Infinite Number of Integers which are Sum of 3 Sixth Powers in 2 Ways

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Theorem

There exist an infinite number of positive integers which can be expressed as the sum of $3$ sixth powers in $2$ different ways.


Proof

There are many parametric solutions to $x^6 + y^6 + z^6 = u^6 + v^6 + w^6$. One is given by:

\(\ds x\) \(=\) \(\ds 2 m^4 + 4 m^3 n - 5 m^2 n^2 - 12 m n^3 - 9 n^4\)
\(\ds y\) \(=\) \(\ds 3 m^4 + 9 m^3 n + 18 m^2 n^2 + 21 m n^3 + 9 n^4\)
\(\ds z\) \(=\) \(\ds -m^4 - 10 m^3 n - 17 m^2 n^2 - 12 m n^3\)
\(\ds u\) \(=\) \(\ds m^4 - 3 m^3 n - 14 m^2 n^2 - 15 m n^3 - 9 n^4\)
\(\ds v\) \(=\) \(\ds 3 m^4 + 8 m^3 n + 9 m^2 n^2\)
\(\ds w\) \(=\) \(\ds 2 m^4 + 12 m^3 n + 19 m^2 n^2 + 18 m n^3 + 9 n^4\)


This set of solutions also satisfy:

\(\ds x^2 + y^2 + z^2\) \(=\) \(\ds u^2 + v^2 + w^2\)
\(\ds 3 x + y + z\) \(=\) \(\ds 3 u + v + w\)


This needs considerable tedious hard slog to complete it.
In particular: For people who enjoy degree $24$ polynomials in $2$ unknowns
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Sources

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