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