# Anomalous Cancellation/Variants/37 + 13 over 37 + 24

## Theorem

$\dfrac {37 + 13} {37 + 24} = \dfrac {37^3 + 13^3} {37^3 + 24^3}$

## Proof

 $\displaystyle 37^3 + 13^3$ $=$ $\displaystyle \paren {37 + 13}^3 - \paren {3 \times 37^2 \times 13 + 3 \times 37 \times 13^2}$ $\displaystyle$ $=$ $\displaystyle \paren {37 + 13}^3 - 3 \times 13 \times 37 \paren {37 + 13}$ $\displaystyle$ $=$ $\displaystyle \paren {37 + 13} \paren {\paren {37 + 13}^2 - 3 \times 13 \times 37}$ $\displaystyle$ $=$ $\displaystyle \paren {37 + 13} \paren {2500 - 1443}$ $\displaystyle$ $=$ $\displaystyle \paren {37 + 13} \times 1057$

 $\displaystyle 37^3 + 24^3$ $=$ $\displaystyle \paren {37 + 24}^3 - \paren {3 \times 37^2 \times 24 + 3 \times 37 \times 24^2}$ $\displaystyle$ $=$ $\displaystyle \paren {37 + 24}^3 - 3 \times 24 \times 37 \paren {37 + 24}$ $\displaystyle$ $=$ $\displaystyle \paren {37 + 24} \paren {\paren {37 + 24}^2 - 3 \times 24 \times 37}$ $\displaystyle$ $=$ $\displaystyle \paren {37 + 24} \paren {3721 - 2664}$ $\displaystyle$ $=$ $\displaystyle \paren {37 + 24} \times 1057$

Hence:

$\dfrac {37^3 + 13^3} {37^3 + 24^3} = \dfrac {1057 \paren {37 + 13} } {1057 \paren {37 + 24} } = \dfrac {37 + 13} {37 + 24}$

$\blacksquare$