Is the sum of two coprime natural numbers prime?

Question

I am just getting started with some basic number theory and I was wondering: given two coprime natural numbers $a$ and $b$, is it true that $a+b$ is a prime number? My intuition says yes, because two coprime numbers by definition share no common factors and so there is nothing that may be factored out of both simultaneously, and thus, there is nothing that can be factored out of their sum. Further, looking at some simple base cases, there is no obvious example (at least to me) where this fails to hold. I am not sure if I have the correct intuition and am just failing to see how to rigorously demonstrate this claim, or if there is something obvious I am missing. Thanks a lot!

Answer 1

No - any two coprime odd numbers (e.g any two primes $\ne 2$) provide a counterexample.

Answer 2

Far from true. $3+5=8,\ 8+7=15{}{}{}{}$

Answer 3

Two odd numbers may be co-prime but their sum will always be an even number. Hence, the given statement does not hold true for co-prime numbers that are both odd.