Translate this back to the language of divisibility. You're asking to find primes $m$, arbitrarily large, such that
$$m \mid x^a - b.$$
But you can choose any $x$ you want! So now ask yourself: can the numbers of the form $x^a - b$ ($a$ and $b$ fixed) just keep magically happen to always be divisible by the same finite set of primes?
Let's think about this from a different direction too. What if I gave you a finite set of primes? Let's say 2, 3, and 5 for concreteness. What kinds of numbers can you make?
In fact, let's play a game with this set of primes. Every time I give you an $x$, you have to give me back a new number made only with the prime factors 2, 3, and 5, and it has to be larger than any number you've given me before.
Key question: regardless of how you answer, how fast does the sequence you give me grow?