AMC 10B 2024 (Problem 24)

Let

P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}.

How many of the values

P(2022)

P(2023)

P(2024)

, and

P(2025)

are integers?

\text{(A)}\;0

\text{(B)}\;1

\text{(C)}\;2

\text{(D)}\;3

\text{(E)}\;4