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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!

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    $\begingroup$ Try some examples, like $3 + 5$ :) If it were so easy to generate primes, they'd be boring. $\endgroup$
    – PM 2Ring
    May 7, 2018 at 11:00
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    $\begingroup$ +1 for questioning intuition. $\endgroup$
    – lisyarus
    May 7, 2018 at 11:02
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    $\begingroup$ 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! $\endgroup$ May 7, 2018 at 11:42
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    $\begingroup$ Downvoted for extreme lack of effort $\endgroup$
    – Lonidard
    May 7, 2018 at 12:28
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    $\begingroup$ 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 $1001 = 500 + 501$. $\endgroup$ May 7, 2018 at 15:04

3 Answers 3

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

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    $\begingroup$ Any two non-two primes, that is :-) $\endgroup$ May 7, 2018 at 11:03
  • $\begingroup$ Ah true, that makes sense. Thanks a lot! $\endgroup$ May 7, 2018 at 11:07
  • $\begingroup$ @PatrickFraser No problem! If you want some intuition, note that since primes are numbers with only two factors and factorisation is related to multiplication not addition, many properties of primes are multiplicative. Additive properties are generally harder to prove or gain intuition from. $\endgroup$
    – John Doe
    May 7, 2018 at 11:13
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Far from true. $3+5=8,\ 8+7=15{}{}{}{}$

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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.

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