## Using the Legendre Symbol to Prove There are Infinitely Many Primes

There are infinitely many primes of the form

\[4k+1\]

.Proof

This is a case of Dirichlet's Theorem which implies infinitely many primes of the form

\[ak+b\]

. Suppose there are infinitely many primes of the form \[4k+1\]

, \[p_1, \; p_2,..., \; p_n\]

. Let \[N=4(p_1p_2...p_n)^2+1\]

. \[N\]

is odd, so must be divisible by some odd prime \[p\]

. The \[N=4(p_1p_2...p_n)^2+1 \equiv 0 \; (mod \; p) \rightarrow (p_1p_2...p_n)^2 \equiv -1 \; (mod \; p)\]

. \[(p_1p_2...p_n)^2\]

is a square number so \[(-1/p)=1\]

then by the properties of The Legendre Symbol, \[p\]

is of the form \[4k+1\]

. The assumption that \[p_1, \; p_2,...,p_n\]

are the only primes of this form forces \[p\]

to be one of these, say \[p_i\]

. But the \[p\]

divides \[N=4(p_1p_2...p_n)^2+1\]

and \[4(p_1p_2...p_n)^2\]

so \[p\]

divides \[N-4(p_1p_2...p_n)^2=1\]

which is impossible, so there are infinitely many primes of the form \[4k+1\]

.