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



theorems like the following : Every integer > 36 is a sum of four squares and 

 four bi quadrates each ={=0; every integer > 14 is a sum of four squares and 

 four cubes =1=0. 



P. F. Teilhet 56 verified that every integer up to 600, except 23, is a sum 

 of two squares and two positive or zero cubes. 



G. Lemaire 57 noted that 3, 6, 7, 11, 15, 19, 22, 23 are not sums of any 

 number of powers of distinct numbers. 



G. Rabinovitch 58 proved that every number >23 is expressible in one 

 of the forms a m +6 w , a m +6 n +c p , , where a, b, are distinct, and m, 

 n,p, exceed unity. 



A. Ge'rardin 59 proved the theorems due to Andre. 53 



EVERY NUMBER A SUM OF THREE RATIONAL CUBES. 



S. Ryley 60 solved a=x 3 +y s +z 3 by taking x = p+q, y = pq, z = m2p. 

 Then 



will, for p = cw 2 /6, equal the square of av av 2 (ra ow 2 /6) if m 3 = 2av(m ay 2 /6) . 

 Let m = dv. Then v Qad/D, where D = 3d 3 + a 2 . Hence 



_ _ 



X ~ 6adD* ' y ~ QadD ' * D 2 



Reference is made to a less simple method in Leed's Correspondent, Quest. 

 211. 



T. Strong 61 showed how to express any number a as sum of three or more 

 rational cubes. Take x, px, mp, r, s, as the roots of the cubes. 

 Thus 



The right member will be the square of 3p(p2iri)+2c if 



p = c 2 /(3<z), c(2ra p)=m 3 +r 3 +s 3 H ---- . 

 Set c = mn, r = mr', s = ws', . The second condition gives 



6an 

 ~3a 2 +n 3 +r /3 +s' 3 H ---- " 



Hence giving any rational values to n, r', s', , we get rational values for 

 x = mc/(3p), m, p, r, s, . Since we can in particular express 4 as a 

 sum of three positive cubes, we can divide unity into three positive parts 

 such that if each be increased by unity the sum is a cube pEvans, 424 Davis, 426 

 and Tebay 428 of Ch. XXI]. 



Wm. Lenhart, 62 to express A as a sum of three cubes, selected any cube 

 r 3 and from Ar* subtracted a cube s 3 chosen by trial such that the difference 



88 L'intermediaire des math., 11, 1904, 16-17. 

 " Ibid., 19, 1912, 218. 

 Ibid., 20, 1913, 157. 

 " Ibid., 22, 1915, 207. 



80 Ladies' Diary, 1825, 35, Quest. 1420. 



81 Amer. Jour. Arts, Sc. (ed., Silliman), 31, 1837, 156-8. 

 M Math. Miscellany, Flushing, N. Y., 1, 1836, 122-8. 



