CHAP, xxvi] FERMAT'S LAST THEOREM. 753 



Gauss' remarks for p = 3 (Werke, 2, 1863,387-391). $mcez=x+y (modp), 

 D=(x+y) p -x p -yp=Q(modp z ), D = pxy(x+y)4>(x, y). 



The equivalent congruence xyz<j>(x, y) = (mod p) is proved insolvable for 

 p = 3, 5, 11, 17 unless at least one of the three unknowns is divisible by p. 

 The method leaves in doubt the case p = 3n+l since the factor x z +xy+y 2 

 of has real roots. 



E. Catalan 124 stated 16 theorems on a n +b n = c n , n a prime > 3. If a is 

 a prime, then a=l (mod n); a n = l (mod rib); every prime factor of c a 

 divides a 1; a+b and ca are relatively prime; also 2a 1 and 26+1; 



rib 71 ' 1 ^a n ^ 



a and b exceed n; a n l is divisible by nb(b+l)(b 2 +b+l). Next, no one 

 of a+b, ca, cb is a prime. If a+ 6 = c", c a = 6", c b = dl, then c is 

 divisible by n. The [of Mathews 123 ] is 



p-*, ff* = i [( P ~ l ) l], 



the sign being plus if k is even. 



Catalan 125 stated the same theorems. Also, if a n +b n = c n , where a, b, c 

 are relatively prime in pairs, and a +b is divisible by n, it is divisible by 

 n n ~ l ; if a+b is divisible by a prime p^n, it is divisible by p n ; if a +b is 

 divisible by a power >n n ~~ 1 of n, it is divisible by n 2 "" 1 ; if a+b is divisible 

 by a power >p n of a prime p 3=n, it is divisible by p 2n . 



L. Gegenbauer 126 proved that 17, 29 and 41 are the only primes p = 4ju+l 

 not dividing a sum of three biquadrates prime to p. Cf. Euler 83 of Ch. 

 XXIII. 



C. de Polignac 127 proved that a n 2*=1 is impossible unless a = 3, 

 n=l or 2. 



A. E. Pellet 128 found by use of inequalities in the theory of roots of 

 unity that x q +y q =z q (mod p), where p is a prime gco+1, has solutions 

 x, y, z each not divisible by p for every co exceeding a certain limit (not 

 specified) for which go? +1 is a prime [Libri 24 ]. 



P. Mansion 129 considered x n +y n =z n , where x, y, z are relatively prime, 

 x<y<z, n an odd prime. By de Jonquieres, 117 y is composite. It is proved 

 here that z is composite. The proof that x is composite is erroneous, as 

 later admitted. 



M. Martone 130 attempted to prove Fermat's last theorem. 



m Bull. Acad. Roy. Sc. Belgique, (3), 12, 1886, 498-500. Reproduced in Oeuvres de Fermat, 

 IV, 156-7. 



125 M4m. Soc. R. Sc. Liege, (2), 13, 1886, 387-397 ( = Melanges Math., 2, 1887, 387-397). 



Proofs of some of these theorems by Lind, 241 pp. 30-31, 41-43, and by S. Roberts, Math. 

 Quest. Educ. Times, 47, 1887, 56-8. 



126 Sitzungsber. Akad. Wiss. Wien (Math.), 95, II, 1887, 842. 



127 Math. Quest. Educ. Times, 46, 1887, 109-110. 



128 Bull. Soc. Math, de France, 15, 1886-7, 80-93. 



129 Bull. Acad. Roy. Sc. Belgique, (3), 13, 1887, 16-17 (correction, p. 225). 



130 Dimostrazione di un celebre teorema del Fermat, Catanzaro, 1887, 21 pp. Napoli, 1888. 



Nota ad una dimostr. . . ., Napoli, 1888 (attempt to complete the proof in the former 

 paper). 



49 



