CHAP, xxvi] FERMAT'S LAST THEOREM. 771 



* D. N. Ranucci wrote a pamphlet, Risoluzione dell'equazione 



x n Ay n = l, 



con una nuova dimostrazione dell' ultimo teorema di Fermat, Roma, 

 1911, 23 pp. 



F. Mercier 2480 noted that we may take x<y<z\i n>l, whence 



x n = z n y n = (z - y) (z n ~ l + yz n ~ 2 -\ ---- ) > (z - y} ny n ~ l > ny n ~ l , 



n/x<(x/y) n ~ 1 <l, n<x. This lemma, instead of helping him to prove Fer- 

 mat's last theorem, led him to commit the error of saying that 3 n +y n = z n 

 is solvable when n is any integer > 1 because it is solvable when n = 2. 



Ph. Furtwangler 249 proved by use of Eisenstein's law of reciprocity for 

 residues of lih powers, where I is an odd prime, that every integral divisor 

 r of Xi satisfies 

 (18) . r l ~ l = l (modi 2 ) 



if Xi, Xz, x z are relatively prime solutions =f=0 of x[-\-x l 2 -i-X3 = Q and x t is 

 prime to I. Since one of the x's is divisible by 2, we have the criterion 

 of Wieferich. Next, every factor r of Xix k satisfies (18) if #+* and 

 Xi Xk are prime to I. Since one of the x's is divisible by 3 unless all three 

 are congruent modulo 3, it follows from the two theorems that, if the x's 

 are all prime to I, (18) holds for r = 3, which is the criterion of Mirimanoff. 



S. Bohnicek 250 proved that integral numbers of the domain of the 2"th 

 roots of unity do not satisfy Fermat's equation with the exponent 2 n ~ l , 

 n>2. 



H. Berliner 251 considered x p = y p -}-z p for x, y, z not divisible by the prime 

 p>2. In Abel's formulas 2x = a p +b p +c p , , we may take a>b>c. 

 Then a = b+c2 k ep, where 2 k is the highest power of 2 dividing abc, while 

 ep is an odd multiple of 3. For every p, a < 3(6+ c); forp^5, a<3&; for 

 p^31, a<3 1/5 (6+c); for p^37, a<3 2/9 6. If p^5, b>3p; if p^37, 

 &>6p+l. 



L. Carlini 252 proved that x n -\-y n = z n (n>2)is not satisfied by three binary 

 forms in u, v, identically in the variables u, v. Hence a like result holds 

 for polynomials in one or more variables. 



J. Plemelj 253 proved x 5 +y 5 +z 5 = Q impossible in R( V5) more simply than 

 had Dirichlet. 20 



* B. Bernstein 254 gave some properties of numbers satisfying x n +y n = z n . 

 The latter is proved impossible under certain assumptions on x, y, z. 



R. D. Carmichael 255 proved that, if x p -\-y p -{-z p = Q has integral solutions 

 each not divisible by the odd prime p, there exists a positive integer 

 s<(p 1)/2 such that 



(modp 3 ). 



2480 Mem. Soc. Nat. Sc. Nat. et Math, de Cherbourg, 38, 1911-12, 729-44. Cf. Grunert. 73 

 249 Sitzungs. Akad. Wiss. Wien (Math.), 121, Ha, 1912, 589-592. 

 Ibid., 727-742. 



251 Archiv Math. Phys., (3), 19, 1912, 60-3. 



252 Periodico di Mat., 27, 1912, 83-8. 



253 Monatshefte Math. Phys., 23, 1912, 305-8. 



254 Math. Unterr., 1912, No. 3, 111-5; No. 4, 150-1 (Russian). 

 266 Bull. Amer. Math. Soc., 19, 1912-3, 233-6. 



