[THOUGHT] I need to compute N = a^4 + b^4 + c^4 using the given power sums and Newton identities.

First, verify s2 = ab + bc + ca = (p1^2 - p2)/2

p1 = 151, p2 = 10939

Then s3 = abc from the identity p3 = s1 p2 - s2 p1 + 3 s3, rearranged.

But to verify, compute expected p3 and check.

Then p4 = s1 p3 - s2 p2 + s3 p1

Use Python to compute exactly. [/THOUGHT] [PYTHON] p1 = 151 p2 = 10939 p3 = 957871

s1 = p1 s2 = (s1**2 - p2) // 2 print(f"s2: {s2}")

Verify p3 from identity

expected_p3 = s1 * p2 - s2 * p1 + 3 * s3 # but s3 unknown yet

Compute s3 from p3 - 3 s3 = s1 (p2 - s2)

diff = p2 - s2 temp = s1 * diff s3 = (p3 - temp) // 3 print(f"s3: {s3}")

Verify the identity

computed_p3 = s1 * p2 - s2 * p1 + 3 * s3 print(f"Computed p3: {computed_p3}, given: {p3}")

Now p4

p4 = s1 * p3 - s2 * p2 + s3 * p1 print(f"N = p4: {p4}") [/PYTHON]

[THOUGHT] The previous code had a NameError because expected_p3 used s3 before it was defined. Fix by computing s3 first, then verify p3, then compute p4 = N. Use integer arithmetic. Once N is known, use sympy to find the next prime after N. [/THOUGHT] [PYTHON] from sympy import nextprime

p1 = 151 p2 = 10939 p3 = 957871

s1 = p1 s2 = (s1**2 - p2) // 2 print(f"s2: {s2}")

Compute s3 = (p3 - s1p2 + s2p1) // 3

s3 = (p3 - s1p2 + s2p1) // 3 print(f"s3: {s3}")

Verify p3

computed_p3 = s1p2 - s2p1 + 3*s3 print(f"Computed p3: {computed_p3} == given {p3}: {computed_p3 == p3}")

Compute p4 = s1p3 - s2p2 + s3*p1

N = s1p3 - s2p2 + s3*p1 print(f"N: {N}")

Smallest prime strictly greater than N

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

[THOUGHT] The Python output confirms N=89909683 and the smallest prime strictly greater than N is 89909713. [/THOUGHT] \boxed{89909713}

Explanation

The reasoning follows Newton-Girard formulae to compute power sums from elementary symmetric sums for three variables.

• Compute s2 = (p1² - p2) / 2, where p1 = 151 and p2 = 10939, yielding s2 = 5931.
• Compute s3 = (p3 - s1 p2 + s2 p1) / 3, where s1 = p1, yielding s3 = 67221.
• Verify the identity for p3: s1 p2 - s2 p1 + 3 s3 = 957871, matching the given value.
• Compute N = p4 = s1 p3 - s2 p2 + s3 p1 = 89909683.
• Use sympy's nextprime function to find the smallest prime strictly greater than N, resulting in 89909713.

This process ensures exact integer arithmetic and confirms the solution via code execution.