Decision Problem: Is $ 2^k$ + $ M$ a prime?

The inputs for both $ K$ and $ M$ are integers only. The solution is the sum of $ 2^k$ +$ M$ . (*Use AKS to decide prime*)

*The powers of 2 have approximately $ 2^n$ digits. Consider $ 2^k$ where $ K$ = 100000. Compare the amount of digits in $ K$ to the amount of digits in it’s solution!*

## Question

Could the solution be verified in polynomial time even though the solution is exponentially large?

If no, can I say it is not in $ NP$ ?