CHAP, xxvi] FERMAT'S LAST THEOREM. 769 



Then 



n 



from which the results of the paper follow. The six ratios of the three 

 solutions prime to p of Fermat's equation satisfy the congruence G m (x) = Q 

 (mod p) of degree m 2. Hence, if m = 2 or 3, G m vanishes identically. 



A. Fleck 239 proved, as an extension of his 227 theorem (iii), that the 

 prime factors of J i} KI, Z/i are of the form Qvp z -\-l. Hence J, , Z/i are all 

 of the form Qnp z -\-l. For any prime factor j of the form 6jup+l of J, 

 (ty) Glt =(tzY"=l (mod j), where = 1 in case A, t = p in case B. A like 

 result is said to hold for the prime factors of K or L. 



E. Dubouis 240 defined, in honor of Sophie Germain, a " sophien " of a 

 prime n to be a prime 6, necessarily of the form kn+1, for which x n = y n +l 

 (mod 6} is impossible in integers prime to 6. He stated that Pepin 109 

 proved that the sophiens of n are finite in number, whereas Pepin proved 

 this only for w = 3. If the resultant of a k = l, (o +!)* = ! is not divisible 

 by 6, then 6 is a sophien of n [Wendt 152 ]. 



B. Lind 241 gave an exposition of various papers dealing with Fermat's 

 last theorem without the use of complex integers or ideals, but unfortunately 

 interpolated careless remarks of his own. Of the results claimed by Lind 

 to be novel, equations (19)-(26) are correct, but long known, while (27) 

 is not proved, viz., that x-\-y 2 = (mod 9) if x n +y n = z n , it being proved 

 only for modulus 3. This error gave rise to later errors in his inequalities 

 (p. 32) and his equations (95), (1066). His attempt (pp. 61-5) to prove by 

 use of congruences Fermat's last theorem contains several serious errors 

 besides the dependence on (27). The bibliography is quite extensive. 



J. Joffroy 242 noted that, if F = x s7 -\-y Z7 2 37 = for integers x<y<z, then 

 = 1919191. For, x* 7 -x = Pm, P = 2-3-5-7-13-19-37; so that 



z, Wi>0. 



T. Hayashi 243 proved that if, for n an odd prime, x n -\-y n = nz n , or if 

 x n +y n = z n for z divisible by n, then 6o+&H ----- \-b s = Q (mod n 2 ), where 

 s = (n 1)/2, and the 6's are the coefficients of the polynomial Y satisfying 

 the identity 



where 



239 Sitzungsber. Berlin Math. Gesell., 9, 1910, 50-3 (with Archiv Math. Phys., 16, 1910). 



240 L'interme'diaire des math., 17, 1910, 103-4. 



241 Abh. Geschichte Math. Wiss., 26, II, 1910, 23-65. Reviewed adversely by A. Fleck, 



Archiv Math. Phys., (3), 16, 1910, 107-9; 18, 1911, 107-8. 



242 Nouv. Ann. Math., (4), 11, 1911, 282-3. Reproduced, Oeuvres de Fermat, IV, 165-6. 



243 Jour. Indian Math. Soc., Madras, 3, 1911, 16-22; 111-4. Same in Science Reports of 



Tohoku University, 1, 1913, 43-50, 51-54. 



50 



