I found out this information on wikipedia: "In mathematics, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as $x^2 ± Dy^2$ (where $x^2$ is relatively prime to $Dy^2$) is a prime power or twice a prime power. In particular, a number that has two distinct representations as a sum of two squares is composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes. "
Do we have this relationship $2n+1=x^2 ± Dy^2=p^k$ if the integer $2n+1$ is expressible in only one way with $D$ an idoneal number ? Is that really a prime power $p^k$ with $k>=1$ ? Or do we only have $k=1$ ? Which values can take the variable $k$ ? is it an odd or an even number or both ? Are there an infinity ?
Does it exist a book or an article about this topic with a proof of this result ?
Thanks for your help