Prime Numbers!!

Algebra Level 3

If $a \text { and } b$ are prime numbers such that their difference and their sum is prime too.

In other words, the four numbers $a, b, a-b, \text { and } a+b$ are all prime.

What can I say about the sum of these four numbers?

The sum is odd and divisible by 7. The sum is even. The sum is odd and divisible by 3. The sum is odd and prime. None of the above.

1 solution

X X
Nov 4, 2018

$a-b$ and $a+b$ are distinct primes, and the difference is $2b$ . So they are both odd or even(they can't be even because 2 is the only even prime).

Since $a+b$ is odd, one of $a$ and $b$ must be odd, and the other must be even(must be 2). Since $a>b$ , $b$ must be 2.

$a-2,a,a+2$ are all primes, but one of them is a multiple of 3, that means one of them( $a-2$ ) is 3.

So, $a=5,b=2$ , the four primes are $2,3,5,7$ , sum equals $17$ which is an odd prime.