Solutions to Approximate Fermat Equation x^3 = y^3 + z^3 Plus or Minus 1

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Theorem

The approximate Fermat equation:

$x^3 = y^3 + z^3 \pm 1$

has the solutions:

\(\ds 9^3\) \(=\) \(\ds 6^3 + 8^3 + 1\)
\(\ds 103^3\) \(=\) \(\ds 64^3 + 94^3 - 1\)


Proof

Performing the arithmetic:

\(\ds 6^3 + 8^3 + 1\) \(=\) \(\ds 216 + 512 + 1\)
\(\ds \) \(=\) \(\ds 729\)
\(\ds \) \(=\) \(\ds 9^3\)


\(\ds 64^3 + 94^3 - 1\) \(=\) \(\ds 262 \, 144 + 830 \, 584 - 1\)
\(\ds \) \(=\) \(\ds 1 \, 092 \, 727\)
\(\ds \) \(=\) \(\ds 103^3\)

$\blacksquare$



Proof that there are infinitely many solutions

$(1 - 9 t^3)^3 + (9 t^4)^3 + (3 t - 9 t^4)^3=1$ holds for all $t$.

$\blacksquare$

Sources