[THOUGHT] I need to find natural numbers $a$, $b$, $c$ satisfying the given system of equations, then compute $N = a^4 + b^4 + c^4$, and finally find the smallest prime strictly greater than $N$.

Let me use the power sum identities. Let:

• $p_1 = a + b + c = 151$
• $p_2 = a^2 + b^2 + c^2 = 10939$
• $p_3 = a^3 + b^3 + c^3 = 957871$

I can find the elementary symmetric polynomials:

• $e_1 = a + b + c = 151$
• $e_2 = ab + bc + ca = \frac{(a+b+c)^2 - (a^2+b^2+c^2)}{2} = \frac{p_1^2 - p_2}{2}$
• $e_3 = abc$ can be found using Newton's identity for $p_3$

Newton's identities:

• $p_1 = e_1$
• $p_2 = e_1 p_1 - 2e_2$
• $p_3 = e_1 p_2 - e_2 p_1 + 3e_3$

From these, I can find $e_1, e_2, e_3$, then $a, b, c$ are roots of $t^3 - e_1 t^2 + e_2 t - e_3 = 0$.

Then I can compute $p_4 = a^4 + b^4 + c^4$ using Newton's identity: $p_4 = e_1 p_3 - e_2 p_2 + e_3 p_1$

Let me compute this step by step. [/THOUGHT] [PYTHON] from sympy import symbols, solve, isprime, nextprime

Given values

p1 = 151 p2 = 10939 p3 = 957871

Elementary symmetric polynomials

e1 = p1 e2 = (p1**2 - p2) // 2 print(f"e2 = {e2}")

From Newton's identity: p3 = e1p2 - e2p1 + 3*e3

So: e3 = (p3 - e1p2 + e2p1) / 3

e3_num = p3 - e1p2 + e2p1 print(f"e3 numerator = {e3_num}") e3 = e3_num // 3 print(f"e3 = {e3}")

Verify: a, b, c are roots of t^3 - e1t^2 + e2t - e3 = 0

t = symbols('t') poly = t3 - e1*t2 + e2*t - e3 roots = solve(poly, t) print(f"Roots: {roots}")

Compute p4 using Newton's identity: p4 = e1p3 - e2p2 + e3*p1

p4 = e1p3 - e2p2 + e3*p1 print(f"N = p4 = {p4}")

Find smallest prime strictly greater than N

next_prime = nextprime(p4) print(f"Smallest prime > N: {next_prime}") [/PYTHON]

[THOUGHT] The calculation shows that the natural numbers are $a = 21$, $b = 33$, $c = 97$ (in some order). The value of $N = a^4 + b^4 + c^4 = 89909683$, and the smallest prime strictly greater than $N$ is $89909713$. [/THOUGHT]

\boxed{89909713}