CHAP, xxiv] INTEGERS WITH EQUAL SUMS OF LIKE POWERS. 713 



E. N. Barisien 49 noted that 1, 5, 9, 11, 15, 16 and 3, 4, 8, 10, 14, 18 and 

 2, 6, 7, 10, 14, 18 and 1, 5, 9, 12, 13, 17 have the same sum and sum of 

 squares; also that 



3, 4, 8, 11, 15, 16 = 2, 6, 7, 12, 13, 17. 



A. Aubry 50 gave known and new solutions of Za = 2a, Za 2 = Za 2 , and 

 proved the impossibility of x, y = t, u, v. 



N. Agronomof 51 noted the case <z+c+3 = 26of(l). 

 A. Gerardin 52 gave a solution of ZA = 2X, SA 3 = SZ 3 : 



A = 2p*-$pq+6q*, B = 2pq, C=pq, X= -p z +9pq-12q z , 

 Y = 2p z -Wpq+12q*, Z = p z -5pq+Qq\ 



N. Agronomof 53 gave an 8 parameter solution of 



I>; = XW (& = 1, 2, 3). 



1=1 t=i 



For any solution of this system, we have 



! (*=i, 2, 3, 4), 



z being arbitrary. Proceeding similarly, we can solve 



By specializing the solution first cited, he obtained solutions of 



i = Z2/; (& = 1, 2, 3; s = l or 2 or 3). 



A. Filippov 53a stated that the specialized solutions just mentioned are 

 trivial since they reduce to Xi=yt or 2/, = 0. 



A. Gerardin 54 noted that Zz = 2a, Zz 2 = Za 2 if a = 3, 6 = 2, c = l, 



z = (2u 2 +8uv+W)/D, D = u 2 +3uv+3v*. 

 R. Goormaghtigh 55 solved the same system by setting 



x = Pg+Qp, y = Ph+Qq, z = P(k+l+m-g-ti)+Qr, 



Then the equation obtained by eliminating z between the proposed equations 

 determines P/Q as follows : 



P = p(k-g}+q(l-h)+r(g+h-k-l), 



49 Mathesis, (4), 3, 1913, 69. 



60 Annaes Sc. Acad. Polyt. do Porto, 9, 1914, 141-151. 



61 Suppl. al Periodico di Mat., 19, 1915, 20. 



62 Nouv. Ann. Math., (4), 15, 1915, 564; I'intermediaire des math., 22, 1915, 130-2 (cor- 



rection for h = 2); 23, 1916, 107-10. Cf. papers 130, 302, 438-40, 442 of Ch. XXI. 

 M T6hoku Math. Jour., 10, 1916, 207-14. 

 530 Ibid., 15, 1919, 143. 



64 L'interme'diaire des math., 24, 1917, 55 (correction, p. 153). 

 Ibid., 25, 1918, 20-21. 



