Can I efficiently verify if given $h$ is the class number of a quadratic field?
Can I efficiently verify if given $h$ is the class number of a quadratic field?
Computing the class number is not tractable.
I tried pari's Qfb and did some experiments with Lidia, but I must be missing something.
What I tried is for random $a$ compute $a^h$ but I don't get the identity and in Lidia I can't check if it is principal.
If i correctly understand the question, the question is as follows: There are given a quadratic number field $K=\mathbb Q(\sqrt D)$ and a to-be-class number integer $h^?$. We want to verify $h^?=h(K)$ efficiently, in particular without calling the computation of the class number.
(So the question is NOT if for a given $h$ there exists a $K$ realizing $h=h(K)$...)
Is it enough / necessary to verify, that $h^?$ is a multiple of $h(K)$? If yes, then one may have "only" to compute $a$ to the power $h^?$ for each $a$ in the Minkowski cage insured by theory. But we are still not able to claim $h^?=h(K)$, but only $h(K)\ |\ h^?$. This may not even be really effective for the half problem.
The question is a sage question or a math one?
If it is a math question, why not use analytic methods?
Remember that checking if a given ( h ) is the class number of a quadratic field can be quite tricky. While tools like PARI/GP’s Qfb and Lidia are powerful, they might not always give straightforward solutions for every scenario. From your experiments, it seems like you’re on the right track by trying different values of ( a ) and computing ( ah ). However, the difficulty in verifying if an ideal is principal in Lidia suggests there might be some nuances or additional steps required. Have you thought about trying out other math software or libraries that might offer different algorithms or methods for this verification? Sometimes, using a different approach or tool can make a significant difference.