial $
f(x) = x^2 + 1
$

Fix $x$ and suppose $
{f\left( x \right)}
$
has a prime divisor $p>2$ then $
x^2 \equiv - 1\left( {\bmod .p} \right)
$

By Fermat's Little Theorem: $1\equiv{x^{p-1}}=
(x^2)^{\tfrac{p-1}{2}} \equiv (-1)^{\tfrac{p-1}{2}}\left( {\bmod .p} \right)
$
and so $
\left. 4 \right|\left( {p - 1} \right)
$
that is $
p \equiv 1\left( {\bmod .4} \right)
$
(1)

Now assume there were finitely many primes $\equiv 1 (\bmod.4)$ let them be $
\theta _1 ,...,\theta _n
$
and consider $f(
2\cdot \theta _1 \cdot ... \cdot \theta _n )
$
you can check that $
f\left( {2\cdot \theta _1 \cdot ... \cdot \theta _n } \right) \equiv 1\left( {\bmod .\theta _i } \right)
$
$
\forall i \in \left\{ {1,...,n} \right\}
$
and so it is not divisible by a prime $\equiv{1}(\bmod.4)$ however this contradicts (1), since $f(2\cdot
\theta _1 \cdot ... \cdot \theta _n )
$
must have at least one odd prime divisor (since it's odd>1). $\square$

b) This one is easier. Suppose there were finitely many $
\rho _1 ,...,\rho _n
$
, and consider $
4 \cdot \left( {\rho _1 \cdot ... \cdot \rho _n } \right) + 3
$

3. Thanks alot.

I understand.