Dear Michael:
I attempt to ask ChatGPT this query.
Run this code to solve Hilbert's thirteenth problem, the septic equation.
Please see attached file for your reference.
Given.
ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx^1+h = 0 and x^2+px+r = 0
Find the values of "p" and "r" in terms of "a, b, c, d, e, f, g, and h."
Code:
ClearAll[a, b, c, d, e, f, g, h]; {a, b, c, d, e, f, g, h} = {a, b, c, d, e,
f, g, h}; Solve[
h + 1/2 g (-p + Sqrt[p^2 - 4 r]) + 1/4 f (-p + Sqrt[p^2 - 4 r])^2 +
1/8 e (-p + Sqrt[p^2 - 4 r])^3 + 1/16 d (-p + Sqrt[p^2 - 4 r])^4 +
1/32 c (-p + Sqrt[p^2 - 4 r])^5 + 1/64 b (-p + Sqrt[p^2 - 4 r])^6 +
1/128 a (-p + Sqrt[p^2 - 4 r])^7 == 0 &&
h + 1/2 g (-p - Sqrt[p^2 - 4 r]) + 1/4 f (-p - Sqrt[p^2 - 4 r])^2 +
1/8 e (-p - Sqrt[p^2 - 4 r])^3 + 1/16 d (-p - Sqrt[p^2 - 4 r])^4 +
1/32 c (-p - Sqrt[p^2 - 4 r])^5 + 1/64 b (-p - Sqrt[p^2 - 4 r])^6 +
1/128 a (-p - Sqrt[p^2 - 4 r])^7 == 0, {p, r}]
Attachments: