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



To make the cubes positive take f <#<. This is the case if r = ll/2, 

 whence x = 5/9. Thus the ratios of 351, 832, 833 to 3136 answer Diophantus' 

 problem. 



A. Ge"rardin, R. Goormaghtigh and others discussed in 1'interme'diaire des mathe'maticiens 

 the following problems in which s is the sum of thte unknowns: 



s 3 -x and s 3 -y cubes, 22, 1915, 222; 23, 1916, 142-4, 210-1. 



s 3 -x, s 3 -y, s 3 -z all cubes or all biquadrates, 22, 1915, 245; 23, 1916, 4-5. 



s 3 -x, s 3 -y, s 3 -z, s 3 -t all cubes or all squares, 23, 1916, 28-9, 52-3. 



s 3 Zi, , s 3 0-5 all cubes or all squares, 100-1. 



s 3 Xi, ' , s 3 x n all cubes or all squares, n odd i^5, 24, 1917, 114-5. 



SYSTEMS OF EQUATIONS OF DEGREE THREE IN FOUR OR MORE UNKNOWNS. 



Alkarkhi 430 (beginning of eleventh century) solved x 1 y 3 = z 2 , x 2 +?/ 3 = < 2 

 by setting x = 2y, z = my, t^ny, whence y = 4 ra 2 = n 2 4, m 2 +n 2 = 8; take 

 w 2 = 4/25, n 2 = 196/25. He treated various similar problems. 



J. Ozanam 431 asked for four numbers such that one obtains a square by 

 adding to the product of the first three the product of any two of the four. 



W. Wright 432 found four numbers the product of any three added to 

 unity being a square. Substitute the value of z from xyz+\ = (pz-\-\Y 

 into vyz-\-l and vxz-\-l. The results are squares if 



F = p' 1 2vyp -\- vxy z = (p qY, 



which determines p, and (r = p 2 2vxp-\-vx z y = D. The latter leads to a 

 quartic in q which is equated to the square of q- 2vxq+2v z xy vxy 2 +2vx-y 

 2t> 2 z 2 , thus determining q. Then vxy+l=n- determines x. J. Baines 

 took wxy+l = a i , wxz +1 = 6 2 , wyz+l = c 2 , which determine w, x, y in terms 

 of z. Take (6 2 -l)(c 2 -l) = l=z. Then zt/z+l = (a 2 -l) 2 +l = (41/9) 2 if 

 a = 7/3. 



J. Anderson 433 found n numbers whose sum is a square such that the 

 square of each exceeds the cube of their sum by a square. Let the numbers 

 be s 2 X{, where Sa? = l. Then shall x\ s 2 = D = (sp a;,-) 2 , say, whence 

 Xi = s(pi + l)/(2p,-). W. Watson used the numbers x,s 3 with the sum s 2 . 

 Then x\ 1 = D = (a;,- w<) 2 gives a;,-. The condition on the sum gives s. 



Several 434 found four numbers x, x-\-y, x-\-2y, x-{-3y in arithmetical 

 progression whose sum s of squares is a square and sum p of the product 

 of the extremes and the product of the means is a cube. Take y = vx. 

 Then s=D if 4+12y+14v 2 =(ry-2) 2 , which gives v. Take r = 4. Then 

 y = 14, p = 478z 2 , which is a cube if # = 478. 



S. Ward 435 found four numbers a, b, c, x such that the product of any 

 three added to unity shall be a square. Set m ab, n = ac, p = bc, and let 

 mx + 1 = ( 1 rx) 2 , whe nee x = (2r + rn) /r 2 . Then shall 



A 2 , r-(px+l) =r~ 



<30 Extrait du Fakhri, French transl. by F. Woepcke, Paris, 1853, 134. 



' Letter to de Billy, May 9, 1676; Bull. Bibl. Storia Sc. Mat., 12, 1879, 517. 



432 The Gentleman's Math. Companion, London, 5, No. 24, 1821, 47-8. 



433 Ibid., 5, No. 27, 1824, 266-8. 



The Math. Diary, New York, 1, 1825, 55-6. 



435 Amer. edition of J. R. Young's Algebra, 1832, 343-5. 



