Problem Analysis #0

[PYTHON] from sympy import nextprime, symbols, solve

def solve_problem(): S1 = 151 S2 = 10939 S3 = 957871

# Calculate e2
# S2 = S1^2 - 2*e2
e2 = (S1**2 - S2) // 2

# Calculate e3
# S3 = e1*S2 - e2*S1 + 3*e3
# 3*e3 = S3 - e1*S2 + e2*S1
e3_numerator = S3 - S1*S2 + e2*S1
e3 = e3_numerator // 3

# Calculate S4
# S4 = e1*S3 - e2*S2 + e3*S1
S4 = S1*S3 - e2*S2 + e3*S1

print(f"{e2=}")
print(f"{e3=}")
print(f"{S4=}")

# Verify roots are natural numbers
x = symbols('x')
roots = solve(x**3 - S1*x**2 + e2*x - e3, x)
print(f"Roots: {roots}")

# Find smallest prime strictly greater than N
next_p = nextprime(S4)
print(f"Next prime: {next_p}")

solve_problem() [/PYTHON]

The problem asks for the smallest prime number strictly greater than $N = a^4 + b^4 + c^4$, given the system of equations for the power sums $S_1, S_2, S_3$.

Using Newton's sums, we determined the elementary symmetric polynomials: $e_1 = a+b+c = 151$ $e_2 = ab+bc+ca = \frac{e_1^2 - S_2}{2} = \frac{151^2 - 10939}{2} = \frac{22801 - 10939}{2} = \frac{11862}{2} = 5931$ $e_3 = abc$. Using $S_3 = e_1 S_2 - e_2 S_1 + 3e_3$, we found $e_3 = 67221$.

Then we calculated $S_4 = a^4 + b^4 + c^4$ using the recurrence $S_4 = e_1 S_3 - e_2 S_2 + e_3 S_1$: $S_4 = 151(957871) - 5931(10939) + 67221(151) = 89909683$.

The roots of the polynomial $x^3 - 151x^2 + 5931x - 67221 = 0$ were found to be $21, 33, 97$, which are natural numbers, confirming the validity of the solution.

Finally, we looked for the smallest prime strictly greater than $N = 89909683$. The Python code determined this prime to be $89909713$.

\boxed{89909713}