[THOUGHT] Compute the elementary symmetric sums s2 = ab + bc + ca and s3 = abc using the power sums.
s1 = 151
s2 = (s1**2 - p2) / 2
s3 = (p3 - s1**3 + 3 * s1 * s2) / 3
Verify they are integers, then use sympy to solve the cubic equation x^3 - s1 x^2 + s2 x - s3 = 0 to find a, b, c.
Once we have a, b, c, compute a^4 + b^4 + c^4. [/THOUGHT] [PYTHON] p1 = 151 p2 = 10939 p3 = 957871 s1 = p1 s2 = (s12 - p2) / 2 s3 = (p3 - s13 + 3 * s1 * s2) / 3 print(f"s2: {s2}") print(f"s3: {s3}") print("Are they integers?", s2.is_integer(), s3.is_integer()) from sympy import symbols, solve, expand x = symbols('x') poly = x3 - s1*x2 + s2*x - s3 roots = solve(poly, x) print("Roots:", roots) [/PYTHON]
[THOUGHT] The roots are 21, 33, 97. Compute 21^4 + 33^4 + 97^4. [/THOUGHT] [PYTHON] print(214 + 334 + 97**4) [/PYTHON]
[THOUGHT] The roots are 21, 33, 97, and their fourth powers sum to 89909683, as verified by Python. [/THOUGHT] \boxed{89909683}
Explanation
The reasoning follows Newton's identities to derive the elementary symmetric sums from the given power sums, then solves for the roots, and finally computes the required fourth power sum.
- Compute the first elementary symmetric sum: s1 = a + b + c = 151 (given as p1).
- Compute the second symmetric sum: s2 = (s1² - p2) / 2 = (151² - 10939) / 2 = 5931, confirmed integer.
- Compute the third symmetric sum: s3 = (p3 - s1³ + 3 × s1 × s2) / 3 = (957871 - 151³ + 3 × 151 × 5931) / 3 = 67221, confirmed integer.
- Form the cubic polynomial: x³ - s1 x² + s2 x - s3 = 0, or x³ - 151x² + 5931x - 67221 = 0.
- Solve the polynomial using SymPy, yielding roots 21, 33, and 97, which are natural numbers.
- Verify these roots satisfy the original equations implicitly through the symmetric sums derivation.
- Compute the fourth power sum: 21⁴ + 33⁴ + 97⁴ = 89909683, obtained via direct Python evaluation.