Please help me to define a multivalued function $ f(x) = 0 $ with integer solutions :
$ x_1 = p(n)$ and $ x_2 = q(n) $
such that
$\dfrac{ p(n) + q(n) } { 2 } = 2 n + 1 $
and
$ \operatorname{Mod}[p(n)^2-q(n)^2, 2 n + 1] = 0 $
$p(n)= \{3, 5, 7, 3, 11, 13, 9, 17, 19, 17, 23, 5, 17, 29, 31, 3, 33, 37, 29, 41, 43, 35, 47, 9, 3, 53, 19, 3, 59, 61, 3, 5, 67, 45, 71, 73, 29, 53, 79, 71,\dots\}$
$q(n)= \{3, 5, 7, 15, 11, 13, 21, 17, 19, 25, 23, 45, 37, 29, 31, 63, 37, 37, 49, 41, 43, 55, 47, 89, 99, 53, 91, 111, 59, 61, 123, 125, 67, 93, 71, 73, 121, 101, 79, 91,\dots\}$
Update :
For $ n=4 $ we get :
$ (p(4),q(4))->(3,15)$
$ \operatorname{Mod}[15^2-3^2, 9] = 0 $
The solution $(9,9)$ is not of interest because 9 is not a prime number.
Only when $ 2 n + 1 $ is an odd prime $ p(n) = q(n) $
Thank you in advance