CHAP, xxvi] FERMAT'S LAST THEOREM. 763 



according as n is even or odd. Thus /(z 2 ) must be divisible by 



Hence /*^7. For /* = 7, /(z 2 ) =z 6 5z 4 +6z 2 1 must be congruent to S(x), 

 whence p = 2. For ju = 8, /(z 2 ) =z 6 6z 4 +10z 2 4, whence p = 17, contrary 

 to n>l. The cases ju = 10, 11, 13 are readily treated. The conclusion is 

 that, if n and p = mn+l are odd primes, m being prime to 3 and m^26, 

 the congruence n + i7 n +f n = (mod p) has no integral solutions each 

 prime to p, except for n = 3, w = 10, 14, 20, 22, 26; w = 5, m = 26; ft = 31, 

 m = 22. A direct examination of (15) was made for m = 28, 32, 40, 56, 64. 

 By use of these results and the theorem of S. Germain, 17 it was shown that 

 Fermat's equation is impossible in integers prime to n for every odd prime 

 exponent n < 1700. 



Dickson 196 proved the last theorem for ft < 7000 by extending the range 

 of the ra's to include all values <74, as well as 76 and 128. 



Dickson 197 factored certain numbers m m l for use in the last paper. 



Dickson 198 discussed the following problem: Given an odd prime n, to 

 find the odd prime moduli p for which x n +y n +z n = Q (mod p) has no solu- 

 tions each prime to p. We may take p = mn-\-l, where m is not divisible 

 by 3, since otherwise such solutions are evident. The general results are 

 applied to the cases n = 3, 5, 7. For n = 3, the only values of p are 7 and 13 

 [cf. Pepin 109 ]. For n = 5, p = ll, 41, 71, 101 [verified up to 1000 by 

 Legendre 17 ]. For n = 7, p = 29, 71, 113, 491. 



Dickson 199 proved, by use of Jacobi's functions of roots of unity, that 

 if e and p are odd primes such that 



pi(e-l) 2 (e-2) 2 +6e-2, 



then x e +y e +z e =Q (mod p} has integral solutions x, y, z, each prime to p. 

 In particular this establishes the conjecture by Libri. 24 Also, x 4 -\-y*=z* 

 (mod p) has solutions prime to p for every prime p = 4/+l exceeding 17 

 [and different 200 from 41]. 



P. Wolfskehl 201 bequeathed to the K. Gesellschaft der Wissenschaften 

 zu Gottingen one hundred thousand marks to be offered as a prize for 

 a complete proof of Fermat's last theorem. It may be noted that 

 Wolfskehl 202 was the author of a paper on the related subject of the class 

 number for complex numbers formed of eleventh or thirteenth roots of unity. 



196 Quar. Jour. Math., 40, 1908, 27-45. The omitted value n = 6857 was later shown in MS. 



to be excluded. 



197 Amer. Math. Monthly, 15, 1908, 217-222. See p. 370 of Vol. I of this History; also, A. 



Cunningham, Messenger of Math., 45, 1915, 49-75. 



198 Jour, fur Math., 135, 1909, 134-141. 



199 Ibid., 135, 1909, 181-8. Cf. Pellet, 128 - 244 Hurwitz, 213 Cornacchia, 217 and Schur. 283 



200 On p. 188, line 11, it is stated that for/ even and <14, p=4/+l is a prime only when 



/=4, p = 17, thus overlooking/ = 10, p=41. The fact that x 4 +y 4 sl (mod 41) has no 

 solutions each prime to 41 was communicated to the author by A. L. Dixon. 



201 Gottingen Nachrichten, 1908, Geschaftliche Mitt., 103. Cf. Jahresbericht d. Deutschen 



Math.-Vereinigung, 17, 1908, Mitteilungen u. Nachrichten, 111-3. Fermat's Oeuvres, 

 IV, 166. Math. Annalen, 66, 1909, 143. 



202 Jour, fur Math., 99, 1886, 173-8. 



