764 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxvi 



No mention will be made here of numerous 203 recent false proofs 204 of 

 Fermat's last theorem, published mostly as pamphlets. Errors in some 

 of these have been noted by A. Fleck, 205 B. Lind 241 (p. 48), J. Neuberg, 206 and 

 D. Mirimanoff. 207 



E. Schonbaum 208 gave a historical introduction to and exposition of the 

 elements of the theory of algebraic numbers; also Kummer's proof, in 

 simplified form, of Fermat's last theorem for the case of regular primes. 



* A. Turtschaninov 209 proved and slightly generalized Abel's 16 theorem 



* F. Ferrari 210 discussed the infinitude of solutions of each of 



X n y n =Z n+l , 



A. Thue 211 stated that there are no [not an infinite number of] integral solu- 

 tions of any of the equations, with n>2, h and k given positive integers, 



x n +(x+k) n = y n , x 2 -W = ky n , (x+h) 3 +x* = ky n , (x+h} 4 -x 4 = ky n . 



These results are consequences of the theorem (pp. 27-30) that, if r>2 

 and a, b, c are any positive integers, c ={=0, there is not an infinitude of pairs 

 of positive integral solutions p, q of bp r aq r = c. 



A. Hurwitz 212 proved that, if m and n are positive integers not both 

 even, x m y n +y m z n +z m x n = Q has integral solutions =|=0 if and only if 

 u t -{-v t -}-w t = Q has such solutions, where t = m 2 mn-\-n 2 . Cf. Bounia- 

 kowsky, 149 Ch. XXIII. 



Hurwitz, 213 after citing Dickson's 199 proof by cyclotomy, gave an ele- 

 mentary, but long, proof that, if a, 6, c are integers +0 and e is an odd prime, 



ax e +by e +cz e =Q (mod p) 

 has A sets of solutions x, y, z each not divisible by the prime p, where 



-(e-l)(e-2)Vp-i?e (17 = 0, 1 or 3). 



. 



p-l 



Hence A > when p exceeds a limit depending on e. 



A. Wieferich 214 proved that if x p +y p +z p = is possible in integers 

 prime to p, where p is an odd prime, then 2 P ~ 1 1 is divisible by p 2 . He 

 deduced this criterion from the conditions (13) derived by Mirimanoff 180 



203 According to W. Lietzmann, Der Pythagoreische Lehrsatz, mit einem Ausblick auf das 



Fermatsche Problem, Leipzig, 1912, 63, more than a thousand false proofs were published 

 during the first three years after the announcement of the large prize. 



204 Titles in Jahrbuch Fortschritte Math., 39, 1908, 261-2; 40, 1909, 258-261; 41, 1910, 248- 



250; 42, 1911, 237-9; 43, 1912, 254, 274-7; 44, 1913, 248-50. 



205 Archiv. Math. Phys., (3), 14, 1909, 284-6, 370-3; 15, 1909, 108-111; 16, 1910, 105-9, 

 372-5; 17, 1911, 108-9, 370-4; 18, 1911, 105-9, 204-6; 25, 1916-7, 267-8. 



206 Mathesis, (3), 8, 1908, 243. 



207 Comptes Rendus Paris, 157, 1913, 491; error of E. Fabry, 156, 1913, 1814-6. L'enseigne- 



ment math., 11, 1909, 126-9. 



208 Casopis, Prag, 37, 1908, 384-506 (Bohemian). 



209 Spaczinski Bote, 1908, No. 454, 194-200 (Russian). 



210 Suppl. al Periodico di Mat., 11, 1908, 40-2. 



211 Skrifter Videnskaba-Selskabet Christiania (Math.), 1908, No. 3, p. 33. 



212 Math. Annalen, 65, 1908, 428-30. Case m =2, n = 1 by Euler 9 and Vandiver, 336 Ch. XXI. 



213 Jour, fur Math., 136, 1909, 272-292. 



214 Ibid., 293-302. For outline of proof, see Dickson, 288 182-3. 



