[THOUGHT] This problem involves finding natural numbers $a, b, c, d, e$ that satisfy a system of power sum equations. Once we find these numbers, we can compute $a^6 + b^6 + c^6 + d^6 + e^6$.

The approach is to use Newton's identities to find the elementary symmetric polynomials from the power sums, then find the roots of the characteristic polynomial.

Let me denote:

• $p_k = a^k + b^k + c^k + d^k + e^k$ (power sums)
• $e_1, e_2, e_3, e_4, e_5$ as the elementary symmetric polynomials

Newton's identities relate power sums to elementary symmetric polynomials:

• $p_1 = e_1$
• $p_2 = e_1 p_1 - 2e_2$
• $p_3 = e_1 p_2 - e_2 p_1 + 3e_3$
• $p_4 = e_1 p_3 - e_2 p_2 + e_3 p_1 - 4e_4$
• $p_5 = e_1 p_4 - e_2 p_3 + e_3 p_2 - e_4 p_1 + 5e_5$

From these, we can solve for $e_1, e_2, e_3, e_4, e_5$ and then the polynomial whose roots are $a, b, c, d, e$ is: $x^5 - e_1 x^4 + e_2 x^3 - e_3 x^2 + e_4 x - e_5 = 0$

Let me compute this step by step. [/THOUGHT] [PYTHON] from sympy import symbols, solve, Rational

Given power sums

p1 = 231 p2 = 14003 p3 = 997227 p4 = 77752499 p5 = 6386215731

Newton's identities to find elementary symmetric polynomials

p1 = e1

e1 = p1

p2 = e1p1 - 2e2 => e2 = (e1*p1 - p2) / 2

e2 = (e1 * p1 - p2) // 2

p3 = e1p2 - e2p1 + 3e3 => e3 = (p3 - e1p2 + e2*p1) / 3

e3 = (p3 - e1 * p2 + e2 * p1) // 3

p4 = e1p3 - e2p2 + e3p1 - 4e4 => e4 = (e1p3 - e2p2 + e3*p1 - p4) / 4

e4 = (e1 * p3 - e2 * p2 + e3 * p1 - p4) // 4

p5 = e1p4 - e2p3 + e3p2 - e4p1 + 5e5 => e5 = (p5 - e1p4 + e2p3 - e3p2 + e4*p1) / 5

e5 = (p5 - e1 * p4 + e2 * p3 - e3 * p2 + e4 * p1) // 5

print(f"e1 = {e1}") print(f"e2 = {e2}") print(f"e3 = {e3}") print(f"e4 = {e4}") print(f"e5 = {e5}")

Now find roots of x^5 - e1x^4 + e2x^3 - e3x^2 + e4x - e5 = 0

import numpy as np

coeffs = [1, -e1, e2, -e3, e4, -e5] roots = np.roots(coeffs) print(f"Roots: {roots}")

Round to nearest integers and verify

roots_int = [int(round(r.real)) for r in roots] print(f"Integer roots: {roots_int}")

Verify the sums

print(f"Sum: {sum(roots_int)}") print(f"Sum of squares: {sum(r2 for r in roots_int)}") print(f"Sum of cubes: {sum(r3 for r in roots_int)}") print(f"Sum of 4th powers: {sum(r4 for r in roots_int)}") print(f"Sum of 5th powers: {sum(r5 for r in roots_int)}")

Compute sum of 6th powers

p6 = sum(r**6 for r in roots_int) print(f"Sum of 6th powers: {p6}") [/PYTHON]

[THOUGHT] The Python code successfully found the natural numbers $a, b, c, d, e$ to be 89, 58, 42, 27, and 15. All the given conditions were verified to be satisfied. The sum of their 6th powers is 540937826363. [/THOUGHT]

\boxed{540937826363}