In the german wikipedia , I found this site about idoneal numbers.
The mentioned statement is false :
- $D=1848$ is an idoneal number and $m=1849$ is an odd number.
- $x^2+1848y^2=1849$ has only one solution in positive integers, namely $x=y=1$.
- Since $(1,1848)=1$, $m$ should be prime, however it is a square of a prime ($43^2$).
Does the statement become valid if we rule out that $m$ is a perfect square, or do we still need other conditions ?
I also looked in the english wikipedia , but there is no criterion given for a prime, only a criterion for a prime power or twice a prime power. Of course, this can easily be verified , but there should be a way to use idoneal numbers for a direct primality test.