[THOUGHT] I will brute-force search over natural numbers a, b, c with a + b + c = 151, and check which triple also satisfies the given equations for squares and cubes. Once I find (a, b, c), I'll compute N = a^4 + b^4 + c^4, then use sympy's nextprime function to find the smallest prime greater than N. [/THOUGHT] [PYTHON] from sympy import nextprime

solutions = [] target_s1 = 151 target_s2 = 10939 target_s3 = 957871

for a in range(1, target_s1): # a >= 1 for b in range(1, target_s1 - a): # b >= 1 c = target_s1 - a - b if c < 1: continue if aa + bb + c*c != target_s2: continue if a3 + b3 + c**3 != target_s3: continue solutions.append((a, b, c))

print("Solutions:", solutions)

if not solutions: raise ValueError("No solutions found")

Use the first (should be only) solution

a, b, c = solutions[0] N = a4 + b4 + c**4 print("a, b, c:", a, b, c) print("N:", N)

Find smallest prime > N

p = nextprime(N) print("Next prime after N:", p)

[THOUGHT] Using the valid Python output, the triple (a,b,c) satisfies the equations and yields N = 89909683. The smallest prime strictly greater than N is 89909713. $\boxed{89909713}$