Question

Prove that there exist infinitely many primes of the form 6n 5

Answer 1

Answer:

Step-by-step explanation:

Primes of the form 6n+5 is particularly easy:

Suppose that there are finitely many primes of the form 6n+5, namely p1,⋯,pn.

Consider p∗=6p1⋯pn−1.

Note that any odd prime other than 3, is of the form 6n+1 or 6n+5.

Thus, prime divisors of p∗ are either of the form 6n+1 or 6n+5.

The prime divisors of p∗ should have at least one prime divisor of the form 6n+5.

This is a contradiction.

For primes of the form 6n+1, use the following:

"Existence of x in x2−x+1≡0 mod p ⟺ p is of the form 6n+1."

Suppose there are only finitely many 6n+1 primes, namely p1,⋯,pn,

Then consider p∗=(p1⋯pn)2−(p1⋯pn)+1.

Prime divisor of p∗ should be of the form 6n+1 according to the above equivalence.

This is a contradiction.