# Is the sum of two coprime natural numbers prime?

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!

• Try some examples, like $3 + 5$ :) If it were so easy to generate primes, they'd be boring. May 7, 2018 at 11:00
• +1 for questioning intuition. May 7, 2018 at 11:02
• As a young teenager, I thought that checking if 2,3,5,7 are factors was enough to test if a number is prime (well, it worked for numbers below 100). Oh boy, did I find many large primes! May 7, 2018 at 11:42
• Downvoted for extreme lack of effort May 7, 2018 at 12:28
• Some examples no-one else has mentioned yet. Since: $$n = 1 + (n-1)$$ and since $1$ is coprime to anything, all numbers $n$ (prime or not) are sums of two coprime integers. For example $1000 = 1 + 999$. In the other extreme, you can try to write an $n$ as the sum of two almost equal terms. If $n$ is divisible by four, you will do: $$4k = (2k-1)+(2k+1)$$ where $2k-1$ and $2k+1$ are coprime (can you see why?), for example $1000=499+501$. And if $n$ is odd, you do: $$2k+1=k+(k+1)$$ where $k$ and $k+1$ are coprime, for example with $1001$ (not prime) you get $1001 = 500 + 501$. May 7, 2018 at 15:04

No - any two coprime odd numbers (e.g any two primes $\ne 2$) provide a counterexample.
Far from true. $3+5=8,\ 8+7=15{}{}{}{}$