CHAPTER XVI. 



TWO QUADRATIC FUNCTIONS OF ONE OR TWO UNKNOWNS 



MADE SQUARES. 



CONGRUENT NUMBERS k] x 2 /j=D BOTH SOLVABLE. 



Diophantus, III, 22, found solutions of (xi-\-x 2 +X3-\-x i )' 1 x l = D [see 

 Ch. VI] and, in V, 9, found solutions of x*(xi+x 2 +Xa) = D. In each 

 case he began with the fact that in any right triangle having the hypotenuse 

 h and legs a, b, the numbers h z 2ab are squares. 



An anonymous Arab manuscript, 1 written before 972, contains the 

 problem [of congruent numbers]: Given an integer k, to find a square z 2 

 such that x 2 k are both squares. The most convenient artifice to solve 

 this problem is stated to be the theorem that if x 2 -\-y' 2 = z 2 , then 

 z 2 2xy=(xy) 2 . [Hence 2xy is a congruent number if x, y are the legs 

 of a right triangle.] It is stated that, if the triangle is primitive and if 

 x*k are squares, the final digits of these squares are 1 or 9, with the express 

 statement that the digit is not 5 [squares of odd numbers end in 1, 5 or 9]. 

 An example is given : Using the primitive right triangle with the sides 3, 4, 

 we get 2xy = 24, 5 2 +24 = 7 2 , 5 2 24 = I 2 . A table gives the expression of the 

 odd numbers 3, , 19 in various ways as sums of two relatively prime 

 parts a, b; also the sides 2ab, a 2 6 2 of a right triangle, and k = 2(2ab)(a 2 6 2 ) ; 

 finally, u and v in z 2 +k = u 2 , z~ k = v i , where z is the corresponding hypote- 

 nuse. The table has 34 such k's. Woepcke noted that if we delete their 

 square factors, we get the following 30 " primitive congruent numbers ": 



Woepcke remarked (p. 352) that there is no indication that the Arabs 

 knew Diophantus prior to the translation by Aboul Wafa (f998), but they 

 may well have derived the problem of congruent numbers from the Hindus 

 who were early acquainted with the indeterminate analysis of Diophantus. 

 Mohammed Ben Alhocain, 2 in an Arab manuscript of the tenth century, 

 stated that the principal object of the theory of rational right triangles is 

 to find a square which when increased or diminished by a certain number k 

 becomes a square. He proved geometrically Diophantus' result that if 

 / 2 = 2 2 then z-2xy = (xy} 2 , so that 2 2 is the required square. Again, 



1 Imperial Library of Paris. French transl. by F. Woepcke, Atti Accad. Pont. Nuovi Lincei, 



14, 1860-1, 250-9 (Recherches sur plusieurs ouv. Leonardo Pise, 1st part, III). Some 

 of the results in the MS. were cited by Woepcke, Annali di Mat., 3, 1860, 206. 



2 French transl. by F. Woepcke, Atti Accad. Pont. Nuovi Lincei, 14, 1860-1, 350-3. 



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