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:
\(\ds 6^3 + 8^3 + 1\) | \(=\) | \(\ds 216 + 512 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 729\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9^3\) |
\(\ds 64^3 + 94^3 - 1\) | \(=\) | \(\ds 262 \, 144 + 830 \, 584 - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \, 092 \, 727\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 103^3\) |
$\blacksquare$
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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
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $729$