CHAP, xxiii] SYSTEMS OF EQUATIONS OF DEGREE n > 4. 703 



A. G6rardin 218 noted cases in which s 5 x, s 5 y, s 5 z are squares, where 

 s = x+y+z', s n x, , s n t are all cubes, where s = x+y+z+t, for 



x, z = T(27p 9 +144p 3 )-108p 6 -63, ?/ = 216p 6 +l, = 126 (s = l). 



He noted (pp. 197-8) cases when Ps n +Qx m +Ry m -\ ---- , Ps n +Qx m +Rz m + 

 - , are all pth powers, where s = x-\-y-\- . 



Ge"rardin 219 gave the general solution of his problem to make sx, 

 s2/, -, s all pth powers, where s=x+y-{- +. 



R. Goormaghtigh 220 gave solutions of x+y+z = s, s 2 x 2 = A p , s z y z = # p , 

 s 2 z 2 = C p , where p is 2 or any odd integer. He 221 stated that, for 

 A < 1000000, A = 1+oH ----- \-x m = l+y-\ ----- \-y n holds only for 



31 = l-r-5+5 2 = l+2+2 2 +2 3 +2 4 , 8191 = 1+2H 



in addition to evident solutions if x or y is negative. 

 Despujols 222 took </>?+02 = (a 2 +& 2 ) n ~ 1 in the identity 



to obtain a congruent number 201 h of order n. He stated that every con- 

 gruent number of order n is of the form 20 2 Xji, where 2 (X 2 H-/* 2 )=:r n , and 

 conversely. 



On xfa&lx*= D (i = l, 2, 3) see p. 174, p. 186. 



PAPERS NOT AVAILABLE FOR REPORT. 



P, Lackerbauer, Lehrsatze und Aufgaben iiber Gleichheiten als Beitrag zur hoheren un- 



bestimmetn Analysis, Progr. Mtinnerstadt, 1834. 

 G. A. Longoni, Sui problem! di analisi indeterminata, Monza, 1840. 

 C. F. Meyer, Ein diophantische Problem, Progr. Potsdam, 1867. 

 Poeschko, Auflosungsmethode unbestimmter Gl., Progr. St. Polten, 1869. 

 J. Slavik, Solution of indeter. equations (Czech), Progr. Koniggratz, 1877. 

 F. M. Costa Lobo, Resolution des Equations indetermin6es, Coimbra, 1885. 

 C. Alasia, Elementi della teoria generale delle equazioni . . . e delle equazioni indetermi- 



nante, Napoli, 1891. 



A. Zinna, L'analisi diofantea, Trapani, 1900. 

 H. Zuschlag, Diophantische Gleich., BerUn, 1908. 



J. Edaljii, Note on indeterminate equations, Jour. Indian Math. Soc., 3, 1911, 115. 

 H. Verhagen, An equation in three unknowns, Nieuw Tijdschrift voor Wiskunde, 3, 1915-6, 



307-14. 



218 L'intermediaire des math., 23, 1916, 169-170. 



219 Ibid., 207-8. 



220 Ibid., 24, 1917, 23-24. 



221 Ibid., 88 (p. 153, correction). 



222 /fad., 26, 1919, 14-15. 



