I find number theory to be closer to a natural, investigative science akin to Physics rather than pure math such as calculus.

IANAM, but I doubt there was ever a modern calculus theorem that was hinted by a brute force search. One can argue that Calculus is "made up" and thus research in this field is finding deeper logical consequences for the initial axioms, while natural numbers just "have properties" and research is Number Theory is trying to uncover these properties by brute force and later trying to prove them, arguably quite unsuccessfully so far. It's an interesting tidbit in the philosophical question of "Does math exist only in our head?"