In the wording of the problem, are you supposed to consider "for all a" being all natural numbers???
Consider the case where p=2. The scary-looking is actually equal to . And just means "is even." Now, does not exist for any . So If is even, does not work, and if , then does not work.
In modular arithmetic, exists if and only if and are relatively prime.
Therefore, we need some restrictions on a. Either way, this problem needs to be clarified before any proof is pursued.