CHAP, xxii] SINGLE EQUATIONS OF DEGREE FOUR. 665 



L. Aubry and H. Brocard 283 solved 2zy+l=z 2 +2/ 2 -f-2 2 for y = 4. 

 Aubry 284 gave a solution involving three parameters of 



ylyl+ylyl+ylyl-yiytysy* = 0. 



Brehm 285 solved xyz(x+yz) = t 2 in integers. Set tq=xyp, where p and 

 q are relatively prime integers. Then the equation gives s(x+yz) = rp 2 x, 

 ys = rq z z, where r and s are relatively prime integers. Hence x, y, t are 

 expressed in terms of t. 



E. Swift 286 proved that x 4 y* = z 3 is impossible for x prime to y. 



R. D. Carmichael 287 noted that if x , yo, U Q) V Q give a solution of 



we can deduce a second solution Rafter performing the operations]: 

 x=xl-ayl+bvl, y = 2x Q y Q u , u=u* 

 F. L. Carmichael 288 obtained the solution 



u = u\ bvl +ay\ abwl, v = 



where 



u 2 = m 2 + (6+a6 2 +a6 3 )n 2 , v z = 2mn+2n 2 ', 



also two simpler solutions, as well as solutions when a/b= D, a = or 6 = 0. 



L. Bastien and L. Aubry 289 found the general solution of 



x 2 =(y 2 



Several 290 treated x 4 -y* = a? 



A. G^rardm 291 discussed 



A. Cunningham 292 treated a 2 +6 2 = c 4 +2d 2 , for b and c given. 



L. Aubry 292a solved (x z -y 2 }(x z +2y*)=x*-2y 2 . 



Ge'rardin 157 of Ch. IV solved z 4 +6zy+z/ 4 = a 4 +6a 2 /3 2 -f /3 4 . On equa- 

 tions quadratic in x and in y, see note 145. On pq(mp z +nq 2 ~) =rs(mr 2 +ns 2 ), 

 see papers 168, 170, 174, 181. 



TO FIND n NUMBERS WHOSE SUM IS A SQUARE AND SUM OF SQUARES IS A 



BIQUADRATE. 



For the case n = 2, see papers 37-63. 



G. W. Leibniz 293 considered the case n = 3. 



283 L'intermediaire des math., 19, 1912, 157-9, 3 (for special solutions). 



284 Ibid., 20, 1913, 95. 



286 Math. Naturw. Mitt., (2), 15, 1913, 20-21. Cf. Kommerell. 270 



286 Amer. Math. Monthly, 22, 1915, 70-1. 



287 Diophantine Analysis, 1915, 46-8. 



288 Amer. Math. Monthly, 23, 1916, 321-9. 



289 L'intermediaire des math., 23, 1916, 36-8. 



290 Ibid., 123-4; 24, 1917, 66, 88, 133-4. 



291 Ibid., 24, 1917, 32. 



292 Ibid., 143-4. 

 *Ibid., 26, 1919, 150-2. 



293 MS. in Bibliothek Hannover, about 1676. Cf. D. Mahnke, Bibliotheca Math., (3), 13, 



1912-3, 39. J. Wallis, Opera Math., 3, 1699, 618, quoted a letter from Leibniz to Olden- 

 burg, Oct. 26, 1674, in which this problem is mentioned (Bull. Bibl. Storia Sc. Mat. e 

 Fis., 12, 1879, 519). 



