Anomalous Cancellation/Variants/37 + 13 over 37 + 24
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Theorem
- $\dfrac {37 + 13} {37 + 24} = \dfrac {37^3 + 13^3} {37^3 + 24^3}$
Proof
\(\ds 37^3 + 13^3\) | \(=\) | \(\ds \paren {37 + 13}^3 - \paren {3 \times 37^2 \times 13 + 3 \times 37 \times 13^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {37 + 13}^3 - 3 \times 13 \times 37 \paren {37 + 13}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {37 + 13} \paren {\paren {37 + 13}^2 - 3 \times 13 \times 37}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {37 + 13} \paren {2500 - 1443}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {37 + 13} \times 1057\) |
\(\ds 37^3 + 24^3\) | \(=\) | \(\ds \paren {37 + 24}^3 - \paren {3 \times 37^2 \times 24 + 3 \times 37 \times 24^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {37 + 24}^3 - 3 \times 24 \times 37 \paren {37 + 24}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {37 + 24} \paren {\paren {37 + 24}^2 - 3 \times 24 \times 37}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {37 + 24} \paren {3721 - 2664}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {37 + 24} \times 1057\) |
Hence:
- $\dfrac {37^3 + 13^3} {37^3 + 24^3} = \dfrac {1057 \paren {37 + 13} } {1057 \paren {37 + 24} } = \dfrac {37 + 13} {37 + 24}$
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $16 / 64$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $16 / 64$