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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