# Fermat's Marginal Notes

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## Fermat's Notes in the Margin of Diophantus's

## Fermat's Notes in the Margin of Diophantus's *Arithmetica*

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.

The purpose of this page is to gather these notes together.

### Fermat's Last Theorem

*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.*

### Integer as Sum of Polygonal Numbers

*Every positive integer is triangular or the sum of $2$ or $3$ triangular numbers; a square or the sum of $2$, $3$ or $4$ squares; a pentagonal number or the sum of $2$, $3$, $4$ or $5$ pentagonal numbers; and so on to infinity, whether it is a question of hexagonal, heptagonal or any polygonal numbers.*

*I cannot give the proof here, for it depends on many abstruse mysteries of numbers; but I intend to devote an entire book to this subject, and to present in this part of number theory astonishing advances beyond previously known boundaries.*

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($\text {1601}$ – $\text {1665}$)