[PYTHON] S1 = 231 S2 = 14003 S3 = 997227 S4 = 77752499 S5 = 6386215731

Calculate elementary symmetric polynomials e1, e2, e3, e4, e5

e1 = S1

e1 = S1

2e2 = e1*S1 - S2

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

3e3 = e2S1 - e1S2 + S3

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

4e4 = e3S1 - e2S2 + e1*S3 - S4

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

5e5 = e4S1 - e3S2 + e2S3 - e1S4 + S5

e5 = (e4 * S1 - e3 * S2 + e2 * S3 - e1 * S4 + S5) // 5

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

Calculate S6

S6 = e1S5 - e2S4 + e3S3 - e4S2 + e5*S1

S6 = e1 * S5 - e2 * S4 + e3 * S3 - e4 * S2 + e5 * S1 print(f"{S6=}") [/PYTHON]

The elementary symmetric polynomials $e_1, e_2, e_3, e_4, e_5$ for the variables $a, b, c, d, e$ can be determined from the power sums $S_k = a^k + b^k + c^k + d^k + e^k$ using Newton's sums. The relations are: \begin{align*} e_1 &= S_1 \ 2e_2 &= e_1 S_1 - S_2 \ 3e_3 &= e_2 S_1 - e_1 S_2 + S_3 \ 4e_4 &= e_3 S_1 - e_2 S_2 + e_1 S_3 - S_4 \ 5e_5 &= e_4 S_1 - e_3 S_2 + e_2 S_3 - e_1 S_4 + S_5 \end{align*} Using the given values: $S_1 = 231$ $S_2 = 14003$ $S_3 = 997227$ $S_4 = 77752499$ $S_5 = 6386215731$

The calculated elementary symmetric polynomials are: $e_1 = 231$ $e_2 = 19679$ $e_3 = 769461$ $e_4 = 13696848$ $e_5 = 87805620$

For $n=5$ variables, the Newton sum for $S_6$ is given by: [ S_6 - e_1 S_5 + e_2 S_4 - e_3 S_3 + e_4 S_2 - e_5 S_1 = 0 ] Thus, [ S_6 = e_1 S_5 - e_2 S_4 + e_3 S_3 - e_4 S_2 + e_5 S_1 ] Substituting the values yields: [ S_6 = 540937826363 ]

\boxed{540937826363}