[THOUGHT] We need to compute S6 = a^6 + b^6 + c^6 + d^6 + e^6 given the sums of powers up to 5. Let’s denote: p1 = a + b + c + d + e = 231 p2 = a^2 + ... + e^2 = 14003 p3 = a^3 + ... + e^3 = 997227 p4 = a^4 + ... + e^4 = 77752499 p5 = a^5 + ... + e^5 = 6386215731 We want p6.

We can use Newton's identities relating power sums pk and elementary symmetric sums ek. For n=5 variables, Newton's identities: k*ek = sum_{i=1}^{k} (-1)^{i-1} ek-i * pi, for k=1..5. We can compute e1, e2, e3, e4, e5 from p1..p5.

Then we can compute p6 using recurrence: p6 = e1p5 - e2p4 + e3p3 - e4p2 + e5p1 - 6e6? Wait careful: For k>n, the recurrence is: pk = e1p{k-1} - e2p{k-2} + e3p{k-3} - e4p{k-4} + e5p{k-5} - ... until e_n p_{k-n}. For n=5, for k=6: p6 = e1p5 - e2p4 + e3p3 - e4p2 + e5p1 - 6e6? Actually e6=0 since we have only 5 variables. So: p6 = e1p5 - e2p4 + e3p3 - e4p2 + e5p1.

Thus we need e1..e5.

Newton's identities: Let e0 = 1. For k=1: 1e1 = p1 → e1 = p1 = 231. For k=2: 2e2 = e1p1 - p2 = 231231 - 14003. Compute that. For k=3: 3e3 = e2p1 - e1p2 + p3. For k=4: 4e4 = e3p1 - e2p2 + e1p3 - p4. For k=5: 5e5 = e4p1 - e3p2 + e2p3 - e1p4 + p5.

We'll compute step by step in Python. [/THOUGHT] [PYTHON] p1 = 231 p2 = 14003 p3 = 997227 p4 = 77752499 p5 = 6386215731

Compute e1..e5

e = [0]*6 e[0] = 1

k=1

e[1] = p1 print("e1 =", e[1])

k=2

e[2] = (e[1]*p1 - p2) / 2 print("e2 =", e[2])

k=3

e[3] = (e[2]*p1 - e[1]*p2 + p3) / 3 print("e3 =", e[3])

k=4

e[4] = (e[3]*p1 - e[2]*p2 + e[1]*p3 - p4) / 4 print("e4 =", e[4])

k=5

e[5] = (e[4]*p1 - e[3]*p2 + e[2]*p3 - e[1]*p4 + p5) / 5 print("e5 =", e[5]) [/PYTHON]

[THOUGHT] Now compute p6 using the recurrence: p6 = e1p5 - e2p4 + e3p3 - e4p2 + e5p1. Plug in the values: e1=231, e2=19679, e3=769461, e4=13696848, e5=87805620. p1=231, p2=14003, p3=997227, p4=77752499, p5=6386215731. Compute step by step: 2316386215731 = ? 1967977752499 = ? 769461997227 = ? 1369684814003 = ? 87805620231 = ? Then combine with alternating signs. Let's compute in Python to ensure accuracy. [/THOUGHT] [PYTHON] e1 = 231 e2 = 19679 e3 = 769461 e4 = 13696848 e5 = 87805620 p1 = 231 p2 = 14003 p3 = 997227 p4 = 77752499 p5 = 6386215731

term1 = e1 * p5 term2 = e2 * p4 term3 = e3 * p3 term4 = e4 * p2 term5 = e5 * p1 print("term1 =", term1) print("term2 =", term2) print("term3 =", term3) print("term4 =", term4) print("term5 =", term5) p6 = term1 - term2 + term3 - term4 + term5 print("p6 =", p6) [/PYTHON]

[THOUGHT] The computed p6 is 540937826363. This is the integer value for a^6 + b^6 + c^6 + d^6 + e^6. [/THOUGHT] \boxed{540937826363}