750 HlSTOKY OF THE THEORY OF NUMBERS. [CnAP. XXVI 



right member remains finite. This argument was called insufficient by 

 E. Netto. 106 



A. Korkine 107 modified the last proof. Let Z be a polynomial in t 

 whose degree ra is not less than the degrees of X and Y. Then one of the 

 latter is of degree m, say Y. Let m X (X^O) be the degree of X. Differ- 

 entiate (Y/X) n +(ZIX) n +l = Q with respect to t. Then, since Y, Z have 

 no common factor, 



is an integral function. As the degrees of the numerators are^|2w X 1 

 and that of the denominators is m(n 1), we have 



A. Lefebure 108 separated into two classes the primes p = 2kn+l. Into 

 the first class, put the p's such that the algebraic sum of any three residues 

 of nth powers modulo p cannot be a multiple of p. Into the second class, 

 put the p's for which the algebraic sum of three residues is a multiple of p. 

 It is claimed that all the p's in the first class are divisors of one of the integers 

 satisfying x n -\-y n =z n , so that every p is a divisor of x, y or 2, or is in the 

 second class. Hence if the first class is infinite, the equation is impossible. 

 But the first class is not finite when the second is infinite [^correction by 

 Pepin 109 ]. 



T. Pepin 109 noted that Libri 24 long ago pronounced judgment on an 

 attempted proof like Lefebure 's. 108 To prove Libri's assertion on 



z 3 +2/ 3 +l=0 (mod p = 3ft+l), 



Pepin showed (by use of Gauss, Disq. Arith., art. 338, on the equation for 

 the three periods of roots of unity) that the number of sets of solutions of 

 the congruence in positive integers <p is p-\-L 8, where L is determined 

 by L 2 +27M 2 = 4p and L=l (mod 3). Hence 7 and 13 are the only primes 

 3/t+l which cannot divide a sum of three cubes without dividing one of 

 them. 



0. Schier 110 claimed to prove x n -\-y n =z n impossible in relatively prime 

 integers if n is an odd prime. We have x +y=z (mod ri). Expand by the 

 binomial theorem 



cancel x n +y n with z n , and divide by the factor n. Thus 



n1 



=z n ~ l nk-\ ----- \-n n ~ l k*. 



Hence also the left member must be divisible by n. It is stated that this 

 divisibility depends on that of the factors xy and x-\-y occurring in every 



108 Jahrbueh Fortschritte Math., 11, 1879, 138. 



107 Comptes Rendus Paris, 90, 1880, 303-4 (Math. Soc., Moscow, 10, 1882, 54-6). 



108 Ibid., 90, 1880, 1406-7. 



109 Ibid., 91, 1880, 366-8. Reprinted, Sphinx-Oedipe, 4, 1909, 30-32. 

 uo Sitzungsber. Akad. Wiss. Wien (Math.), 81, II, 1880, 392-8. 



