770 



HlSTOEY OF THE THEORY OF NUMBERS. 



[CHAP. XXVI 



while 



/y 5* f**'y* f )t& 1 _-... /*, / y*2yj/s 2 I I / 1 \ g 1 ^> ^s 

 vt/^ t/o*^M OjU/ W I \ ^^ * / ^3 I**' 



are such that ?7 2 ( l) s n(z) 2 has as divisors only 2 and numbers of the 

 form r 2 ( l) s n 2 . The initial equations are both impossible if n = 5 or 13. 

 A. E. Pellet 244 considered for a prime p = hn-\-l, having g as a primitive 

 root, the number hN s of times that 



g in +g in +g kn =Q (mod p) (i,j,k = Q,l,--,h-l). 



By use of the equation for the n periods of the pih roots of unity it is shown 

 that pN 3 has the limits /i 2 



whence [error 245 ] the inferior limit 

 is positive if h>n^n. Hence in that case, x n -\-y n -\-l = Q (mod p) has 

 solutions prime to p. Cf. Libri. 24 



D. Mirimanoff 246 reproduced his 237 paper and used his first formula to 

 obtain results concerning g(5) and q(7). Also he proved that 0p_i(0 is 

 divisible by p not only when t is one of the six ratios r = x/y, , but also 

 for t=r and t= r 2 . Finally, he proved Sylvester's formula for q(m) 

 [Vol. I, Ch. IV of this History]. 



A. Thue 247 proved that, if n is a prime >3, and e is an imaginary nth 

 root of unity, and each Bi is an integer numerically ^. 



tan 



if not every Bi = 0. Next, for R an integer, let PQ = R n , where 



i=0 



Then for a suitably chosen k and integers /,-, gt such that 



we have P/7^ = B/A, where A = 2/ie 1 ', B = Sgfe*. It is stated that applica- 

 tion can be made to Fermat's equation 



If a"+& n = c n for relatively prime integers (p. 15), we can find positive 

 integers p, q, r, each < V3c, such that pa j rqb = rc. Hence 



(ar) n + (6r) " = (pa+qfy n , 



whence q n r n is divisible by a. 



Thue 248 proved that if y n = x n -{-l, n>3, the most general solution of 



A n +B n = (co+c 1 ?/^ ----- hen-it/"- 1 ) 71 , 

 where A, B and each c are integral functions of x, is 



/*+(/*); =(/</)" 



where / is an arbitrarj^ integral function of x. 



244 L'interm6diaire des math., 18, 1911, 81-2. 

 246 This deduction fails if n = 5, h = 20. 



246 Jour, fiir Math., 139, 1911, 309-324. 



247 Skrifter Vidcnskapsselskapet I Kristiania (Math.), 1, 1911, No. 4. 



248 Ibid., 2, 1911, No. 12, 13 pp. For his paper, ibid., No. 20, see 178 Ch. XXIII. 



