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



x+y+z, and x n y n +x n z n +y n z n is divisible by (xyz) 3 if n^5. Proofs 184 

 have been given of the first result and the fact that, if x+y+z = 0, s is a 

 function of xyz and xy -\-xz-\-yz, 



* G. Cornacchia 185 treated the congruence x n +y n =z n (mod p). 



P. A. MacMahon 186 noted that the integral solutions of x n ay n = z 

 may be obtained by the development of a lln into a continued fraction. 



F. Lindemann 187 again 170 proved Abel's formulas and, after treating at 

 great length each of the three cases, concluded that Fermat's equation is 

 impossible in integers. A. Fleck 188 pointed out a serious error and various 

 minor errors. I. I. Iwanov 189 noted errors, also in Lindemann's 170 first 

 proof, where in (67) the modulus n 6 should be n 5 . 



A. Bottari 190 proved that if x, y, z are positive integers in arithmetical 

 progression such that x n +y n = z n , then either n = l and x = y/2 z/3 or 

 n = 2 and #/3=2//4=2/5. If x, y, z } t are positive integers in arithmetical 

 progression such that x n +y n -\-z n = t n , then n = 3, or/3 = y/4. = 2/5 = t/Q. Cf. 

 Cattaneo. 192 



J. Sommer 191 omitted the restriction that n is a regular prime in stating 

 that Kummer proved that x n -\-y n = z n , for n>2, is not solvable in complex 

 integers based on an nth root of unity. He gave the proof for n = 3 and 



71 = 4. 



P. Cattaneo 192 gave a brief proof of the results of Bottari, 190 but included 

 the false solution n = l,x = y/2 = 2/3 = /4. 



A. S. Werebrusow 193 failed in his proof of Fermat's last theorem, the 

 error being indicated by L. E. Dickson and others (ibid., pp. 174-7). 



Werebrusow 194 stated that (x+y+z) n x n y n z n has, for n odd, the 

 factor n(x+y)(x-\-z)(y-t-z). While this is true for n an odd prime, it fails 

 for n = 9, x = y=z = l (ibid., 16, 1909, 79-80). 



L. E. Dickson 195 noted that, if a is a common root of the congruences 



(15) z m =l, (z+l) TO =l (modp) 



of Wendt, 152 the numbers (9) are common roots and are distinct if 2 m 1 

 is not divisible by p. They are the roots of a sextic in z which is unaltered 

 when z is replaced by l/z or by 12. The sextic must divide z m 1 

 modulo p. Set #=2+1/2, m=2/t. The sextic becomes 



C(x) =z 3 +3z 2 +/to+2/3-5. 

 From z" l/z ft = vw get /(a: 2 ) =0, where /( 05) is of degree %p \ or (/* !) /2 



184 L'interme'diaire des math., 13, 1906, 142; 14, 1907, 22-23, 36-39, 92-95, 258. 



186 Sulla Congruenza x n +y n =z n (mod p), Tempio (Tortu), 1907, 18 pp. 



186 Proc. London Math. Soc., (2), 5, 1907, 45-58. For z = l, G. Cornacchia, Rivista di 

 fisica, mat. sc. nat., Pavia, 8, II, 1907, 221-230. 



187 Sitzungsber. Akad. Wiss. Miinchen (Math.), 37, 1907, 287-352. 



188 Archiv Math. Phys., (3), 15, 1909, 108-111. 



189 Kagans Bote, 1910, No. 507, 69-70. 



190 Periodic di Mat., 22, 1907, 156-168. 



191 Vorlesungen iiber Z&hlentheorie, 1907, 184. Revised French ed. by A. L4vy, 1911, 192. 



192 Periodico di Mat., 23, 1908, 219-20. 



193 L'interme'diaire des math., 15, 1908, 79-81. 



194 Ibid., p. 125. Case n = 3, in I'e'ducation math., 1889, p. 16. 



195 Messenger of Math., (2), 38, 1908, 14-32. 



