# How did Euler show that number is idoneal?

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.

• Your definition is unclear. You should make it clear that $n$ is idoneal if it satisfies that property. Right now you don't use the word idoneal anywhere in the definition.
– jgon
Dec 16 '18 at 14:51
• Sorry. Yes, we say that $n$ is idoneal if satisfies that property. Dec 16 '18 at 15:14

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.
• This may indeed be true of each non-idoneal $m$, but it's true of some idoneal $m$, too. E.g. $m=2, n=6=x^2+my^2$ only if $x=2, y=1$. I take it $x\geqslant 0, y\geqslant 0$. Feb 3 '19 at 13:57