CHAP. XIX] SUMS OF PRODUCTS AND LlNEAR TERMS SQUARES. 523 



integers whose product is n v 2 . Then 



xy+n = v 1 , xz+n = (x-\-v) 2 , yz+n=(y+v) z , xy+xz+yz+n=(x+y+v) 2 , 



the right members being the reduced values of P 2 /4, for the respective P's. 

 To solve an interesting related problem ( 35-39), take 



f=x z +y z +z z -2xy-2yz-2xz-2a(x+y+z)-b=Q 



and P = x+yza for the four combinations of signs. Then 

 4(xy+xz+yz) +la(x+y+z) +a 2 +b, 4(xy+xz+yz} +a 2 



and the expression obtained from the last two by permuting the variables, 

 are all squares. Now/=0 if z = x+y+av, provided x and y make F = v 2 . 

 The latter is the case if x~\-a and y+a are two numbers whose product is 

 (v 2 6+3a 2 )/4. In particular, if a = l, b= 1, we see how to find three 

 numbers x, y, z such that 



xy+z, xz+y, yz+x, xy+x+y, xz+x+z, yz+y+z, 



v = xy+xz+yz, <r+x+y+z 



are all squares. The simplest solution is x = l, y 4, 2 = 12. Solutions in 

 which also the numbers themselves are squares are 



9 25 4JL. .2.5 64 196 

 64> 64> 16 j 9 > 9 > 9 



Euler 120 asked for numbers p, q, r, such that the product of each by 

 the sum of the remaining numbers is a square. Hence if S be their sum, 

 p(S p},q(S q), - are to be squares. Take p(S p] =f 2 p 2 , etc. Hence 

 the desired numbers are 



V1+/ 2 

 Take /= a/a, etc. For three numbers, let them be 



a z -\-a~ a a 



The sum of the last two is l^aabpfd. The sum of all three is therefore 

 unity if a 2 (& 2 +/3 2 ) =4a6|8, whence a : a = 46/3 : 6 2 +|8 2 . Taking a = 46/3 

 and multiplying the initial numbers by d, we get the solution 



166 2 /3 2 (6 2 +/3 2 ) , /3 2 (36 2 - /3 2 ) 2 , 6 2 (3/3 2 - 6 2 ) 2 . 



For four numbers, Euler gave the solutions (1, 2, 2, 5), (1, 10, 34, 125), 

 (5, 9, 26, 90), (5, 32, 61, 512) and solutions involving two parameters. For 

 five numbers, he gave 2, 40, 45, 58, 145. 



Euler 121 gave a special method of treating the last problem. Select any 

 number, like = 130, which is in several ways a sum of two parts whose 

 product is a square, viz., 



p= 2, 5, 13, 26, 32, 40, 49, 65, 



S-p = 128, 125, 117, 104, 98, 90, 81, 65. 



120 Novi Comm. Acad. Petrop., 17, 1772, 24; Coinm. Arith., I, 459-66; Op. Om., (1), III, 188. 



121 Opera postuma, 1, 1862, 260 (about 1769). 



