Problem Analysis #5

GPT-5.1 (high) · Run 1

13.25s
3,904 toks
Problem Statement

Find

\na6+b6+c6+d6+e6\na^6 + b^6 + c^6 + d^6 + e^6

\nfor natural numbers a,b,c,d,ea,b,c,d,e satisfying

{\na+b+c+d+e=231\na2+b2+c2+d2+e2=14003\na3+b3+c3+d3+e3=997227\na4+b4+c4+d4+e4=77752499\na5+b5+c5+d5+e5=6386215731\begin{cases}\na + b + c + d + e = 231 \\\na^{2} + b^{2} + c^{2} + d^{2} + e^{2} = 14003 \\\na^{3} + b^{3} + c^{3} + d^{3} + e^{3} = 997227 \\\na^{4} + b^{4} + c^{4} + d^{4} + e^{4} = 77752499 \\\na^{5} + b^{5} + c^{5} + d^{5} + e^{5} = 6386215731 \end{cases}
Ground Truth
540937826363
Execution Trace
Reasoning Process

[THOUGHT] I will:

  1. Store the given power sums p1..p5.
  2. Use Newton's identities to compute the elementary symmetric sums e1..e5:
    • e1 = p1
    • p2 - e1p1 + 2e2 = 0, etc.
  3. Then use the Newton recurrence for k>5: p6 = e1p5 - e2p4 + e3p3 - e4p2 + e5*p1
  4. Print p6, which is a^6 + ... + e^6. [/THOUGHT] [PYTHON] from fractions import Fraction

Given power sums

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

Compute elementary symmetric sums using Newton's identities

e1 = p1

p2 - e1p1 + 2e2 = 0 -> e2

e2 = Fraction(e1*p1 - p2, 2)

p3 - e1p2 + e2p1 - 3*e3 = 0 -> e3

e3 = Fraction(p3 - e1p2 + e2p1, 3)

p4 - e1p3 + e2p2 - e3p1 + 4e4 = 0 -> e4

e4 = Fraction(e1p3 - e2p2 + e3*p1 - p4, 4)

p5 - e1p4 + e2p3 - e3p2 + e4p1 - 5*e5 = 0 -> e5

e5 = Fraction(p5 - e1p4 + e2p3 - e3p2 + e4p1, 5)

print("e1..e5:", e1, e2, e3, e4, e5)

Now compute p6 using recurrence for k>5:

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

from math import isfinite p6 = e1p5 - e2p4 + e3p3 - e4p2 + e5*p1 print("p6 as Fraction:", p6, "is integer?", p6.denominator == 1) print("p6 integer:", int(p6)) [/PYTHON]

Tool Output
[PYTHON OUTPUT] e1..e5: 231 19679 769461 13696848 87805620 p6 as Fraction: 540937826363 is integer? True p6 integer: 540937826363 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] Using Newton's identities on the given power sums, I computed the elementary symmetric sums and then applied the recurrence to obtain the 6th power sum, which matches the Python output.

a6+b6+c6+d6+e6=540937826363a^6 + b^6 + c^6 + d^6 + e^6 = 540937826363

540937826363\boxed{540937826363}

Final Model Output
540937826363

Comparative Analysis (P5) (Average across runs)

Avg Time Taken

Solar-Open-100B
529.7
Deepseek-V3.2
55.2
Gemini-3-Pro-Preview
47.5
Grok-4.1-fast
46.2
K-EXAONE-236B-A23B
42.5
Claude-Opus-4.5
18.4
GPT-5.1 (high)
15.4

Avg Token Usage

Solar-Open-100B
20,191
Gemini-3-Pro-Preview
11,089
K-EXAONE-236B-A23B
10,356
Grok-4.1-fast
8,582
Claude-Opus-4.5
5,049
Deepseek-V3.2
4,802
GPT-5.1 (high)
3,897
    Open · Problem 5 · GPT-5.1 (high) | EntropyMath