Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418{,}195{,}493$.

[i]Proposed by Eugene Chen[/i]
1. We start with the function \( f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5 \) and need to find the prime \( p \) such that \( f(p) = 418,195,493 \).

2. First, we observe that \( 418,195,493 \) is a large number, and we need to find a prime \( p \) such that \( f(p) \) equals this number. 

3. We can simplify our task by considering the polynomial inside the function. Let's denote it as \( g(x) = x^4 + 2x^3 + 4x^2 + 2x + 1 \). We need to find \( p \) such that \( g(p)^5 = 418,195,493 \).

4. To find \( p \), we first need to estimate the value of \( g(p) \). Since \( 418,195,493 \) is a large number, we can take the fifth root of it to get an approximate value for \( g(p) \):
   \[
   \sqrt[5]{418,195,493} \approx 33
   \]

5. Now, we need to find a prime \( p \) such that \( g(p) \approx 33 \). We can test small prime numbers to see if they satisfy this condition.

6. Let's start with \( p = 2 \):
   \[
   g(2) = 2^4 + 2 \cdot 2^3 + 4 \cdot 2^2 + 2 \cdot 2 + 1 = 16 + 16 + 16 + 4 + 1 = 53
   \]
   \[
   f(2) = 53^5 = 418,195,493
   \]

7. We see that \( g(2) = 53 \) and \( 53^5 = 418,195,493 \), which matches the given value.

8. Therefore, the prime \( p \) that satisfies \( f(p) = 418,195,493 \) is \( p = 2 \).

The final answer is \( \boxed{2} \)