706 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. XXIV 



M. Frolov 7 noted that Sa fc = S6 fc , 2aJ = 26j, k = l, , w, imply 



= 2(6+60*. 



For n = 2 there must be at least 3 terms a; for w = 3, at least 4. For n = 3, 

 the least terms are stated incorrectly 70 to be 1, 5, 8, 12 and 2, 3, 10, 11. For 

 w = 3, there are examples when the a's and 6's together give 1, 2, , 2m. 

 J. W. Nicholson 8 noted the identities 



3a+36, 2a+46, a, 6 = 3a+46, a+36, 2a+6; 

 5a+106, 4a+116, 3a+56, 2a+86, 3a+36, 2a+66, a, b 



= 5a+116, 4a+66, 3a+106, 3a+86, a+56, 2a+36, 2a+6, 



there being one more term on the left than on the right. But for n = l, 

 , 5, the sum of the nih powers of the ten numbers a32, a24, a18, 

 a10, a4 equals the sum of the nih powers of the ten a30, a28, 

 a16, a8, a6. 



A. Martin 9 noted the special case of (1) : 



a, 6, 2a+26 = a+26, 2a+6. 

 Also, 



p, q, 2p+2q, 3p+3g f = 3p+2g, 2p-\-3q, p+a; 



a+6+c, a+6 c, a 6+c, a+6+c = 2a, 26, 2c. 

 R. W. D. Christie 10 noted that, if t = e+f+g+h, 



s+e, s+/, s+g, s+h, s t = se, sf, sg, sh, s+t. 



[Since we may reduce each term by s, we obtain an evident identity.] 



A. Cunningham 11 noted that x+y, 6, c=x, y, 6+c if xy = bc. Next, if 

 a, 6, c==x, y, z, then 



a, 6, c-\-kz, kc=x, y, z-\-kc, kz. 



Similarly a solution in two sets of n numbers yields one in two sets of n+1 



numbers. J. H. Taylor noted that if ai+a 3 + +a 2r -i = a 2 +a 4 H h2r, 



then 



2 

 CL i | -L Ct 2 j Gt 3 "~}~ J. 0X4 % *""^2r """ ^* 1 > ^2 ~t~ J- 1 ^ 3 j ^4 ~T" -^ j * 



If 6iH \-bzr = 2r(nr)r, then 



61, , 6 2r , n=6i+l, , 6 2r +l, n 2r. 

 H. M. Taylor noted the generalization of (1) : 



J. _ 2 ^ 



n 



R. W. D. Christie noted that a6+cd, 6c, ad = 6c+ad, a6, cd, and 

 n1, n2, n+3, n4, w+5, n+6, n 7 



=n+l, n+2, n3, n+4, n 5, n 6, w+7. 



7 Bull. Soc. Math. France, 17, 1888-9, 69-83; 20, 1892, 69-84. The second was reprinted 



in Sphinx-Oedipe, 4, 1909, 81-89. 



70 On the proof-sheets Escott noted that 5, 1, 4, 8 = 2, 2, 7, 7 has smaller terms. It is de- 

 rived from 3,-l, 2, 6 J= 0, 0, 5, 5 of Escott 63 by increasing each term by 2. 



8 Amer. Math. Monthly, 1, 1894, 187. 



9 Math. Magazine, 2, 1898, 212-3, 220. 



10 Math. Quest. Educ. Times, (2), 2, 1902, 40. His condition s = o+6+c+d is unnecessary. 

 " Ibid., (2), 4, 1903, 98-100. 



