428 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xm 



M. Weill 143 obtained solutions x = a+\8, y = b+\'8, z = l + 8, of 



by finding 6 rationally in terms of a, fr, X, X'. Again, a 



has solutions of the form # = 1+X5, ?/ = l+X'5, 2 = 1 + 5. To one of these 



two is reduced the solution of ax 2 -\-by 2 = z 2 when a+6 or ab is a square. 



E. Cahen 144 noted that Weill's formulas do not give all solutions and 

 showed how to find all solutions of x--\-y z = 5z 2 . 



E. Turriere 145 noted that, if a, b, c are the sides of a triangle two of whose 

 medians are perpendicular, then a 2 +6 2 = 5c 2 , whose solutions are expressed 

 rationally in two parameters. 



A. Desboves 38 gave all solutions of # 2 +?/ 2 = (m 2 +n 2 )^ 2 . Cf. papers 133-5, 

 142-5 above; Catalan 63 of Ch. VII; G. F. Malfatti 19 of Ch. VIII; papers 

 191, 252, 294, 307, 311 of Ch. XII; and 225 of Ch. XXII. 



R. Hoppe 29 of Ch. V solved p 2 -3g 2 = r 2 ; Euler 109 of Ch. XXII solved 

 a 2 +3/3 2 =D. 



FURTHER SINGLE QUADRATIC EQUATIONS IN THREE OR MORE UNKNOWNS. 



Bhascara 146 (born 1114) found four distinct numbers whose sum equals 

 the sum of their squares. Take as the numbers y, 2y, 3y, 4y. Then 



C. F. Gauss 147 considered the solution in integers of 

 (1) f^ax z +aixl+a2xl+2bxiXz+2biXX2+2 



If a = 0, a; is determined rationally in terms of x\, x z ', to obtain integral 

 solutions, multiply the three x's by the denominator of x. Next, let a={=0. 

 We derive the equivalent equation 



L 2 A 2 Xi +2Bx iX 2 A 1X2 = 0, L ax+b 2 Xi+biX 2 , 



If ^2 = 0, B^r 0, we can give arbitrary values to x z and L and determine 

 x and Xi rationally. If ^L 2 = -S = 0, either A : is not a square and 0^2 = ^ = 

 or Ai = k z and L = /cx 2 . Finally, let a 2 + 0, A 2 +0. Then 



where D is the determinant of/, whence Da = B 2 AiA 2 . If D = 0, we have 

 linear factors. If D +0, criteria for solvability were given by Gauss. 116 



Given one solution a, i, 2 of /=0, we can transform / into a like form 

 with a = (treated above). In fact, determine integers /?, , 72 so that 



Then the desired transformation is 



143 Nouv. Ann. Math., (4), 16, 1916, 351-5. 



144 Ibid., (4), 17, 1917, 463-5. 



146 L'cnseigncment math., 18, 1916, 89-90. 



146 Vija-ganita, 119; Colebrooke, 1 p. 200. 



147 Disq. Arith., 1801, art. 299; Werke, I, 1863, 358; German transl., Maser, 344-6. 



