Teorema de fermat

NOTES ON FERMAT'S LAST THEOREM I
A. J. van der Poorten
Centre for Number Theory Research , Macquarie University

ceNTRe Macquarie University NSW 2109 Australia

The story of `Fermat's Last Theorem' has been told so often it hardly bears retelling. H. M. Edwards

Dramatis Person Euclid of Alexandria Diophantus of Alexandria Pierre de Fermat Leonhard Euler Joseph Louis Lagrange SophieGermain Carl Friedrich Gauss Augustin Louis Cauchy Gabriel Lame Peter Gustav Lejeune Dirichlet Joseph Liouville Ernst Eduard Kummer Harry Schultz Vandiver Gerhard Frey Kenneth A. Ribet Andrew Wiles 1953{ 250 1601{1655 1707{1783 1736{1813 1776{1831 1777{1855 1789{1857 1795{1870 1805{1859 1809{1882 1810{1893 1882{1973

; 300

Fermat's Last Theorem states that there are no positive integers x , y andz with

xn + yn = zn
if n is an integer greater than two. For n equals two there are many solutions: 32 + 42 = 52 52 + 122 = 132 82 + 152 = 172

:::

1991 Mathematics Subject Classi cation. 11D41, 11-01. Work supported in part by grants from the Australian Research Council and a research agreement with Digital Equipment Corporation. Typeset by AMS-TEX
Draft at August 18, 1993 14:38 ]

2Alf van der Poorten

the Pythagorean triples. In the margin of his copy of Diophantus' Arithmetica the French jurist Fermat wrote c 1637 that for greater n no such triples can be found he added that he had a marvellous proof for this, which, however, the margin was too small to contain: Cubum autem in duos cubos, aut quadrato-quadratum in duos quadratoquadratos, et generaliter nullam in innitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caparet. Every other result which Fermat had announced in like manner had long ago been dealt with only this one, the last , remained. Problem 8 in Book II of Claude Bachet's translation of Diophantus asks for a rule for writing a square as the sumof two squares. The resulting equation z2 = y2 + x2 is that of the Theorem of Pythogoras, which says that in every right-angled triangle the square on the hypotenuse is the sum of the squares on the other two sides. The logo of Macquarie University's ceNTRe for Number Theory Research

provides a graphical proof, showing in particular that Pythogoras' Theorem has only the depth of the identity (x+ y)2 = x2 + 2xy + y2 . It is a little more di cult to nd all solutions in integers, but not much more. If x2 + y2 = z2 , we can suppose that x , y and z pairwise have no common factor, for such a factor would be common to all three quantities and can be factored out, leaving an equation of the original shape. Thus at least two of x , y and z must be odd. But the square of an odd number (so, ofthe shape 2m +1) leaves a remainder of 1 on division by 4, whilst the square of an even number (so, of the shape 2m ) leaves a remainder of 0 on division by 4. It follows that z must be odd and that one of x and y , say x = 2x0 , must be even. Then we obtain 4x0 2 = z2 ; y2 = (z + y)(z ; y) so x0 2 = 1 (z + y) 1 (z ; y) : 2 2 But if the product of two numbers that have no factor in common is asquare, then each of the two numbers is a square. This is clear on splitting the two numbers into their prime factors and checking the contribution of each distinct prime. To apply the principle we need only note 1 that both 2 (z + y) and 1 (z ; y) are integers, because both z and y are odd and 2 that they have no common factor. The latter is clear, because if d were a common factor then d is a factorboth of their sum z , and their di erence y . Yet we began by determining that y and z are relatively prime | that they have no common 1 factor. So both 2 (z + y) and 1 (z ; y) are squares, say, 2 1 (z + y ) = u2 and 1 (z ; y) = v2 : 2 2
Draft at August 18, 1993 14:38 ]

Notes on Fermat's Last Theorem

3

Thus x0 2 = u2v2 , and summarising, we have

x = 2uv y = u2 ; v2

and z = u2 +...