Is the sum of two prime numbers always a prime numer?
It is not.
Of the infinitely many primes only one (the number 2) is even,
the rest are all odd. The sum of any two primes other than 2 is
even and therefore not a prime. If one of the primes in the sum is
2 then the sum is a prime only if the other is the lower of a pair
of twin primes.
So, while it is possible, it is certainly more likely that the
sum is a composite.