Prime Pickle

Number Theory Level 5

Let $p$ and $q$ be prime numbers and $r$ be a whole number, such that $(p)(p+3)+(q)(q+3)=(r)(r+3)$ . Find the sum of all possible values of $p$ .

Details:

$\bullet$ $r$ needn't be a prime number.

$\bullet$ $1$ is nether prime nor composite.

$\bullet$ You may use the fact that all primes except $2$ and $3$ can be expressed in the form $6k \pm 1$ .

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