# Fermat's Last Theorem

## Theorem

- $\forall a, b, c, n \in \Z_{>0}, \; n > 2$, the equation $a^n + b^n = c^n$ has no solutions.

## Proof

The proof of this theorem is beyond the current scope of $\mathsf{Pr} \infty \mathsf{fWiki}$, and indeed, is beyond the understanding of many high level mathematicians.

For the curious reader, the proof can be found here, in a paper published by Andrew Wiles, entitled *Modular elliptic curves and Fermat's Last Theorem*, in volume 141, issue 3, pages 443 through 551 of the *Annals of Mathematics*.

It is worth noting that Wiles' proof was indirect in that he proved a special case of the Taniyama-Shimura Conjecture, which then along with the already proved Epsilon Conjecture implied that integral solutions of the theorem were impossible.

## Cubic

The Diophantine equation $a^3 + b^3 = c^3$ has no solutions in strictly positive integers.

## Source of Name

This entry was named for Pierre de Fermat.

## Historical Note

Many of Fermat's theorems were stated, mostly without proof, in the margin of his copy of Bachet's translation of Diophantus's *Arithmetica*.

In $1670$, his son Samuel published an edition of this, complete with Fermat's marginal notes.

### Fermat's Note

As Fermat himself put it, sometime around $1637$:

*Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.*

Loosely translated from the Latin, that means:

*The equation $x^n + y^n = z^n$ has no integral solutions when $n > 2$. I have discovered a perfectly marvellous proof, but this margin is not big enough to hold it.*

Nobody managed to find such a proof, until it was finally proved by Andrew Wiles in $1994$. It is seriously doubted that Fermat actually had found a general proof of it, and it is almost impossible that he found Wiles' proof since it uses areas of mathematics that were not yet invented in Fermat's time.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 1.6$: The Brachistochrone. Fermat and the Bernoullis - 1973: Donald E. Knuth:
*The Art of Computer Programming: Volume 3: Sorting and Searching*: Notes on the Exercises: Exercise $3$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($1601$ – $1665$) - 1994: Andrew John Wiles:
*Modular elliptic curves and Fermat's Last Theorem*(*Ann. Math.***Vol. 141**,*no. 3*: 443 – 551) - 1997: Gary Cornell, Joseph H. Silverman and Glenn Stevens:
*Modular Forms and Fermat's Last Theorem* - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): Notes on the Exercises: Exercise $4$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $7$: Patterns in Numbers