# Arithmetica 2.8

## The original text by Diophantus (AD 3)

To divide a given square into a sum of two squares.

To divide 16 into a sum of two squares.

Let the first summand be x^2, and thus the second 16 - x^2. The la
tter is to be a square. I form the square of the difference of an
arbitrary multiple of x diminished by the root [of] 16, that is, d
iminished by 4. I form, for example, the square of 2x - 4. It is 4
*2 + 16 - 16x. I put this expression equal to 16 - x^2. I add to b
oth sides x^2 + 16x and subtract 16. In this way I obtain 5x^2 = 1
6x, hence x = 16/5.

Thus one number is 256/25 and the other 144/25. The sum of these n
umbers is 16 and each summand is a square.




## Fermat's reaction (1637)

It is impossible to separate a cube into two cubes, or a fourth po
wer into two fourth powers, or in general, any power higher than t
he second, into two like powers. I have discovered a truly marvelo
us proof of this, which this margin is too narrow to contain.




## The structure of Wiles' proof (1995)

Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last T
heorem". Annals of Mathematics. 141 (3): 443-551.

Introduction
    443

Chapter 1
    455 1. Deformations of Galois representations
    472 2. Some computations of cohomology groups
    475 3. Some results on subgroups of GL2(k)

Chapter 2
    479 1. The Gorenstein property
    489 2. Congruences between Hecke rings
    503 3. The main conjectures

Chapter 3
    517 Estimates for the Selmer group

Chapter 4
    525 1. The ordinary CM case
    533 2. Calculation of eta.

Chapter 5
    541 Application to elliptic curves

Appendix
    545 Gorenstein rings and local complete intersections