CHAP, xxi] EQUATIONS OF DEGREE THREE IN THREE UNKNOWNS. 603 



E. Fauquembergue 378 called the numbers x/y, x, xy. Then 

 (1) x*+x 2 y 2 , x 3 +x 2 , x*+x 2 !y 2 



are made squares by removing the factors x 2 , and making the product 

 (x+y 2 }(x+ l)(x+l/y 2 ) a square by Fermat's 156 method. Setting y = af(3, 

 we get x = ]V/(4a 4 /3 4 ), where 



N=(a 2 + aj3+l3 2 )(a 2 +a(3-(3 2 )(a 2 -at3+!3 2 )(-a 2 + 

 Then (1) are the squares of 



These fractions are said to be arithmetically irreducible. The sums by 

 twos of the numerators are 2Na*, 2Na 2 fi 2 , 2JV/3 4 , which are in G. P., but are 

 not made cubes as required. 



L. Euler 379 desired three rational numbers whose sum, product and sum 

 of products by twos are all squares. Denote the numbers by nx, ny, nz. 

 Then 



xyz(x+y+z) = D = v 2 (x+y+z) 2 , z = v 2 (x+ij)/(xy-v 2 ). 



Then n 3 xyz=E3 requires n = m 2 xy(x-\-y)(xy v 2 }. By the sum of products 

 by twos, 



+ ^|! =a 



xy v 2 

 Set xy v 2 = u 2 , x = tv. Then the preceding condition becomes 



P+l-s 2 



u 2 ~2s(t 2 +l)-3t 2 - 2' 

 Set s = t r and multiply numerator and denominator by 2 +l s 2 . Thus 



v 2rt-r 2 +l u 2 +v 2 



ni 



__ _ 



u~ Q tv u 2 



Rational values of t are found from r = l, 3/2, 3, 9. The simplest numbers 

 derived from r = 3/2, < = 60/19, are 705600/rf, 196/4157, 361/557, where 

 d = 231 5449. The corresponding integral solutions are 705600d, 109172d, 

 1500677c?. Euler 380 had expressed his belief that these give the least integers. 

 E. Fauquembergue 381 used a simpler method and obtained 



4a 2 6 4 (a 2 + 6 2 ) , (a 4 - 6 4 ) 2 , 4a 4 6 2 (a 2 + 6 2 ) , 



whose product is a square, sum is (a 2 +6 2 ) 4 , and sum of products by twos is 

 4a 2 & 2 (a 2 +& 2 ) 2 (<z 4 -f-& 4 ) 2 . For a = 2, 6 = 1, we get 80, 225, 320. 



878 L'intermediaire des math., 5, 1898, 86-7. 



379 Novi Comm. Acad. Petrop., 8, 1760-1, 64; Comm. Arith., I, 239; Op. Om., (1), II, 519. 



380 Corresp. Math. Phys. (ed., Fuss), 1, 1843, 631, Aug. 23, 1755. 

 881 L'intermediaire des math., 6, 1899, 95-96. 



