# Fermat's Marginal Notes

## 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.

### 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.