Question: How can I query the question database? For example, at this moment I would like to see the questions with 0 answers for the tag elementary-number-theory. Is there a query syntax? Are there more query possibilities when I use the API to this site?

Answer: 
You don't really need to check the database per se (which I interpret to mean either the [SEDE](http://data.stackexchange.com/), or the public data dumps). The search options available on the main site should be sufficient.

* `[elementary-number-theory]` will return posts tagged [elementary-number-theory](https://math.stackexchange.com/questions/tagged/elementary-number-theory "show questions tagged 'elementary-number-theory'").
* `answers:0` will return questions with no answers. (Really, truly zero answers; not just "unanswered", [which has a different meaning](https://meta.stackexchange.com/q/18870/214632).)

More information about search options can be found [in the Help Center](https://math.stackexchange.com/help/searching).
You can type the tag `[elementary-number-theory]` and click "unanswered" (directly below "ASK QUESTION") in order to see the unanswered questions for the tag. However, the results are then automatically sorted by vote count, from highest to lowest.

You're probably better off clicking "newest" (below and between "TAGS" and "USERS") and scrolling down if the question was relatively recent (within the past few days).