CHAPTER XXII. 



EQUATIONS OF DEGREE FOUR. 



SUM OR DIFFERENCE OF TWO BIQUADRATES NEVER A SQUARE; AREA OF A 

 RATIONAL RIGHT TRIANGLE NEVER A SQUARE. 



Leonardo Pisano 1 recognized the fact, but gave an incomplete proof, 

 that no square is a congruent number (i. e., x 2 +y 2 and x~ y 2 are not both 

 squares), while the latter is the area of a rational right triangle. Four 

 centuries later, Fermat 2 stated and proved the result thus implied by 

 Leonardo : no right triangle with rational sides equals a square with a rational 

 side. The occasion was the twentieth of Bachet's problems inserted at the 

 end of Book VI of Diophantus: to find a right triangle whose area is a 

 given number A. The necessary and sufficient condition given by Bachet 

 was that (2A) 2 +K 4 =D for a suitable K. For, this condition implies 

 that 2A/K and K are legs of a right triangle of area A ; while, conversely, if 

 K and H are legs of a right triangle of area A, they are proportional to K 2 

 and 2A, which are therefore legs of a right triangle. He quoted two condi- 

 tions given by F. Vieta, Zetetica, 1591, IV, 16, of which the first is that the 

 area increased by some biquadrate should be a biquadrate, and expressed 

 doubt as to the necessity of the conditions. 



Fermat's proof is of especial interest as it illustrates in detail his method 

 of infinite descent and as it presents the only instance of a detailed proof 

 left by 'him. In the left column is given a translation of Fermat's account 

 and in the right column proofs 3 of the statements. 



" If the area of a right triangle 

 were a square, there would be two 

 biquadrates whose difference is a 

 square, and hence two squares whose 

 sum and difference are squares. 

 Thus there would be a square equal 

 to the sum of a square and the 

 double of a square, such that the 

 sum of the two component squares 



If the sides have a common 

 factor, the area has a square factor 

 which may be removed. Since we 

 may therefore assume that the sides 

 x, y, z are relatively prime, we may 

 apply the rule of Diophantus and 

 set x = 2mn, y = m 2 n 2 , where m and 

 n are relatively prime integers not 

 both odd. Then mn(m 2 n 2 ) shall 



1 Tre Scritti, 1854, 98; Scritti, 2, 1862, 272. See Leonardo 17 of Ch. XVI. 



2 Fermat's marginal notes in his copy of Bachet's edition of Diophantus' Arithmetical 



Oeuvres de Fermat, Paris, 1, 1891, 340; 3, 1896, 271. 



3 Cf. H. G. Zeuthen, Geschichte der Math, in XVI and XVII Jahrhundert, 1903, 163. In 



the elaboration of Fermat's proof by A. M. Legendre, Theorie des nombres, 1798, 401-4; 

 ed. 2, 1808, 340-3, use is made of the theory of quadratic forms to show that = ? -2 -}-2s 2 ; 

 while P. Bachmann, Niedere Zahlentheorie, 2, 1910, 451-4, employed the uniqueness of 

 factorization of the integral algebraic numbers a+b-J 2. Both completed the final 

 step in the proof by comparing the areas of the initial and new triangles. H. Dutordoir, 

 Annales de la Societe Sc. de Bruxelles, 17, 1892-3, I, 49, announced in eight lines that 

 he could fill in an elementary manner the gaps left in this proof by Fermat. For the 

 elaboration used in the text, see L. E. Dickson, Bull. Amer. Math. Soc., (2), 17, 1911, 

 531-2. 



615 



