CHAP, xxiii] SINGLE EQUATIONS OF DEGREE n>4. 699 



sufficient condition 181 being that x+y and N=(x*+y 5 )f(x-{-y) be EL 

 Since N=(x 2 3xy+y-}-+5xy(x y} 2 , set z = 2 , t/ = 5?? 2 and make x+y= ED. 

 E. Miot 182 took x 5 +y b = 2 k pqr 2 , where p is a prime 4n+ 1, whence 2 k p = s 2 +< 2 , 

 and multiplied the initial equation by q 5 . L. Aubry obtained an infinitude 

 of solutions by setting 



x l = an, y l = bn, t l = cn-\-dn 2 , u l=en+fn 2 . 



" V. G. Tariste " 183 noted that, if x, y, z are < 10, 



x n +y n +z n +xyz = WQx+Wy-\-z 



holds only for n = 3 and then x, y, z are the digits of 370, 407 or 952. A. H. 

 Holmes 184 obtained special solutions with n = l or 2 by assuming that 

 yz = 100 or xz = 10. 



A. Cunningham 185 noted that every prime p = -X"" F n , with n = 12m +7, 

 can be expressed in the forms (x 3 y 3 ]f(xy). Cf. Cunningham. 187 



G. Frobenius 186 proved that x 2 +y 2 +z 2 = kxyz is solvable in positive 

 integers only for k 3 and k = l, while the latter case reduces to the former 

 by the substitution x = 3X, y = 3 Y, z = 3Z. Cf . Hurwitz. 174 



Cunningham 187 noted that, if n > 3, X n Y n = x 2 +xy-\-y 2 has an infinitude 

 of positive integral solutions. He noted (24, 1913, 85-6) cases when 

 x*y z or x 7 y 7 is expressible in the form Q 2 +l. He expressed (26, 1914, 50) 

 the product of two numbers of type x~-{-x-\-l and (27, 1915, 102) the 

 product of three such factors in the form A 2 +35 2 in several ways. 



T. Kojima 188 proved that if a rational function of several variables 

 with integral coefficients equals an nth power for all integral values of the 

 variables, it is an exact nth power. 



H. Brocard 189 stated that x = y = l is the only integral solution of 

 x x -}-y v =x+y, and that x x +y y = xy has no positive integral solution. These 

 problems were proposed by G. W. Leibniz. 190 



A. Cunningham 191 gave several solutions of 11(2* +#,+!) = 2 3 . 



E. Fauquembergue 192 noted that the only solutions of (4z 4 1) (4x 1) = y 2 

 in integers are x = Q, 1, 2; y = l, 3, 21. 



M. Rignaux 193 gave two identities x*+y 6 = z*+w 2 . 



W. Mantel 194 proved that x 2 -{-y 2 -\-z 2 = x 2 y 2 t 2 is impossible in integers; 



that, if n = 2, 6, 9, 11, 12, x\-\ \-xl=XiX 2 - -x n has no positive integral 



solutions, and gave the least solutions for n = 3 (3, 3, 3), n = 4 (2, 2, 2, 2), 



181 Republished, 1'intermediaire des math., 19, 1912, 227-8. 



182 Ibid., 119-120. 



183 Ibid., 133. 



184 Amer. Math. Monthly, 18, 1911, 69-70. 



185 L'intermediaire des math., 20, 1913, 3. Proof by Aubry, p. 120; by Welsch, p. 184. 



186 Sitzungsber. Akad. Wiss. Berlin, 1913, 458-87. 



187 Math. Quest. Educ. Times, 23, 1913, 31-32. 



188 T6hoku Math. Jour., 8, 1915, 24. 



189 L'intermediaire des math., 22, 1915, 61-2; 21, 1914, 101. 



190 Opera omnia (ed., L. Dutens), III, 85-6; letter to Oldenbourg, June 21, 1677. 



191 L'interme'diaire des math., 23, 1916, 41-2. 



192 Ibid., 24, 1917, 41-42. 



193 Ibid., 25, 1918, 7. For z 3 +y 6 = 2 3 +w> 6 , see Gerardin 86 of Ch. XXI. 



194 Wiskundige Opgaven, 12, 1917, 305-9. 



