Show that the sum of twin primes p and p2 is divisible by 12

Show that the sum of twin primes p and p+2 is divisible by 12 provided that p>3.

Solution

Given if p, p + 2 is a twin prime pair with p > 3, then as we proved in class, p + 1 is divisible by 3.

Also, we know that p will be an odd number if p is any prime number bigger than 2.

if p is odd then p + 1 will be even. Thus, p + 1 is divisible by both 3 and 2, it\'s means p+1 is divisible by 6.

Now, p + p + 2 = 2(p + 1).

So, if p + 1 is divisible by 6, then 2(p + 1) is divisible by 12.

hence the sum of twin primes p and p+2 is divisible by 12.