Problem Analysis #4

[PYTHON] import sympy

def solve_problem(): S1 = 151 S2 = 7989 S3 = 511201 S4 = 35858673

# Newton sums
# e1 = S1
e1 = S1

# 2*e2 = e1*S1 - S2
e2 = (e1 * S1 - S2) // 2

# 3*e3 = e2*S1 - e1*S2 + S3
e3 = (e2 * S1 - e1 * S2 + S3) // 3

# 4*e4 = e3*S1 - e2*S2 + e1*S3 - S4
e4 = (e3 * S1 - e2 * S2 + e1 * S3 - S4) // 4

print(f"{e1=}")
print(f"{e2=}")
print(f"{e3=}")
print(f"{e4=}")

# Calculate S5
# S5 - e1*S4 + e2*S3 - e3*S2 + e4*S1 = 0
S5 = e1 * S4 - e2 * S3 + e3 * S2 - e4 * S1

print(f"{S5=}")

# Verify roots
x = sympy.symbols('x')
poly = x**4 - e1*x**3 + e2*x**2 - e3*x + e4
roots = sympy.solve(poly, x)
print(f"Roots: {roots}")

# Check if roots are natural numbers
# And check the sum of 5th powers
calculated_S5 = sum([r**5 for r in roots])
print(f"Calculated S5 from roots: {calculated_S5}")

solve_problem() [/PYTHON]

The problem asks for the value of $S_5 = a^5 + b^5 + c^5 + d^5$ given the power sums $S_1, S_2, S_3, S_4$. Using Newton's sums relations, we determined the elementary symmetric polynomials $e_1, e_2, e_3, e_4$ for the variables $a, b, c, d$. The polynomial with roots $a, b, c, d$ is $P(x) = x^4 - e_1 x^3 + e_2 x^2 - e_3 x + e_4$. From the Python code, we found: $e_1 = 151$ $e_2 = 7406$ $e_3 = 141056$ $e_4 = 866400$

The roots were found to be $\{12, 25, 38, 76\}$, which are natural numbers satisfying the given conditions. Using the recurrence relation for power sums $S_k - e_1 S_{k-1} + e_2 S_{k-2} - e_3 S_{k-3} + e_4 S_{k-4} = 0$ for $k=5$: $S_5 = e_1 S_4 - e_2 S_3 + e_3 S_2 - e_4 S_1$ Substituting the values: $S_5 = 151(35858673) - 7406(511201) + 141056(7989) - 866400(151)$ $S_5 = 2624775001$

\boxed{2624775001}