732 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxvi 



take any integer k and set p(qs)=k(pr). Then pqrs = (p r) (s + k) . 

 The second case hi which a z and x z are not divisible by z differs from the 

 preceding only as to signs. 



L. Euler's 5 theorems on the linear forms of the divisors of a m b m are 

 cited under Euler 5 ' 6 of Ch. XVI of Vol. I of this History. 



Lagrange's 142 method for r n As n = q m is given in Ch. XXIII. 



A. J. Lexell 6 considered a 5 +6 5 = c 5 . Set x+y = a?, xy = b 5 . Then 



L 5 /t^rw 



Since the factors are relatively prime, x = p 2 , x 5 4z> = q*. Hence 



p w g 2 



N. Fuss I 7 noted that, if ld=4 n =D is possible in rational numbers, 

 r n -\-p n = q n would be possible in integers. To reduce the former to integers, 

 set x = pq/r 2 ; then r 2n 4p"g n = D, say the square of r n +2v, where v is 

 prime to r. Then p n q n = v(r n -}-v), whence v = p n , r n -\-v = q n . 



L. Euler 8 multiplied a n +b n = c n by 4a n and added 6 2n . Thus 



(2a n +b n y = 4a n c n +b 2n = D. 



Euler 9 noted that he had failed in attempts to prove x n -\-y n = z n im- 

 possible if n>2. 



C. F. Kausler 10 proved that x 6 -\-y e = z* is impossible in integers. For, 

 if possible, set x = mn, where m is a prime. Of the forty cases, all are imme- 

 diately excluded except two : 



z*-\-z 2 y z -\-y* = m 6 n 6 or mn 6 , z 2 y' 2 = l or m 5 . 



For the second alternatives, eliminate z 2 . Then 



and m is a factor of 3?/ 4 . If y is divisible by m, z is, and x, y, z have a 

 common factor. There remains the case m = 3; then z-\-y, zy are 3 5 , 1 

 or 3 4 , 3 or 3 3 , 3 2 , cases readily excluded. The first alternative is excluded 

 by the lemma : There are no integers y, z for which 



Sophie Germain 11 (1776-1831) stated in her first letter to Gauss, Nov. 

 21, 1804, that she could prove that x n +y n = z n is impossible if n p 1, 



6 Comm. Arith., I, 50-6, 269; II, 533-5. 



6 Euler's Opera postuma, 1, 1862, 231-2 (about 1768). 



7 Ibid., 241 (about 1778). Cf. Euler. 8 



8 Ibid., 242 (about 1782). 



9 Ibid., 587; letter to Lagrange, March 23, 1775. Corresp. Math. Phys. (ed., P. H. Fuss), 



1, 1843, 618, 623, letters to Goldbach, Aug. 4, 1753, May 17, 1755. Novi Comm. Acad. 

 Petrop., 8, 1760-1, 105; Comm. Arith. Coll., I, 296. 



10 Nova Acta Acad. Sc. Petrop., 15, 1806, ad annos 1799-1802, 146-155. 



11 The first and third letters were published in Oeuvres philosophiques de S. Germain, Paris, 



1879, 298. Cinq lettres de Sophie Germain a C. F. Gauss, publics par B. Boncompagni, 

 Berlin, 1880, 24 pp. Reproduced in Archiv Math. Phys., 65, 1880, Litt. Bericht 259, 

 pp. 27-31; 66, 1881, Litt. Bericht 261, pp. 3-10. Reviewed, with Gauss, 13 by S. Gun- 

 ther, Zeitschr. Math. Phys., 26, 1881, Hist.-Lit. Abt., pp. 19-26; Italian transl., Bull. 

 Bibl. Storia Sc. Mat. e Fis., 15, 1882, 174-9. 



