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



A. Gerardin 17 noted that 

 1, ra+3, 2m-2, 4w+2, 5m-3, 6m-l 



= 2, w-1, 2m+3, 4m-3, 5m+l, 6w-2, 

 x, x+3, x+5, x+6, z+9, z+10, x+12, x+15 



=x+l, x+2, z+4, x+7, x+S, x+11, x+13, z+14, 

 also the result due to G. Tarry : 

 c, a+36, 2abc, 4a+56 3c, 5a+6 4c, 6a+46 5c 



e 



= 6+c, ab, 2a+46 c, 4a 3c, 5a-f-56 4c, 6&+36 5c. 



Ge"rardin 18 noted that 6 2 +a6- a 2 , a 2 +2a6-46 2 , 46 2 and 46 2 have the 

 same sum and sum of cubes as a?+ab 6 2 , 46 2 +2a6 a 2 , 6 2 and 6 2 . 

 G. Tarry 19 gave 



i+26 5c, 2a+46 7c, 3a 66+c, 3a 46 c, 4a 6 6c, 

 i-96, 6a+56-16c,8a-116-4c, 9a+36-20c, 10a-106-9c, 

 10a-5&-14c, lla-26-19c, lla-21c, 12a-106-13c, 12a-86-15c, 

 13a-36-22c, 14a-76-20c 



=c, a+36-4c, 2a-56+2c, 2a-36, 3a+26 7c, 3a+46 9c, 4a 76, 

 4a 26 5c, 5a+56 14c, 6a 106 c, 8a+46 19c, 9a 116 6c, 

 10a-46-15c, 10a+6-20c, lla-106-llc, lla-86-13c, 12a-36-20c, 

 12a-6-22c, 13a-96-16c, 14a-66-21c. 



Welsch 20 stated that the general solution of (2) is 



1 ( ~ ~V I \ \ 1 f ~ V \ \ 1 1 ' ~ 



ry _ "" 1/7 \ -T A 1 0^ "~~ ( /i A ^ A I Q I * "~~~ I it ^ 

 ^ 7l*~-l ^^ 2 \ I / J 71 2 \ J J *y 7& 1 ^^ 2 \ 



with a:,-, 2/i (i = l, , n 2) arbitrary, where 



n 2 n 2 



i=l i=l t=l i=l 



and X, ju are of the same parity as a X, a Y. E. B. Escott (pp. 213-4) 

 noted that one can proceed as he 16 had done for w = 3. 

 H. B. Mathieu 21 asked if the general solution is 



o 



2su uv-\-st, st-\-tv, su2uv-\-tv = stuv, 2su2uv-\-st-\-tv, su-\-tv. 



Numerical solutions not of this type were cited in reply. 22 



A. Gerardin 23 noted three cases of (1) in which c = 2a+2b = t [Martin 9 ], 



and that 4p 2 3mp, 3w 2 +4wp 4p 2 have the same sum and sum of cubes 



as 6m 2 3 mp, 2p 2 -f- 4mp 6m 2 , 3m 2 2p 2 . 



U. Bini 24 set y s =x 8 +r a in (2), whence Sr s = 0. By the latter, r m is 



eliminated from the quadratic equation, which is then treated as a quadratic 



for ri. Next, let 



(4) x n -\-y n -\-z n = u n +v nJ rw n (n = l, 2, 4), 



17 Sphinx-Oedipe, 1908-9, 96; errata, 144. 



18 Ibid., 4, 1909, 44. 



19 Ibid., 176. 



20 L'interm&liaire des math., 16, 1909, 89-90. For n = 3, ibid., 15, 1908, 280-1. 



21 Ibid., 16, 1909, 219-220. 



22 Ibid., 17, 1910, 72, 165. 



23 Assoc. frans. av. ec., 38, 1909, 143-5. 



24 Mathesis, (3), 9, 1909, 113-8; same method in Periodico di Mat., 25, 1910, 119-128. 



