CHAP, xxni] SINGLE EQUATIONS OF DEGREE n>4. 697 



diophantine equations in n unknowns such that all the unknowns are 

 expressible rationally in n 1 parameters. An instance is the method of 

 solving x 3 +y 3 +z 3 +u 3 = Q used by Schwering 73 and Kuhne 74 of Ch. XXI. 

 A. Cunningham 170 found solutions of 



(3) (z 3 +?/ 3 )(X 3 +P)= 3 +7 7 3 



by expressing n = (x 3 +y 3 )/(x+y) in the form i 2 +3w 2 in one of the three 

 ways: (%x-y) 2 +3(%x) 2 for x even, (rc-^/) 2 +3Q?/) 2 for y even, 



(x+y\* 



~ 







for x > y both 



and by expressing N= (X s + Y 3 )[(X+ F) in the form T 2 +3C/ 2 . Then 



nN = A 2 +3B 2 , A = tT^3uU, B = tUuT. 

 But A 2 +3 2 is expressible in the form ( 3 +? 3 ) /(+??) in one of three ways. 



Hence (3) is reduced to (x+y)(X+Y) = +v. R. W. D. Christie 171 noted 

 the special solution 



He 172 noted that 10 3 +30 2 = (3 3 +7 2 )(3 2 +4 2 ). Cunningham noted that 



is satisfied if A=A\, a = a 2 , A?=a 8 cT6d, B = a 3 dbc. 



*P. S. Frolov 173 found the least solution of (4) for re = 1. 



A. Hurwitz 174 discussed the positive integral solutions rci, , x n of 



(4) xl-i ----- \-xl=xxiX Z - -x n , n^3, 



where x is an integer. If j- = (x, Xi, , x n } is a solution, then evidently 

 '=(#, rci, #2, , rc n ) is a solution when rei+rci = rerc 2 - -x n . Similarly, 

 " = (#, rci, x' 2j x 3 , --, x n ) is a solution when x' 2 -\-Xz=xx i x 3 - - -x n . Call 

 these solutions ', ", , (n) " neighbors " to ^. Build the neighbors to 

 each of these, etc. Then all such solutions are said to be " derived " 

 from ^. Call a " fundamental ' ' solution if no one of its n neighbors 

 has a smaller sum rci+ -\-x n . It is proved that is a fundamental 

 solution if and only if 2x\ ^xx\ -x n for i=l, - , n; that every solution is 

 either a fundamental solution or can be derived from another one; that 

 there is no positive integral solution of (4) when re is a given integer >n; 

 that all positive integral solutions with x = n can be derived from 

 rci= =rc n = l (the case n = 3 being due to Markoff 157 ). If n^5 and if 

 re, Xi, -, x n form a fundamental solution of (4) with Xi^x^ = - =x n) the 

 last n 2 k of the rc/s have the value unity, where k is determined by 



E. B. Escott 175 cited two numerical equations rc 7 +rrc 5 +s:r 3 +te+& = 

 with rational roots [see Ch. XXIV 63 ]. "Charbonier" (18, 1911, 62-3) 

 employed the roots a, 6, a 6, c, d, e, c d e. 



170 Math. Quest. Educ. Times, (2), 5, 1904, 76. [Cf . 27, 1915, 17-18.] 



171 Ibid., 100. 



Ibid., (2), 6, 1904, 115. 



173 Vest, opytn. fiziki (Spacinski's Bote Math.), Odessa, 1906, Nos. 419-20, pp. 243-55. 



174 Archiv Math. Phys., (3), 11, 1907, 185-96. Cf. papers 173, 186, 194, 195o. 

 176 L'interme'diaire des math., 16, 1909, 242. 



