For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers?

$ \textbf{(A)}\ 794\qquad 
\textbf{(B)}\ 796\qquad 
\textbf{(C)}\ 798\qquad 
\textbf{(D)}\ 800\qquad 
\textbf{(E)}\ 802$
To solve the problem, we need to find the sum of all values of \( f(n) \) that are prime numbers, where \( f(n) = n^4 - 360n^2 + 400 \).

1. **Rewrite the function**:
   \[
   f(n) = n^4 - 360n^2 + 400
   \]
   We need to determine when \( f(n) \) is a prime number.

2. **Consider the quadratic form**:
   Let \( x = n^2 \). Then the function becomes:
   \[
   f(n) = x^2 - 360x + 400
   \]
   We need to find values of \( x \) such that \( f(n) \) is prime.

3. **Solve the quadratic equation**:
   We solve the quadratic equation \( x^2 - 360x + 400 = p \), where \( p \) is a prime number:
   \[
   x^2 - 360x + (400 - p) = 0
   \]
   The discriminant of this quadratic equation must be a perfect square for \( x \) to be an integer:
   \[
   \Delta = 360^2 - 4(400 - p)
   \]
   Simplify the discriminant:
   \[
   \Delta = 129600 - 1600 + 4p = 128000 + 4p
   \]
   For \( \Delta \) to be a perfect square, let \( \Delta = k^2 \):
   \[
   k^2 = 128000 + 4p
   \]
   Rearrange to solve for \( p \):
   \[
   4p = k^2 - 128000
   \]
   \[
   p = \frac{k^2 - 128000}{4}
   \]
   Since \( p \) must be a prime number, \( k^2 - 128000 \) must be divisible by 4.

4. **Check for integer solutions**:
   We need \( k^2 - 128000 \) to be divisible by 4. Let’s test some values of \( k \) to find prime \( p \).

   - For \( k = 128 \):
     \[
     k^2 = 128^2 = 16384
     \]
     \[
     4p = 16384 - 128000 = -111616 \quad \text{(not valid as } p \text{ must be positive)}
     \]

   - For \( k = 360 \):
     \[
     k^2 = 360^2 = 129600
     \]
     \[
     4p = 129600 - 128000 = 1600
     \]
     \[
     p = \frac{1600}{4} = 400 \quad \text{(not a prime number)}
     \]

   - For \( k = 362 \):
     \[
     k^2 = 362^2 = 131044
     \]
     \[
     4p = 131044 - 128000 = 3044
     \]
     \[
     p = \frac{3044}{4} = 761 \quad \text{(prime number)}
     \]

   - For \( k = 358 \):
     \[
     k^2 = 358^2 = 128164
     \]
     \[
     4p = 128164 - 128000 = 164
     \]
     \[
     p = \frac{164}{4} = 41 \quad \text{(prime number)}
     \]

5. **Sum the prime values**:
   The prime values of \( f(n) \) are 761 and 41. Their sum is:
   \[
   761 + 41 = 802
   \]

The final answer is \(\boxed{802}\).