CHAP. XXV] SUM OF THREE RATIONAL CUBES. 727 



is a number t found in his 186 table (Ch. XXI) of numbers expressible as a 

 sum of two positive rational cubes. Or, let Ar 3 +s 3 = J = a 3 +6 3 . Then A 

 is the sum of the cubes of ax, bx, ex if c = p s and 



1 t+c* p 3 - 22 z 



, / , ps z V 



H-r <J =l r-\-- 7) = 



V VA/' 



z 3 A A VA/' r 3 A-s 3 ' 



Hence 



s 3 ) 6(r 3 A-s 3 ) s(2r 3 A+s 3 ) 



j- -, 6z = - -, cz= -+, d = 

 rd rd rd 



As an application, 2 and 4 are expressed as sums of three positive rational 

 cubes. The same table is used tentatively to express n-f-lorn lasa sum 

 of n cubes each >1 or each <1, with examples when n = 4, 5, 6. 



Several 63 expressed any number n as the sum of three rational cubes. 

 Let their roots be (lz)/(2z), (ax 2 l)/z. The sum of their cubes is n if 



Assuming that 2 = 1 2az 2 +ftiz 3 , we get z = 6an/(n 2 +3a 3 ). 



EVERY POSITIVE NUMBER A SUM OF FOUR POSITIVE RATIONAL CUBES, ETC. 



G. Libri 64 noted that if m, n, r are solutions of az 3 +fo/ 3 +c2 3 = 0, then 

 aX 3 -\-bY 3 +cZ 3 = d is solvable for d arbitrary. Set X = mp+q, Y = np+s, 

 Z = rp+t. The new equation lacks p 3 and will lack p 2 and hence determine 

 p rationally in terms of s, t, if we take q= (?M 2 s+cr 2 )/(am 2 ). 



If A is a multiple of 24, it is a sum of four cubes [not necessarily positive] : 



Next, let A = 24z+6, 0<6<24. If b is one of the numbers 1, 3, 5, 7, 8, 9, 

 11, 13, 15, 16, 17, 19, 21, 23, 6 3 -6 is a multiple 24i/ of 24, whence A = 6 3 +s, 

 where s = 24( u) is a sum of four cubes, so that A is a sum of five cubes. 

 If 6 is not one of the above numbers, 61 is one of them. Hence every 

 integer is a sum of six cubes one of which is or 1. If 



we have the identity in r, s, t, 

 (1) 



Every integer is the algebraic sum of 17 biquadrates, taken positively or 

 negatively. The proof, similar to the above for cubes, follows from 



12r 3 



61 Math. Quest. Educ. Times, 13, 1870, 63-4. 



M Memoria sopra la teoria del numeri, Firenze, 1820, 17-23. 



