CHAP. XX] aiXi~-\ ----- \-a n x n z MADE A CUBE OR HIGHER POWER. 543 



lies on the cone xlxl~ 2 4x1 = 0, which every plane x 3 = jurc 4 cuts in two lines 

 having x\ = 2 V/x"^. We get rational coordinates of the general point 

 P on C by taking /x to be a square if n is odd. For example, let n = 2m. 

 The general point on the line joining P=(2fj. m , 0, n, 1) and (p, 1, 0, 0) is 

 (2ju m + 8p, 5, /u, 1) = (x), which is on the surface if 8 = 4pn m /(D p 2 ) and gives 

 rational solutions xt. The same problem was treated by others. 85 



F. Ferrari 86 made/=z 2 -\-axy -{-by- an nth power. Let/= (x + ay) (x -\-fiy). 

 A sufficient condition is x-{-ay = (r-\-as} n . The latter becomes linear in a 

 by use of a 2 aa-\-b = Q. Hence we get x, y as polynomials in r, s, a, 6. 



G. Candido 87 used Lucas' u k , v k satisfying 



f> l Jr & 1* 



- n tii n * 



to show that for p = 2X+a/i, # = X 2 +ajuX+fyr, an infinitude of solutions of 

 x 2 +axy+by 2 = z k is given by 2x = v ]c a}j.u k , y = fj.u k , z = q. The explicit 

 formulas are given in the cases k = 2, 3, 4 and for a = or 6 = 0. 



F. Ferrari 88 used, as had Lagrange, the expansion of (ai+fatVa n to 

 find A's such that Au+ad.M=(a?+aai)*. 



E. Swift 89 proved that the number n(n 3)/2 of diagonals of an n-gon 

 is not a biquadrate. 



By Thue 211 of Ch. XXVI, x 2 -h 2 = ky n (n>2) has only a finite number 

 of solutions. On l+y z ^x n , see Lebesgue 68 of Ch. VI. On l+2y 2 = 3 k , see 

 Fauquembergue 158 of Ch. XXIII. On l4z n = D, see papers 7, 8, 169 of 

 Ch. XXVI. 



if 



0,\X\-\- -+a n xl MADE A CUBE OR HIGHER POWER. 



S. R4alis 90 noted that ulu 2 = a(uiX2 raa) 2 +/3(wi2/ 2 



J. Neuberg 91 took x z = x\, 2/2 = 2/1, 2:2= z\ in the preceding result to get 



E. N. Barisien 92 noted that any sixth power is the sum of two squares 

 diminished by a third : 



86 L'interm6diaire des math., 18, 1911, 35. 



86 Periodico di Mat., 25, 1909-10, 59-66. Cf. Lagrange. 63 



87 Ibid., 27, 1912, 265-8. Cf. Candido." 



88 Ibid., 28, 1913, 71-8. 



89 Amer. Math. Monthly, 23, 1916, 261-2. 



90 Nouv. Corresp. Math., 4, 1878, 325. 



91 Mathesis, 1, 1881, 74. 



92 Le matematiche pure ed applicate, 2, 1902, 35-36. 



