$P, Q$ and $R$ are prime numbers. $P + Q = R$ and $1 < P < Q.$ What is the value of $P$?

Question

Attempt:

$P + Q = R$

$P + Q - R = 0 $

$1 < P < Q$

$1 + Q < P + Q < 2Q$

$1 + Q < R < 2Q$

I am lost...

The sum of two primes minus a third = 0 could be anything!

Answer 1

Hint: If both $P$ and $Q$ are odd, then $R$ is an even prime greater than $2$. That is impossible, so one of $P$ and $Q$ must be an even prime. Can you continue from there?

Answer 2

Since sum of two primes is $P$ and $Q$ prime greater than 2, then $R$ is odd, so one of $P$ and $Q$ them must be even. Since only even prime is $2$ and $P$ is smaller one (betven $P$ and $Q$) we have $P=2$.

Answer 3

$P = 2. \tag 1$

For, with

$P > 2, \tag 2$

we have that $P$ is odd; since $Q > P$, $Q$ is also odd. Then since

$R = P + Q, \tag 3$

$R$ must be even; but there is no even prime greater than $2$.