[THOUGHT] We are given three equations for natural numbers a, b, c:

1. a + b + c = 151
2. a^2 + b^2 + c^2 = 10939
3. a^3 + b^3 + c^3 = 957871

We need to compute N = a^4 + b^4 + c^4, then find the smallest prime number strictly greater than N.

We can use symmetric polynomial relations to find a, b, c individually or compute N directly using Newton's identities.

Let: s1 = a + b + c = 151 s2 = ab + bc + ca s3 = abc

We have: a^2 + b^2 + c^2 = s1^2 - 2 s2 = 10939 => s2 = (s1^2 - (a^2+b^2+c^2)) / 2 = (151^2 - 10939)/2

Compute s2.

Also, for cubes: a^3 + b^3 + c^3 = s1^3 - 3 s1 s2 + 3 s3 = 957871 => s3 = (a^3+b^3+c^3 - s1^3 + 3 s1 s2) / 3

Then we can compute a^4+b^4+c^4 using Newton's identities: p4 = a^4+b^4+c^4 = s1 * p3 - s2 * p2 + s3 * p1 where p1 = s1, p2 = a^2+b^2+c^2, p3 = a^3+b^3+c^3.

Alternatively, we can compute directly.

Let's compute s2 and s3 first, then compute N, then find next prime.

We'll use Python to compute exactly (integers). Since a,b,c are natural numbers (positive integers), we can also solve for them via solving cubic: t^3 - s1 t^2 + s2 t - s3 = 0, but we don't need them individually for N.

Let's implement.

[/THOUGHT] [PYTHON] import sympy

Given values

s1 = 151 p2 = 10939 p3 = 957871

Compute s2 = ab+bc+ca

s2 = (s1**2 - p2) // 2 print("s2 =", s2)

Compute s3 = abc

p3 = s1^3 - 3s1s2 + 3*s3

=> 3s3 = p3 - s1^3 + 3s1*s2

s3_num = p3 - s1**3 + 3s1s2 s3 = s3_num // 3 print("s3 =", s3)

Check if s3 is integer

if s3_num % 3 != 0: print("Error: s3 not integer") else: # Compute N = a^4+b^4+c^4 # Using Newton's identities: p4 = s1p3 - s2p2 + s3p1 p4 = s1p3 - s2p2 + s3s1 print("N = p4 =", p4)

# Now find smallest prime > p4
# Use sympy's nextprime
next_p = sympy.nextprime(p4)
print("Smallest prime > N:", next_p)
print("Final answer:", next_p)

[/PYTHON]

[THOUGHT] The Python output gives N = 89909683 and the smallest prime greater than N is 89909713. This is the final answer.

\boxed{89909713}