Find all primes of the form n61 where n is any positive inte

Find all primes of the form n⁶1, where n is any positive integer (note: the equivalent statement is: determine for what kind of positive integers n that n⁶ 1 is composite). You need to prove your statement.

Solution

There are infinite primes of the form 6n-1

Suppose not. Let there be only finitely many primes, say p1, p2 , pk. Let

P=6*p1*p2*p3pk1

Now every prime is either of the form 6n1 or 6n+1 and product of any two numbers of the form 6n+1 is also of the form 6n+1. So the question is

What are the prime dividers of P?

They all can\'t be of the form 6n+1 since P is of the form 6n1. So it must have at least one prime factor of the form 6n1. Clearly p is not divisible by any of the primes p1, p2, pk. So there has to be a prime of the form 6n1 which is different from these primes. Hence, there has to be infinitely many primes of the form 6n1.