[THOUGHT] We have three equations:

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

We need N = a^4 + b^4 + c^4, then find the smallest prime > N.

We can use Newton's identities (power sums). Let: S1 = a + b + c = 151 S2 = a^2 + b^2 + c^2 = 10939 S3 = a^3 + b^3 + c^3 = 957871 We want S4 = a^4 + b^4 + c^4.

Let e1 = a + b + c = S1 e2 = ab + bc + ca e3 = abc

Newton's identities: S1 = e1 S2 = e1S1 - 2e2 => 10939 = 151151 - 2e2 S3 = e1S2 - e2S1 + 3e3 => 957871 = 15110939 - e2151 + 3e3

We can solve for e2, e3, then compute S4 using: S4 = e1S3 - e2S2 + e3S1 - 4e4? Wait careful: For three variables, the recurrence is: S4 = e1S3 - e2S2 + e3S1 - 4e4, but e4 = 0 because there are only three variables (product of four distinct variables is zero). Actually e4 = 0 because we have at most three variables. So S4 = e1S3 - e2S2 + e3*S1.

Let's compute step by step with Python. [/THOUGHT] [PYTHON] import sympy

S1 = 151 S2 = 10939 S3 = 957871

From S2 = e1S1 - 2e2

e1 = S1 e2 = (e1*S1 - S2) / 2 print("e2 =", e2)

From S3 = e1S2 - e2S1 + 3*e3

e3 = (S3 - e1S2 + e2S1) / 3 print("e3 =", e3)

Compute S4 = e1S3 - e2S2 + e3*S1

S4 = e1S3 - e2S2 + e3*S1 print("S4 = a^4 + b^4 + c^4 =", S4) print("S4 as integer:", int(S4) if S4.is_integer() else S4) [/PYTHON]

[THOUGHT] We have computed ( N = a^4 + b^4 + c^4 = 89909683 ). Now we need the smallest prime number strictly greater than ( N ). We can check primes using a simple primality test. Since ( N ) is large, we can iterate numbers above ( N ) until we find a prime. We'll use sympy's nextprime function. [/THOUGHT] [PYTHON] import sympy N = 89909683 next_prime = sympy.nextprime(N) print("Next prime after N:", next_prime) [/PYTHON]

[THOUGHT] The smallest prime strictly greater than ( N = 89909683 ) is ( 89909713 ), as computed by Python. [/THOUGHT] \boxed{89909713}