CHAPTER XIX. 



SYSTEMS OF THREE OR MORE EQUATIONS OF DEGREE TWO IN 



THREE OR MORE UNKNOWNS. 



2 , 2 -f2 2 , ?/ 2 +2 2 ALL SQUARES. 



Paul Halcke 1 gave the solution x = 44, // = 240, = 



N. Saunderson 2 satisfied # 2 +z 2 = D by expressing z 2 as a product of two 

 factors aw and z*/(aw} and taking half their difference as x. Similarly for 

 2 / 2 +2 2 =D. Takea 2 +b 2 = c 2 . Then 



x 



If z 2 \ If. z 2 \ 



= -[aw - I, 2/==kol &w; 7 I, 

 2\ aw/ 2\ ow/ 



"1 9 4 



O O ^ I 



Equate the sum of the last two terms to zero. Hence w = czf(2ab}. To 

 obtain integers, let z = 4abc. Then x = a(46 2 c 2 ), y = b(4a- c 2 ). For a = 3, 

 6 = 4, we get a: = 117, ?/ = 44, 2 = 240. 



L. Euler 3 made the last two sums squares by taking 



x p 2 l ii o n - l 



z~ 2p ' z 2q 

 Then the first sum will be a square if 



First, let q - 1 = p + 1 . Then must 2p 4 + Sp 3 + Qp- - 4p +4 = D . Since 4 is 

 a square, the condition is satisfied in the usual manner if p= 24. Next, 

 g-l = 2(p+l) leads to the solution p = 48/31, and g-l=4(p-l)/3 to 

 p = 2/13. For 



both (p+1) 2 and (p I) 2 may be cancelled and the condition becomes 



say the square of tp-+(t-+l)p t. Hence p= 4//( 2 +l), where t is arbi- 

 trary. If x = a, ?/ = 6, 2 = c is one solution of our problem, x = ab, y = bc, 

 z = ac is another. 



Euler 4 made S-A 2 , S-B 2 , squares, where S = A 2 +B 2 -\ . Thus 



S is to be expressed as a (21 in several ways, the most general way being 



| 2 2fx-(f*-l}y 



if S = x-+y 2 is one way. For three numbers A=x, B, C, take as C the 



1 Deliciae Mathematicae, Oder Math. Sinnen-Confect, Hamburg, 1719, 265. 



2 The Elements of Algebra, 2, 1740, 429-431. 



3 Algebra, 2, 1770, art. 238; French transl., 2, 1774, pp. 327-335. Opera Omnia, (1), I, 



477-82. Cf. Fuss 95 and Schwering 160 of Ch. V. 



4 Novi Comm. Acad. Petrop., 17, 1772, 24; Comm. Arith., I, 467; Op. Omnia, (1), III, 201. 



33 497 



