# Fermat’s last theorem (analytic form of)

Let $x$, $y$, $z$ be positive real numbers.

For each positive integer $r$, let

${a}_{r}=({x}^{r}+{y}^{r})/r!$ and ${b}_{r}={z}^{r}/r!$.

For $s$ divisible by 4, let

${A}_{s}={a}_{2}-{a}_{4}+{a}_{6}-\mathrm{\cdots}+{a}_{s-2}-{a}_{s}$,

${B}_{s}={b}_{2}-{b}_{4}+{b}_{6}-\mathrm{\cdots}+{b}_{s-2}-{b}_{s}$.

Then Fermat’s last theorem is equivalent^{} (by elementary means) to:

Theorem If ${a}_{n}={b}_{n}$ for some odd integer $n>2$, then either

(i) ${A}_{N}>0$ for some $N>x,y$,

or

(ii) ${B}_{M}>0$ for some $M>z$.

For a proof that these theorems are equivalent see:

proof of equivalence of Fermat’s Last Theorem to its analytic form

Title | Fermat’s last theorem (analytic form of) |
---|---|

Canonical name | FermatsLastTheoremanalyticFormOf |

Date of creation | 2013-03-22 16:17:34 |

Last modified on | 2013-03-22 16:17:34 |

Owner | whm22 (2009) |

Last modified by | whm22 (2009) |

Numerical id | 8 |

Author | whm22 (2009) |

Entry type | Theorem |

Classification | msc 11D41 |