Euler famously showed that there are at least 65 idoneal (convenient) numbers. This was Euler's definition of idoneal number:
Number $n$ is idoneal if 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 $x,y\ge0$, then $m$ is a prime number.
How did Euler prove that for example $15$ or $168$, or any other, is in fact idoneal?
I am not interested in proof with Gauss's genus theory, or anything sophisticated. I am interested in techniques that were available to Euler.
The basic idea, as Fueter explains here (p. 19-20), is showing that for each number $m$ that is not idoneal there exists a composite number smaller than $4m$ that has a unique representation in the form $x^2 + my^2$. In modern terms, this corresponds to the result that each ideal class (or ring class when working orders) in ${\mathbb Q}(\sqrt{m})$ contains an element of norm $< 4m$. As Weil writes, Euler's presentation of this topic is highly confusing, so even he did not dare to spell out the gaps in Euler's proof.