CHAP, xxvi] FERMAT'S LAST THEOREM. 733 



where p is a prime Sk+7. In her 12 fourth letter, Feb. 20, 1807, she stated 

 that if the sum of the nth powers of any two numbers is of the form h z +nf 2 , 

 the sum of these two numbers is of that form. Gauss 13 replied, April 30, 

 1807, that this is false, as shown by 15 u +8 u = /i 2 +ll/ 2 , whereas 



C. F. Gauss 14 gave a sketch of a proof of the impossibility of a 5 +6 5 +c 5 = 

 and noted that the method is not applicable to seventh powers. 



P. Barlow 15 proved that if n is a prime and x n y n =z n is solvable in 

 integers prime in pairs, then one of the four sets of conditions 



must hold. For, (x n y n }l(xy) is not divisible by a factor +n of xy, 

 and if divisible by n, the quotient is prime to xy and to n. Hence z n is 

 divisible by xy, and, if n is a factor of x y, by n(x y), while the quotient 

 is prime to wand to xy. In the first case, xy = r n . In the second case, 

 n(xy)=r n = n n r"i, xy = n n ~ l r[. 



His attempt to prove x n y n =z n impossible if n>2 involves the error 

 (cf. Smith, 79 Talbot 84 ) that a sum of fractions in their lowest terms is not 

 an integer if the denominator of each fraction has a factor not dividing 

 all the remaining denominators. 



N. H. Abel 16 stated that, if n is a prime >2, a n = b n -\-c n is impossible 

 in integers when one or more of the numbers a, b, c, a +6, a-f-c, bc, 

 a llm , b llm , c llm are primes [cf. Talbot 77 , de Jonquieres 117 ]. If the equation 

 is possible, then a, b, c have factors x, y, z, respectively, such that either 

 [cf. Barlow 15 ] 



= x n +y n +z n , 2b = x n +y n z n , 2c=x n +z n y n ; 



= n n ~ l x n +y n +z n , 2b = n n ~ l x n +y n z n , 2c = n n ~ l x n +z n y n ; 



n n - 1 (x n +y n )+z n , 2b = n n - 1 (x n +y n )z n , 2c = n n ~ l (x n -y n }+z n ', 



or values derived from the second set by permuting a, b, and x, y, and 

 changing the signs of c and z; or values derived from the third set by 

 replacing a by b, b by c, c by a, x by y, y by z, and z by x. Thus 2a 

 must have one of the three forms listed, where x, y, z have no common 

 factor. Finally, 2a^9 n +5 n +4 n ; the least one of a, b, c cannot be less than 

 (9 n 5 n +4")/2. The editor, L. Sylow, remarked p. 338 that these theorems 

 appear to contain some inaccuracies. 



12 Published by E. Sobering, Abb. Gesell. Wiss. Gottingen, 22, 1877, 31-32. 



13 Lettera inedita di C. F. Gauss a Sofia Germain, publicata da B. Boncompagni, Firenze, 



1879. Reproduced in Archiv Math. Phys., 65, 1880, Litt. Bericht 257, pp. 5-9. 



14 Werke, II, 1863, 390-1, posth. paper. 



"Appendix to English transl. of Euler's Algebra. Proof "completed" by Barlow in Jour. 



Nat. Phil. Chem. and Arts (ed., Nicholson), 27, 1810, 193, and reproduced in Barlow's 



Theory of Numbers, London, 1811, 160-9. 

 18 Oeuvres, 1839, 264-5; nouv. eU, 2, 1881, 254-5; letter to Holmboe, Aug. 3, 1823. 



