TITLE: Prove that the number $2a+b+c$ is not a prime number
QUESTION [8 upvotes]: Let $a,b,c$ be positive integers such that ${ a }^{ 2 }-bc$ is a perfect square.  prove that the number $2a+b+c$ is not a prime number.

I have no idea about the solution . How can I start to prove it .

REPLY [4 votes]: Let us assume that $p=2a+b+c -$ prime. Then $a^2-bc=a^2-b(p-2a-b)=(a+b)^2-pb$. If this number is equal to $n^2$, it is less $a^2$, and $n < a$. It turns out that $pb=(a+b+n)(a+b-n)$, where both factors  exceed  $b$. Therefore, each of them is strictly less than $p$, ie $p$ is not divided. This contradicts the fact that the product is divided into simple $p$.