698 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxm 



E. N. Barisien 176 noted that x=f(ri), y = ^>(n) give solutions (but not 

 necessarily all solutions) of the equation F(x, y) = Q obtained by eliminating 

 n. Similarly when x, y, z are functions of n, m. A. Cunningham 177 gave 

 the least solution 3, 4, 5, and the general solution of 



E. B. Escott 1770 noted that, if X=x*+l, 



A. Thue 178 considered solutions x, y, z, relatively prime in pairs, of 



Ax n +By n +Cz n -xyzU(x, y, z)=0, 



where U is a homogeneous polynomial of degree n 3 whose coefficients, 

 as well as A, B, C, are integers. Let n be odd. Let p, q, r be integers, not 

 all zero, such that px+qy+rz = Q. Then 



C 



(Ar n -Cp n )x n +(Br n -Cq n )y n =xyE 1 , Ei = r n zU -- {(px) n +(qy} n +(rz} n } } 



xy 



with two similar equations derived by permuting x, y, z and p, q, r. Then 

 ax = Br n Cq n , by = Cp n Ar n , cz = Aq n Bp n . 



Hence Aax-\-Bby+Ccz = Q, so that we have a second linear relation. Also 

 ay n ~ 1 bx n ~ 1 =Ei, with two similar equations. Let u be the greatest of 

 x, y, z numerically; X the greatest of p, q, r; I of A, B, C; m the greatest of 

 the coefficients of U, and d = %(n 2)(n l)m+(2 n ~ 1 +l)L He proved the 

 following theorems. If AjBC + 0, n^3, and if p, q, r can be found such that 

 A n ~ 1 < / w/(76), then a = b = c = 0. If our given function of degree n is irre- 

 ducible, we can determine a function Kn^ld of A, B, C and the coefficients 

 of U, such that no numbers p, q, r exist for which \ n ~ l <ujK. If 



Ax n +By n +Cz n = Q 



has relatively prime solutions and if n is odd and >1, there do not exist 

 solutions p, q, r not all zero of px + qy + rz for which X n ~ l < uf { (2 n ~ l + 1 ) I 2 } . 

 G. Candido 179 considered a polynomial f(x, y) with the factors L = x+ay 

 and <j> (x, y) , where a is rational . Set x + ay = z n , < = A . Then f(x, y)=Az n 

 has the solutions 



x = %v n (p, q), y = %u n (p, q), z = X+aju, p = 2X+a/z, g=X 2 +aX/i, 



where u k) v k satisfy (%v k ) 2 (\p z q)ul = q k . Similarly, if / has the factor 

 Q = o; 2 -|-|&r2/+72 2 , where /5, 7 are rational, take it as z n . Each method is 

 applied in detail to solve LQ=Az z ; in the particular case x z -\-y 3 = : Az 3 ) the 

 solutions are those obtained by Lucas 198 of Ch. XXI. 



A. Cunningham 180 proved that if 4x 3 y 3 = 3x 2 yz 2 in positive integers, 

 then x = y, z = l. He discussed (p. 28) x 5 +y b = t 2 +u 2 , a necessary and 



176 Sphinx-Oedipe, 5, 1910, 76-77. 



177 Math. Quest. Educ. Times, (2), 15, 1909, 49; (2), 18, 1910, 101-2. 

 1770 Ibid., (2), 17, 1910, 57. 



178 Skrifter Videnskapsselsk. Kristiania (Math.), 2, 1911, No. 20. 



178 Periodico di Mat., 27, 1912, 265-273. 



180 Math. Quest. Educat. Times, (2), 22, 1912, 69-70. 



