CHAP, xxvi] FERMAT'S LAST THEOREM. 755 



A. Rieke 137 again attempted to prove x p +y p = z p impossible, but again 

 confused (pp. 251-2) algebraic and arithmetical divisibility, even for 

 p = 3 (p. 253). 



E. Lucas 138 proved (p. 267, p. 275) the theorem of Cauchy, 29 and (p. 370- 

 1) the formulas (1), (3), (4) of Legendre 17 , with the aim to show that, when 

 x, y, z are relatively prime in pairs, no one of them is a prime or a power of a 

 prime [cf. Markoff 157 ]. He proved (p. 341) the first result due to Jaquemet. 3 



D. Mirimanoff 139 found in terms of the units a necessary and sufficient 

 condition that the second factor [Rummer 61 ] of the class number be divisible 

 by X. He treated in detail the case A = 37. 



J. Rothholz 140 used the theorem of Kummer 25 on the divisors of a n 

 to show (?) that # 2n ?/ 2n = 2 2n has no integral solutions if n is a prime 

 or if one of the numbers x, y, z is a prime and n is an odd prime; x n -\-y n = z n 

 is impossible if x, y or z is a power of a prime, the prime not being =1 

 (mod n), while n is an odd prime; x n +y n = (2p) n is impossible if n and p 

 are odd primes; x n y n = z n is impossible if x, y or z has one of the values 

 1, , 202. The history of the theorem is discussed at length. On p. 29 

 are pointed out two errors in the proof by Rieke. 134 



* W. L. A. Tafelmacher 141 proved Abel's formulas and congruencial 

 corollaries from them. In the second paper he proved that Fermat's 

 equation is impossible for n = 3, 5, 11, 17, 23, 29 and, in case x+y 2 = 

 (mod 7i 4 ) for n = 7, 13, 19, 31 [but with proofs valid only when no one of 

 x, y, z is divisible by n, since the argument pp. 273-8 does not suffice to 

 exclude the case in which one of these numbers is divisible by n~\. 



H. Teege 142 proved that x 5 +y 5 = l has no rational solutions by setting 

 = t, t+ljt = z, (qlpY = s. Then 



Since z is rational, (4s+l) 2 4(s l)(4s+l)=ra 2 . Set ra = 5ju. Then 

 4s +1 = 5/A Let n = b/a, where a and b are relatively prime. Thus 



Hence a 2 divides 5p 5 . The impossibility of this equation is proved by 

 considering the cases a divisible or not divisible by 5. 



H. W. Curjel 143 proved that if x s y i = l and x, y are primes, then z is a 

 prime, t is a power of 2, and x or y equals 2. 



Several 144 proved by use of cube roots of unity the known result that, 

 if n is odd and not a multiple of 3, (x+y) n x n y n is divisible by x*+xy+y 2 . 



S. Levanen 145 discussed x 5 -\-y 5 = 2 m z 5 , for x, y, z without common factor, 



137 Zeitschr. Math. Phys., 36, 1891, 249-254. Error indicated in 37, 1892, 57, 64. 



138 Th6orie des nombres, 1891. References in Introduction, p. xxix, where it is stated falsely 



that Kummer proved Fermat's theorem for all even exponents. 



139 Jour, fur Math., 109, 1892, 82-88. 



140 Beitrage zum Fermatschen Lehrsatz. Diss. (Giessen), Berlin, 1892. 



141 Anales de la Universidad de Chile, Santiago, 82, 1892, 271-300, 415-37. Report from 



Lind, 241 p. 50. 



142 Zeitschr. Math. Naturw. Unterricht, 24, 1893, 272-3. 



143 Math. Quest. Educ. Times, 58, 1893, 25 (quest, by J. J. Sylvester). 

 144 /6zd., 112. 



146 Ofversigt af Finska Vetenskaps-Soc. Forhandlingar, Helsingfors, 35, 1892-3, 69-78. 



