Find all Primes

we have

p² + q² +7pq = k²

k² - (p + q)² = 5pq

(k + p + q)(k - p - q) = 5pq

CASE 1 ; k - p -q = 5 , pq = k + p +q
pq = 2 (p + q) + 5

p = (2q + 5)/(q-2) = 2 + 9/(q-2)

9/q-2 must be an integer so q=3,5,11 p = 11,5,3

CASE 2 :k + p+q = 5 , pq = k - p - q

and similarly you can see other cases

THNKS for the solution

(i assumed there can't be any solution and was trying to find contradictions only by congruences ;(

also p=q=any prime are more solutions :):)

omg i missed even that !!!