CHAP, xxvi] FERMAT'S LAST THEOREM. 765 



from Kummer's criterion. Shorter proofs have since been given by 

 Mirimanoff 223 and Frobenius. 228 



P. Mulder 215 noted that if n is an odd prime and a n +b n is divisible by n, 

 it is divisible by ft 2 . Proof as by Kummer. 25 



Chr. Ries 216 argued that a 2n +6 2n = c 2n (n>l) is impossible in integers 

 by considering the two factors of a~ n whose difference is 26 n , but assumed 

 that every prime factor of 26" divides 6. 



G. Cornacchia 217 employed the theory of roots of unity to investigate 

 the number of sets of solutions of x n -{-y n = l (mod p), where p is a prime 

 of the form nk+1. There are proper solutions for ft = 3 if p=t=7, 13; for 

 n = 4, if p*5, 13, 17, 41; for n = 6, if p* 7, 13, 19, 43, 61, 97, 157, 277; for 

 n = 8 > if ,, 4=17, 41, 113; for n any odd prime if p> (n-2) 2 n(n-l)+2(n+3). 

 For p a prime nk+1, x n -\-y n -\-z n = (mod p) has proper solutions for ft = 4 

 if p+5, 17, 29, 41 [Gegenbauer 126 ]; for ft = 6, if p=t=13, 61, 97, 157, 277, 

 31, 223, 7, 67, 79, 139; for n = 8 if p + 17, 41, 113, 89, 233, 137, 761. He 

 proved a theorem like that of Dickson, 199 but with a limit 



p>(e-2) 2 e(e-l)+2(e+3) 



which is larger than Dickson's if e>3. 



A. Flechsenhaar 218 considered, for n a prime > 3, 



(16) x n +y n -z n = Q (mod n 2 ) 



for x, y, z prime to n. We may set x<n, y<n, x+y z. Multiply (16) 

 by pi' and pi in turn, where p\x = l, p?y = l (mod ft). Hence the solvability 

 of (16) implies that of 



(17) l+6 B -(6+l) w =0, c +i_( c +i)=o (mod ft 2 ), 



where b^p 2 x, c = piy, whence 6c=l (mod ft). These conditions continue to 

 hold after 6 is replaced by 6 ft, and c by c n. We get 



l + (ft--l) n -(ft-O n =0, t = borc. 



Since these have the form of (17), it is stated that (n 6 1) (nc 1) = 1, 

 whence 6+c+l = (mod ft), by a false analogy, as no proof had been given 

 that, for every pair of solutions 6, c of (17), we have 6c = l. 



Admitting 6-f c+l=0, 6c=l, 6^c, we have ft = 6ra+l. Solutions 6, c 

 then exist and are tabulated for n a prime ^307. But (p. 274) for n a 

 prune 6m 1, (16) has no solutions prune to ft. 



J. Nemeth 219 noted that x k +y k = z k , x l +y l =z l have no common sets of 

 positive solutions if k, I are distinct positive integers. 



J. Kleiber 220 stated that if ft is an odd prime, x, y, z are relatively prime, 

 and y, z not divisible by ft, x n +y n = z n implies that 



x+e i y=(p+e i q) n (i = 0, 1, .,n-l; e* = l), 



which readily give y = 0. But he had assumed that the laws of factorization 

 of integers hold for numbers involving e, had not specified the kind of 



216 Wiskundige Opgaven, Amsterdam, 10, 1909, 273-4. 



216 Math. Naturw. Blatter, 6, 1909, 61-3. 



217 Giornale di mat., 47, 1909, 219-268. See Cornacchia 185 and the references under Libri. 24 



218 Zeitschr. Math. Naturw. Unterricht, 40, 1909, 265-275. 



219 Math. 4s Phys. Lapok, Budapest, 18, 1909, 229-230 (Hungarian). 



220 Zeitsch. Math. Naturw. Unterricht, 40, 1909, 45-47. 



