616 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxn 



is a square. But if a square is the be a square, whence m = a 2 , n = b z , 



sum of a square and the double of a 4 & 4 =D, where a and b are rela- 



a square, its root is likewise the tively prime, one even and the other 



sum of a square and the double of odd. Thus a 2 +& 2 and a 2 b 2 are 



a square, which I can easily prove, relatively prime. Hence a 2 +6 2 = 2 , 



It follows that this root is the sum a 2 6 2 = ?7 2 , and rj being odd in- 



of the two legs of a right triangle, tegers. Also 2 = ?7 2 +2& 2 . Set 



one of the squares forming the base e = (+?7)/2, /=( 7?)/2. 



and the double of the other square Then e and / are integers and 



the height. This right triangle will e/=6 2 /2. A common factor of e and 



therefore be formed from two squares / would divide , rj, 6 2 and a 2 . Hence 



whose sum and difference are squares, e and / are relatively prime. We 



But 4 both of these squares can be may take e odd (changing if neces- 



shown to be smaller than the squares sary the sign of 77). Thus e = r 2 , 



of which it was assumed that the /=2s 2 , 2rs = b, where r and s are 



sum and difference are squares, integers. Hence = e+/=r 2 +2s 2 , 



Similarly, we would have smaller 77 = r 2 2s 2 . Also a 2 = 6 2 +^ 2 = r 4 +4s 4 . 



and smaller integers satisfying the The right triangle with the legs r 2 



same conditions. But this is im- and 2s 2 has the area rV. It is there- 



possible, since there is not an infini- fore formed (in the sense of Diophan- 



tude of positive integers smaller tus, as above) from two squares 



than a given one. The margin is mi=al and ni = bl, its sides being 



too narrow for the complete demon- 2miniandm 2 l ^n 2 l . Thus2mi7ii = 2s 2 , 



st ration and all its developments." m\ nl = r z . By miUi s 2 , we get 



dibi = s, a factor of b 2rs. Hence 5 

 i and &i are each less than b and 

 hence less than a. 



Fermat's 6 observations on Diophantus II, 8 and V, 32 includes the 

 statements that the sum of two bi quadrates is never a bi quadrate or a 

 square. 



Fermat had proposed, Sept., 1636, to Sainte-Croix that he find a right 

 triangle whose area is a square (Oeuvres, II, 65; III, 287); to Frenicle, 

 May (?), 1640, (Oeuvres, II, 195); to Wallis, Apr. 7, 1658 (Oeuvres, II, 

 376). Fermat stated that the problem is impossible in a letter to Pascal, 

 Sept. 25, 1654 (Oeuvres, II, 313). The attempted 7 proof by J. Wallis, 

 June 30, 1658 (Oeuvres de Fermat, III, 599) goes no further than a proof 

 of the rule of Diophantus for the sides of a right triangle. Fermat referred 

 in a letter to Carcavi, Aug., 1659 (Oeuvres, II, 431-6, see 436) to proofs by 

 the " descente inde*finie " which he had sent to Carcavi and Frenicle con- 



4 As translated by Heath, Diophantus of Alex., ed. 2, 1910, 6 Or, by a\ +b^a\ +b* =a. 



293-5. Tannery (Oeuvres de Fermat, III, 272) gave the 



incorrect reading: But the sum of these two squares can 



be shown to be smaller than that of the first two of which 



it was assumed that the sum and difference are squares. 

 8 Oeuvres, I, 291, 327; III, 241, 264. 

 7 Criticized by Frenicle, Oeuvres de Fermat, III, 606, 609. 



