Learn Russian: Я не знаю номер.

In Russian, you don't need helper verbs like "do" or "does" to make a negative sentence. You simply put the negative word не directly before the verb. So, "I do not know" is just Я не знаю (literally "I not know").

Номер receives the action, so it's the direct object. Shouldn't it be in the accusative case and change its ending?

It is in the accusative case! However, for masculine nouns that are inanimate (things, not people or animals) like номер, the accusative form looks exactly the same as the nominative (dictionary) form. That's why the ending doesn't change here.

Does номер mean any kind of number, like in math?

No, номер is used specifically for numbers that act as labels or identifiers, like a phone number, a hotel room, or a license plate. If you are talking about a mathematical number or a quantity, you would use a different word, like число.

What is the dictionary form of знаю?

The dictionary form (infinitive) is знать (to know). Знаю is the "I" form in the present tense. It follows a very regular pattern for verbs ending in -ать, where you drop the -ть and add -ю for the "I" form.