522 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. XIX 



FURTHER EQUATIONS WHOSE QUADRATIC TERMS ARE SUMS OF PRODUCTS. 



Bhascara 116 (born 1114) treated the problem to make iv+2, +2, y+2, 

 z+2 the squares of numbers in A. P., and wx+lS, xy + lS, 2/2+18 all 

 squares, such that the sum of the roots of the seven squares when increased 

 by 11 gives 13 2 Since 18/2 is the square of 3, the roots of the first four 

 squares are y, ?/+3, ?/+6, ?/+9. Then the roots of wx + 18, etc., are found 

 to be ?/+37/-2, 7/ 2 +9i/+16, y 2 +15y+52. The sum of the roots plus 11 

 gives 3?/ 2 +3l7/+95 = 13 2 , y = 2. 



Diophantus, IV, 16, solved z(x+y)=a, y(x+z)=b, x(y+z) = c, when 

 a = 35, 6 = 32, c = 27, by assuming that #=15/2, y = 2Qfz, whence 2 = 5. 



Rallier des Ourmes 117 obtained 2xz = a-}-c b, etc., by elimination from 

 Diophantus' equations. From yz = m, xz = n,xy = p follows y= Vpra/n, etc. 

 For a = 24, 6 = 45, c = 49, we get m = 10, w=14, p = 35, whence x = 7, y = 5, 

 z = 2. He gave also a solution by listing the pairs of complementary factors 

 of the smallest two, 24 and 45, of the three given numbers : 



24 = 1-24 = 2- 12 = 3 -8 = 4-6, 45 = 1-45 = 3-15 = 5-9. 

 From each list select a pair of factors with a common sum, as 2-12, 5-9, 

 and select by trial one of a pair as one unknown and the cofactor as the sum 

 of the other two unknowns. 



To find n numbers, given the product of each by the sum of all the others, 

 list the pairs of cof actors of each of the smallest n 1 of the n given num- 

 bers and select those pairs, one from each list, which have the same sum 

 (the sum of the unknowns) . The smallest cofactor of each pair is one of 

 the smallest n 1 of the unknowns and their sum subtracted from the 

 total sum gives the largest unknown. For n = 5, use 180 = 4 45, 294 = 7 42, 

 418 = 11-38, 444 = 12-37; the unknowns are 4, 7, 11, 12, 



15 = 49-(4+7+ll + 12). 



S. Jones 118 took x(y+z) = a 2 x 2 , y(x+z) =6 2 , z = ax+b, which give x, y, z. 

 Then z(x+y) = \3 if a 2 +2a-l= D = (a-n}~ and a 2 -2a+3=D. The 

 latter becomes a quartic in n which is a square if n= 2/3. 



L. Euler 119 developed a method to make various functions simultaneously 

 equal to squares. The method will be explained for his problem ( 31-34) : 

 Given an integer n, find integers x, y, z such that xy+n, xz+n, yz-\-n, 

 xy+xz+yz-\-n are all squares. For any set of solutions of 



/ = x 2 + y 2 + z 2 2xy 2xz 2yz 4n = 



and for any function P, P 2 / is a square. Taking P = x+y z, we find 

 that 4(xy+ri) is a square. Taking P = x y+z, we find that 4(xz+ri) is 

 a square. Similarly, yz+ n is a square. Taking P = x+y +2, we find that 

 (xy+xz+yz+ri) is a square. Now/=0 if z = x+y+2v, where v* = xy+n. 

 To satisfy the latter take any integer for v and take x and y to be any pair of 



116 Vfja-ganita, 143^4. Algebra with arith . . . from Sanskrit ... of Bhdscara, transl. 



by Colebrooke, 1817, 218-9. 



117 M6m. de Math&natique et de Physique, Paris, 5, 1768, 479-84. 



8 The Gentleman's Math. Companion, London, 3, No. 15, 1812, 348-9. 

 118 Novi Comm. Acad. Petrop., 6, 1756-7, 85-114; Comm. Arith., I, 245-259; Opera 

 Omnia, (1), II, 399-427. French transl., Sphinx-Oedipe, 8, 1913, 97-109. 



