What Is Number Theory?

I'm interested in kind of enumerating or listing the kinds of evidence that fall short of a proof. A huristic might be something like an argument, except at a much lower standard of rigor. Sometimes one might have a model that we think captures the essential elements of the mathematical system we're trying to understand. And so then you would conjecture that your system has the same behaviour as your model. Ok, at some point i want to hear some examples of models and conjectures, and, you know, the extent to which they work or don't work on some questions or not others. But if you don't mind, i'd like to go back just for a few little personal