718 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxv 



But any number is of the form 6p+r, r = 0, , 5, while p is a sum 

 nl+ -+nl of four squares. By the earlier remark, 6p is a sum of 48 

 bi quadrates. Hence .4 =48 +5. 

 E. Lucas 60 gave the identity 



(1) 6(xl+xl+xl+xly = -2(xi+x,y+2(xi- x 3 Y (i,j = 1, -, 4; i<j). 

 [It becomes Liou vine's 6 identity for Xi = x+y, x^ x y, Xz = z-\-t, x = z 

 Lucas also gave the incorrect identity 



Assuming that every integer is a sum of nine cubes, he stated incorrectly 

 that it follows that every integer is a sum of at most 26 sixth powers. 

 Lucas 7 noted the identities 



the second being erroneous [Fleck 23 ], since the left member exceeds the 

 right by 60(a; 2 t/V+x 2 ?/ 2 w 2 +x 2 2: 2 tt 2 +2/Vtt 2 ). 



S. Realis 8 proved that 47 biquadrates are sufficient by using the result 

 that any integer is a sum of 4 squares, one of which is arbitrary (under 

 certain restrictions) and hence may be chosen a biquadrate. 



E. Lucas 9 reduced the number to 45 as follows. Let k = Qp-\-r. If 

 p = 8h+j O'=l, 2, 3, 5 or 6), p is a ED, and, by (1), k a sum of 3-12+5 

 biquadrates. If p = 8h or 8/&+4, p27 is a S; then 



so that at most 3-12+2+5 biquadrates are needed. Finally, if p = Sh+7, 

 p 14 is a O, so that 



& = 6ni+6tt2+6tt3+3 4 +3+r, N 4 =i3- 12+4+5. 



Lucas 10 obtained the lower value JV 4 =41. Since Sh+j (j = l, 2, 3, 5, 

 or 6) is a EJ, 4S/&+6;' is a sum of 36 biquadrates. By subtracting at most 

 five of the biquadrates I 4 , 2 4 , 3 4 from any given number, we obtain one of 

 these numbers 48h+t (t = 6, 12, 18, 30, 36). By the tables our theorem is 

 true for numbers ^=5-3 4 . 



E. Maillet 11 proved that every positive integer is a sum of 21 or fewer 

 cubes ^0, five or more of which are or 1. He employed the identity 



to conclude that 6a(o: 2 +m) is a sum of at most six positive cubes if ^m^a 2 

 and if m is a sum of three squares, i. e., if m 4= 4^(8^+7). Under the similar 

 conditions on m', 6A = 6a(a 2 +w)+6a'(a /2 +ra / ) is a sum of at most twelve 



60 Nouv. Corresp. Math., 2, 1876, 101. 



7 Jour, de math. 61em. et sp<c., 1, 1877, 126-7, Probs. 38, 39. Quoted by C. A. Laisant, 



Recueil de problemes de math., algebre, 1895, 125. 



8 Nouv. Corresp. Math., 4, 1878, 209-210. 

 Ibid., 323-5. 



"Nouv. Ann. Math., (2), 17, 1878, 536-7. 

 u Assoc. franc., av. BC., 24, II, 1895, 242-7. 



