710 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxiv 



Tarry 34 gave (1) and noted that, for x arbitrary, 



a, b, , h = p, q, - , t 

 imply 



a, ,h,p+x, -, -,t+x n =p, "-,1, a+x, ,h+x. 



By use of this lemma he found 



6a-36-8c, 5a-9c, 4a-46-3c, 2a+2&-5c, a-2b+c, b 



= 6a-26-9c, 5a-46-5c, 4a+6-8c, 2a-36, a+26-3c, c. 

 H. B. Mathieu 35 gave as the general solution of (2), for n = 3, 



L. Aubry (p. 234) noted that z+?/+2 = w+y+w; implies xyz = uvw. 

 O. Birck 36 noted that, if x+y+z = Q, 



(ix ky)*+ (iy kz) n -\-(iz kx) n = (iy kx) n + (iz ky) n + (ix kz) n , 



n = 0, 1, 2,4. 

 " V. G. Tariste " 37 noted that 



(23n+57Z) e +(407i-6Z) e +(177i-630 e 



e = 2, 4. 

 Further such cases were given by E. B. Escott and A. Ge"rardin. 38 



E. Miot 39 stated that any 2"(2a+l) numbers in arithmetical progression 

 can be separated into two equal sets having the same sum of fth powers 

 for = l, -, n, if a>0, n>l; while t = l, -, n 1 if a = 0. Hence, if in 

 Tarry 's 33 example we replace x by a-}-(x l)r, we get 



a, a+2r, o+6r, , a+23r = a+7-, a+3r, , a+22r. 



Tarry 40 noted that the number of terms in each member of the equations 

 deduced in his 34 lemma is 2k d, if k is the number of terms in each member 

 of the given equations, while x is expressible in d ways as a difference of 

 two numbers belonging to the same member. Given 



1, 5, 10, 16, 27, 28, 38, 39 = 2, 3, 13, 14, 25, 31, 36, 40, 

 take 



x = 11 = 16-5 = 27-16 = 38-27 = 39-28 = 13-2 = 14-3 = 25-14 = 36-25. 

 Thusd = 8, 



1, 5, 10, 24, 28, 42, 47, 51 =2, 3, 12, 21, 31, 40, 49, 50. 



E. Miot (p. 85) noted that 

 1+n, 2+7i, 10+n, 12+w, 20+n, 21+n 



= w, 5+rc, 6+7i, 16+rc, 17+n, 22+n. 



34 L'intermSdiaire des math., 19, 1912, 219-221. Cf. Tarry. 46 

 38 Ibid., 225. 



88 Ibid., 19, 1912, 252-5. Cf . Birck 21 " of Ch. XXII. 

 87 Ibid., 129; cf. 201, 250. 



3 *Ibid., 21, 1914, 126-9. 



89 Ibid., 20, 1913, 64-5. Generalization of Tarry. 38 

 40 Ibid., 68-70. 



