Comment by skinner_

If subtraction is allowed but negative numbers are not, that's known to be undecidable, even without exponentiation. There are several ways to deal with a subtraction with a negative result, but each variant is undecidable.

If we allow negatives but disallow exponentiation, that's decidable. If we allow negatives and partial exponentiation that stays within the integers, that's undecidable again.