Question

Given any integer n, if n > 3, could n, n + 2, and n + 4 all be prime? Prove or give a counterexample.

Answer 1

We will try to prove the result using method of contradiction.

Solution:

If possible, let n, n+2 and n+4, all three numbers are prime.

This means that the numbers have no other factors apart from one and the number itself......................(1)

Now

Consider divisibility by number 3

Dividing by three gives remainders 0, 1 and 2

This implies that out of any three continuous set of natural numbers, at least one should be divisible by 3

In our case, as n is not divisible by 3 implies that one of n+1 and n+2 must be divisible by 3

Case 1: 3 divides n+2

This is in contradction to (1)

Case 2: 3 divides n+1

This implies that [(n+1)+3=n+4] is divisible by 3

Again this is in contraction to (1)

So our assumption was wrong

Thus n, n+2 and n+4 can not all be prime.