580 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxi 



where a is not divisible by 3, while 77 = 2 or 1 according as a, b are both 

 odd or not both odd. 



C. Richaud 230 solved (z+1) 3 z* = y- for x and made the radical rational. 

 Thus (22/) 2 -l = 3r 2 , whence x = Q,y = l; x = 7,y = 13; z = 104, y = lSl; etc. 

 The same solutions were given by Moret-Blanc, 231 who remarked that 

 x 3 +(x+iy = y 2 only for z = 0, 1 (cf. E. Lucas, Mathesis, 1887, 200). 



W. J. Greenfield 232 gave numerical solutions of x 3 y 3 = D. 



M. Weill 233 noted that (-3a 2 ) 3 +(a 3 +4) 3 =(l+ 3 )(a 3 -8) 2 . 



P. F. Teilhet 234 gave the solutions 65, 56, 671; 5985, 5896, 647569. 



E. Fauquembergue 235 reproduced Euler's 225 formulas with p = n, q = m. 

 Replacing p by /3 a, q by or, we obtain the formulas of Axel Thue 236 

 for x 3 +2/ 3 = 2 2 , who noted that, if zis not divisible by 3, then x 2 xy-\-y 2 = B 2 . 

 Thus, for relatively prime p and q, px = q(B+xy), qy = p(Bx-\-y}, since 

 the product of the second factors is xy. Eliminating B, we get 



xfy=(q z -2pq)l(p z -~2pq). 



In case the numerator and denominator have a common factor, it is 3, and 

 p-2q = 3pi', seiqi = q+2pi', we get 



^ ' y = q 2 i-2piqi : p 2 1 -2p l q i . 

 Hence in every case we may set 



x=(q 2 -2pq), y=(p 2 -2pq\ B = ^(p 2 -pq+q 2 ). 



Now x+y must be a square, A 2 . Hence (q 2p) 2 3p 2 ==bA 2 , so that the 

 lower sign is excluded. From 2pq=(p q) 2 A 2 } we get 



where a, (3 are relatively prime. Any common factor of the numerator 

 and denominator divides 6. If it be 3, we reduce to a like fraction as above. 

 If it be 2, then |8 and hence p and y are even; but we may assume that if 

 either x or y is even, x is even. Thus in every case we may set 



/3), q=(2a(3-2a 2 ), 



It follows that X 6 +Y 3 =z 2 is impossible in integers if z is not divisible by 

 3. For, if the preceding x or y be a square, a = k 2 , a 3 (3 3 = h 2 , or $ = k\, 

 /3 3 -f8o: 3 = /ii, respectively; in either case, X\-{-Y\=z\ in smaller integers. 



Multiplying x and a by ^A, y and ft by tfB, we see that Ax 3 +By* = z- 

 has the integral solutions 



230 Atti Ac. Pont. Nuovi Lincei, 19, 1865-6, 185. 



231 Nouv. Ann. Math., (3), 1, 1882, 364; cf. (2), 20, 1881, 515; I'interin&liaire dea math., 



9, 1902, 329; 10, 1903, 133. 



232 Math. Quest. Educ. Times, 23, 1875, 85-6. 



233 Nouv. Ann. Math., (3), 4, 1885, 184. Cf. GtJrardin. 242 



234 L'interme'diaire des math., 3, 1896, 246. 



236 Ibid., 4, 1897, 110-12. Cf. the remarks, 112-15. 



238 Ibid., 5, 1898, 95; Det Kgl. Norske Videnskabera Selskabs Skrifter, 1896, No. 7. 



