Primes $p$ such that $p^2$ divides $x^2 + y^2 + 1$ Call a prime $p$ awesome if there exist positive integers $x$ and $y$ such that $p^2$ divides $x^2+y^2+1$. 
Observation: $2$ is not awesome, because $x^2+y^2+1\not\equiv 0$ (mod $4$). But $3$ is awesome, because $9$ divides $27=5^{2}+1^{2}+1$. So my question is:

Are there infinitely many awesome primes? Can we find all awesome primes? 


Motivation: It is true that for every prime $p$, there exists positive integer $x$ and $y$ such that $p$ divides $x^2+y^2+1$. The proof can be found here. (This is actually a nice result; for example, it is used in a proof of Lagrange's $4$-square theorem). 

If this is too trivial, what can we say if $p^2$ is replaced by $p^{k}$? :)

Well time factor here is probably only 3.
Written in the ideal equation: $X^2+Y^2+1=3Z^3$
Solutions can be written using the following equation Pell. $p^2-2(k^2+k+1)s^2=1$
Then the solutions are of the form:
$Y=2p^2+2(k+2)ps-1$
$X=2p^2+2(2k+1)ps-1$
$Z=2p^2+2(k+1)ps-1$
$k$ - what some integer.
