Fermat's Last Theorem in $ \mathbf {Z}_{10}$

An amusing observation, which people often argued about on USENET news back in the 1990s, is that Fermat's last theorem is false in $ \mathbf {Z}_{10}$. For example, $ x^3 + y^3 = z^3$ has a nontrivial solution, namely $ x = 1$, $ y=2$, and $ z=\ldots60569$. Here $ z$ is a cube root of $ 9$ in $ \mathbf {Z}_{10}$. Note that it takes some work to prove that there is a cube root of $ 9$ in $ \mathbf {Z}_{10}$ (see Exercise 10).

William Stein 2012-09-24