Why is the sum of two prime numbers greater than two never prime?

Answer 1

Consider prime numbers p1, p2 greater than 2. Since p1 and p2 are prime and greater than 2, they are both necessarily odd. Hence, they are of the form:

p1 = 2k+1, p2 = 2j+1, where j and k are positive integers. Their sum is then:

p1+p2 = (2k+1)+(2j+1) = 2k+2j+2 = 2 (k+j+1). So 2 divides their sum, hence the sum can't be prime