[THOUGHT] We are given the sums of powers for five natural numbers a,b,c,d,e. We need to compute S6 = a^6 + b^6 + c^6 + d^6 + e^6.

We can use Newton's identities (power sums and elementary symmetric polynomials) to find S6. Let p_k = a^k + b^k + c^k + d^k + e^k for k=1..6.

We know p1, p2, p3, p4, p5. We need p6.

Newton's identities for n=5 variables:

Let e1 = sum a, e2 = sum_{i<j} a_i a_j, e3 = sum_{i<j<k} a_i a_j a_k, e4 = sum_{i<j<k<l} a_i a_j a_k a_l, e5 = a b c d e.

Then: p1 = e1 p2 = e1 p1 - 2 e2 p3 = e1 p2 - e2 p1 + 3 e3 p4 = e1 p3 - e2 p2 + e3 p1 - 4 e4 p5 = e1 p4 - e2 p3 + e3 p2 - e4 p1 + 5 e5 p6 = e1 p5 - e2 p4 + e3 p3 - e4 p2 + e5 p1 - 6 e6? Wait, for n=5, e6 = 0 because we have only 5 variables. Actually, Newton's identities for n variables:

For k <= n: p_k - e1 p_{k-1} + e2 p_{k-2} - ... + (-1)^{k-1} e_{k-1} p_1 + (-1)^k k e_k = 0.

For k > n: p_k - e1 p_{k-1} + e2 p_{k-2} - ... + (-1)^n e_n p_{k-n} = 0.

We have n=5. For k=6 > 5: p6 - e1 p5 + e2 p4 - e3 p3 + e4 p2 - e5 p1 = 0.

So p6 = e1 p5 - e2 p4 + e3 p3 - e4 p2 + e5 p1.

Thus we need e1, e2, e3, e4, e5. We can compute them from p1..p5 using Newton's identities for k=1..5.

Let's compute step by step with Python.

We'll set up equations:

e1 = p1 = 231.

For k=2: p2 = e1 p1 - 2 e2 => e2 = (e1 p1 - p2)/2.

For k=3: p3 = e1 p2 - e2 p1 + 3 e3 => e3 = (p3 - e1 p2 + e2 p1)/3.

For k=4: p4 = e1 p3 - e2 p2 + e3 p1 - 4 e4 => e4 = (e1 p3 - e2 p2 + e3 p1 - p4)/4.

For k=5: p5 = e1 p4 - e2 p3 + e3 p2 - e4 p1 + 5 e5 => e5 = (p5 - e1 p4 + e2 p3 - e3 p2 + e4 p1)/5.

Then p6 = e1 p5 - e2 p4 + e3 p3 - e4 p2 + e5 p1.

We'll compute with exact integers.

Let's code it. </think>

[PYTHON] import sympy as sp

Given sums

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

Compute elementary symmetric sums e1..e5

e1 = p1

e2 = (e1 * p1 - p2) // 2 print("e2 =", e2)

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

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

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

Now compute p6 = e1p5 - e2p4 + e3p3 - e4p2 + e5*p1

p6 = e1 * p5 - e2 * p4 + e3 * p3 - e4 * p2 + e5 * p1 print("p6 =", p6)

Verify integer division gave integer results; if not, we'd have a problem.

Let's double-check using sympy rational arithmetic to avoid integer division issues.

print("\nChecking with exact rational arithmetic:") p1_s = sp.Integer(p1) p2_s = sp.Integer(p2) p3_s = sp.Integer(p3) p4_s = sp.Integer(p4) p5_s = sp.Integer(p5)

e1_s = p1_s e2_s = (e1_s * p1_s - p2_s) // 2 e3_s = (p3_s - e1_s * p2_s + e2_s * p1_s) // 3 e4_s = (e1_s * p3_s - e2_s * p2_s + e3_s * p1_s - p4_s) // 4 e5_s = (p5_s - e1_s * p4_s + e2_s * p3_s - e3_s * p2_s + e4_s * p1_s) // 5

p6_s = e1_s * p5_s - e2_s * p4_s + e3_s * p3_s - e4_s * p2_s + e5_s * p1_s print("p6 (exact) =", p6_s) print("p6 (integer) =", int(p6_s)) [/PYTHON]

[THOUGHT] The Python computation shows that ( a^6 + b^6 + c^6 + d^6 + e^6 = 540937826363 ). This is the exact integer result from the symmetric sum relations.

\boxed{540937826363}