748 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxvi 



F. Lukas 97 set y = xa, z = xb, a<b, in y n +z n =x n , n>2. Hence 



Let wi, - , w n be its roots, all positive. Then 



2wi = n(a-\-b), - 2w i = a 2 + 6 2 + 2nab = integer, 



n 



which are said to be impossible if n>2. This error was noted in Jahrbuch 

 Fortschritte der Math., 7, 1875, 100. 



T. Pepin 98 proved that x 7 +y 7 +z 7 = G is not satisfied by integers not 

 divisible by 7, by use of the fact that v?=x*-\-7 3 y' i has no integral solutions 

 with 2/=|=0 (proved by descent). He proved (pp. 743-7) that the first 

 equation has no solution in which one of the unknowns is divisible by 7. 



J. W. L. Glaisher" expressed Cauchy's 29 theorem in a new form. Let 

 n be odd and set x = c b, y a c. Then 



Then E n is algebraically divisible by Ez = 3xy(x+y). If n = 6ml, E n is 

 divisible by E 2 = 2(x z +xy+y*). If w = 6m+l, E n is divisible by E* = %El 



Glaisher 100 expressed (x-\-y) n x n y n , for n odd sil3, in terms of 

 P=x z -}-xy-\-y' 2 and y=xy(x-}-y). [^Earlier by Cauchy. 33 "] 



T. Muir 101 noted that x, y, xy are the roots of w 3 

 Hence by Waring' s formula for the sum of like powers of the roots, 



(m-2)(m-3) 



2m+l 1-2-3 



(m-3)...(m-6) 



1-2-3-4-5 

 He gave a similar formula for (x+yY m +x 2m -}-y 2m . For three variables, set 



j3=Sz 2 -|-2z?/, y = '2x 2 y '-\-2xyz, d=xyz(x+y+z). 

 Then x, y, z, xyz are the roots of w* (3w 2 +yw 5 = 0. Thus 



/ o 



summed for all integral solutions ^0 of 2r+3s+4 = 2ra+l. Since s>0, 

 the sum has the factor y = %{(x -\-y-\-z~) 3 x 3 y* z 3 }. 



Glaisher 102 noted that Newton's identities give a recursion formula 

 for x\-\ ----- haC, extended Cauchy's theorem to negative exponents, and 

 gave recursion formulas for and factors of the sum of the pth powers of all 

 the quantities i=b dba n in which r of the signs are negative. 



87 Archiv Math. Phys., 58, 1876, 109-112. 



98 Comptes Rendus Paris, 82, 1876, 676-9. 



99 Quar. Jour. Math., 15, 1878, 365-6. 



100 Messenger Math., 8, 1878-9, 47, 53. 



101 Quar. Jour. Math., 16, 1879, 9-14. 



102 Ibid., 89-98. 



