[puzzledobserver]

I am not a mathematician, but here is a motivation I read somewhere some years ago.

There are basically two ways to produce big numbers: add two small numbers, or multiply two small numbers. You can produce all positive integers by starting with zero and repeatedly adding one. You can almost do the same thing with multiplication too, except for these pesky primes, which are somehow atomic. Naturally then, one might ask: (a) How many primes are there? (b) How frequently do they occur? (c) Can we look at a number and determine whether it is a prime? Now consider: Despite being among the oldest of the mathematical disciplines, there are still open problems about primes that can be explained to high school students.

Also, multiplication and addition are not simply operations that are of interest with respect to integers, but similar ideas apply to a bunch of other domains too. Polynomials, for example. So primality and primality-like ideas are like catnip for mathematicians.