Euler famously showed that there are at least 65 idoneal (convenient) numbers. This was Euler's definition of idoneal number:
The number $n$ is idoneal if the following holds: Let $m>1$ be an odd number relatively prime to n which can be written in the form $x^2+ny^2$ with $x,y$ relatively prime. If the equation $m=x^2+ny^2$ has only one solution with $x,y≥0$, then $m$ is a prime number.
Moreover, it is allegedly said it exists infinitely many primes for each idoneal number. Is it true ?
With $n=k^2$ an even square, does that mean that $x,y$ are odd numbers ? So, with $k=1$, it exists infinitely many primes m with $m=x^2+4y^2$ and with $x,y$ odd numbers ?
The form $4 x^2 + 4xy + 5 y^2$ represents infinitely many primes (in the long run, one quarter of them). See the book by Cox, Primes of the Form $x^2 + n y^2.$
Next, to get primes we need that $y$ to be odd. Finally $$ 4 x^2 + 4 xy + 5 y^2 = (2x+y)^2 + 4 y^2. $$