150 REPORT 1860. 



may write the equation in the symmetrical formx K +y K +z K =0. The impos- 

 sibility of solving this equation has been demonstrated by M. Kummer, first, 

 for all values of A not included among the exceptional primes* ; and secondly, 

 for all exceptional primes which satisfy the three following conditions : — 



(1.) That the first factor of H, though divisible by X, is not divisible by 

 X 2 (see art. 50). 



(2.) That a complex modulus can be assigned, for which a certain definite 

 complex unit is not congruous to a perfect Xth power. 



(3.) That B^ is not divisible by X 3 , B* representing that Bernoullian 

 number [c< /u — 1 ] which is divisible by Xf. 



Three numbers below 100, viz. 37, 59, 67, are, as we have seen, excep- 

 tional primes. But it has been ascertained by M. Kummer that the three 

 conditions just given are satisfied in the case of each of those numbers; so 

 that the impossibility of Fermat's equation has been demonstrated for all 

 values of the exponent up to 100. Indeed, it would probably be difficult to 

 find an exceptional prime not satisfying the three conditions, and conse- 

 quently excluded from M. Kummer's demonstration. 



We must confine ourselves here to an indication of the principles on which 

 the demonstration rests in the case of the non-exceptional primes J. 



demonstrated (Crelle, vol. xvii. p. 203), is nevertheless of great interest for the number of 

 auxiliary propositions contained in it. Of the same character are the notes by MM. Lebesgue 

 and Liouville, in Liouville's Journal, vol. v. pp. 184 & 360, and a few theorems given with- 

 out demonstration by Abel, (Euvres, vol. ii. p. 264. 



In the year 1847, M. Lame presented to the Academy at Paris a memoir containing a 

 general demonstration of Fermat's Theorem, based on the properties of complex numbers 

 (Comptes Rendus, vol. xxiv. p. 310; Liouville, vol. xii. pp. 137 & 172). It was, however, 

 observed by M. Liouville (Comptes Rendus, vol. xxiv. p. 315), that this demonstration is 

 defective, as it assumes, without proof, the proposition that a complex number can be repre- 

 sented, and in one way only, as the product of powers of complex primes — a proposition 

 which, as we have seen, is untrue, unless we admit ideal as well as actual complex primes. 

 The discussion on M. Lame's memoir attracted Cauchy's attention to Fermat's Theorem; and 

 the 24th and 25th volumes of the Comptes Rendus contain several communications from 

 him on the subject of complex numbers [or polynomes radicaux, as he has preferred to term 

 them]. In the earlier papers of this series, Cauchy attempts to prove a proposition which, 

 as we have already observed (see art. 41), is untrue for complex numbers considered gene- 

 rally, viz. that the norm of the remainder in the division of one complex number by another 

 can be rendered less than the norm of the divisor (see Comptes Rendus, vol. xxiv. pp. 517, 

 633 & 661). Elsewhere (ibid. p. 579) he assumes the proposition as a hypothesis, and 

 deduces from it conclusions which are erroneous (pp. 581, 582). But at p. 1029 he recognizes 

 and demonstrates its inaccuracy. The results at which he arrives iu his subsequent papers 

 on the same subject are, for the most part, comprehended in M. Kummer's general theory 

 (Comptes Rendus, vol. xxv. pp. 37, 46, 93, 132, 177). In one place, however (p. 181), he 

 enunciates, though without demonstrating, the following important result : — 



" If the equation x x -{-i/ K +z K = be resoluble, x, y, z denoting integral numbers prime to 



X, the sum 



1^+2^+3^+.... +(X^ip 



is divisible by X." 



(Compare M. Kummer's memoir in the Berlin Transactions for 1857, p. 64.) 



The investigation of the Last Theorem of Fermat has been twice proposed as a prize- 

 question by the Academy of Paris — first at some time previous to 1823 (see Legendre's 

 memoir already cited, in vol. vi. of the Memoires de l'Academie des Sciences, p. 2), and again 

 in 1850 (Comptes Rendus, vol. xxx. p. 263) : at neither time was the prize adjudged to any 

 of the memoirs received. On the last occasion, after several postponements of the date 

 originally fixed for the award, the prize was ultimately, in 1857 \ib. vol. xliv. p. 158), con- 

 ferred on M. Kummer, who had not been a competitor, for his researches on complex num- 

 bers. 



* Liouville, vol. xvi. p. 488, or Crelle, vol. xl. p. 131. 



t See the memoir No. 15 in the list of art. 41. 



J When X is not an exceptional prime, the equation v x -\-y x +z K =Q is irresoluble not only 



