MathNet 2020 — Math Olympiad Vault — Part 04 Question 20

Find the total number of primes $p \lt 100$ such that $\lfloor (2 + \sqrt{5})^p \rfloor - 2^{p+1}$ is divisible by $p$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.