nt.number theory – solutions in prime numbers of the equation $ , 3p ^ 2 + q ^ 2 = r ^ 2 + 3 $

Let us consider the diophantine equation $ , 3p ^ 2 + q ^ 2 = r ^ 2 + 3 $,

Actually, I'm only interested in tern solutions $ , (p, q, r) , $ of prime numbers.

It is easy to prove that if $ , (p, q) , $ is a pair of twin prime numbers $ , (p lt q) $, then $ , (p, q) , $ solves the equation if $ , p , $ is a Sophie Germain Prime and $ , r , $ represents its safe prime number (that is $ , r = 2p + 1 $).

Likewise if $ , (p, q) , $ is a pair of twin primes with $ , p gt q $, then $ , (p, q) , $ solves the equation if $ , r = 2p-1 $,

So here are some solutions to the given equation:

$ (3,5,7) ; ; (5,7,11) ; ; (11,13,23) ; ; (29,31,59) ; ; (41.43, 83) ; ; … $

$ (7,5,13) ; ; (19,17,37) ; ; (31,29,61) ; ; … $

We force to find other classes of solutions $ , r = p + 2q $, This restriction means that only the pairs are taken into account $ , (p, q) , $ meet the following condition:

$ 5 (q ^ 2 + 1) = 2 ((p-q) ^ 2 + 1) ; ; ; ; ; ; ; ; ; (*) $

A tern satisfactory $ , (*) , $ is $ , (13,5,23) $,

I ask you to find other, possibly more general, classes of solutions to the given equation.

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