Question

• Show that there are infinitely many primes p≅ 1(mod5).

Answer 1

Let us show that there are infinitely many primes $p\equiv 1(mod\ 5)$. Let us prove by contraposition. Suppose that there are finitely many such primes, and let these primes be $p_i$ for $1\le i\le n$. Let $m=5p_1p_2...p_n+1$. Then $m\equiv 1(mod\ 5)$ and $m>p_i$ for all  $1\le i\le n$. Therefore, $m$ is a composite number, and thus $p_s$ divides $m$ for some $1\le s\le n$. Consequently, $m=kp_s$, that is $p_s(5p_1...p_{s-1}p_{s+1}...p_n)+1=kp_s$. It follows that $p_s$ divides $kp_s-p_s(5p_1...p_{s-1}p_{s+1}...p_n)=1$. It is impossible because of $p_s>1.$ This contradictions proves that there are infinitely many primes $p\equiv 1(mod\ 5)$.