I found a pattern that looks quite interesting.
$$\begin{align} 2(4 + 2) + 13^3 &= 47^2 \\ 2(4 +5) + 7^3 &= 19^2 \\ 2(4 + 8) + 1^3 &= 5^2.\end{align}$$ It seems at first that if $p$ is some prime number and $n\geqslant0$ is a whole number, then $$\begin{align} &2(4 + 2 + 3n) + (13 - 6n)^3 \\ =\space &6(n+2) + (13 - 6n)^3 \\ =\space &p^2.\end{align}$$ However, this seemingly only works for $n = 0, 1, 2$. Are there any other values of $n$ that satisfy the equation above? I know that $n \leqslant 2$ because otherwise $p^2 < 0$ and that's impossible (for $p$ that is). The equation is a squared number $k^2$ when $n = -27$, but $k = 5359225 = 5^2\times 463^2$.
Thanks to other users, $n$ has been tested down to $-10^8$.
Thank you in advance.