PREFACE. xix 



sequent three centuries. At intervals during the past century, leading 

 scientific academies offered one of their prizes for a proof. The dignity of 

 this famous theorem was injured by the offer of a very large prize in 1908. 

 Since only printed proofs may compete, the gain thus far has gone to the 

 printers ; in this history no mention will be made of the very numerous false 

 proofs called forth by this last prize. 



Fermat's last theorem is not of special importance in itself, and the 

 publication of a complete proof would deprive it of its chief claim to atten- 

 tion for its own sake. But the theorem has acquired an important position 

 in the history of mathematics on account of its having afforded the inspira- 

 tion which led Kummer to his invention of his ideal numbers, out of which 

 grew the general theory of algebraic numbers, which is one of the most 

 important branches of modern mathematics. 



Although Gauss had proved in 1832 that the laws of elementary arith- 

 metic hold also for complex integers (numbers like 5+7 V--T) and made a 

 brilliant application of them in his investigation of biquadratic residues, 

 the theory of algebraic numbers was really born in the year 1847. For it 

 was then (pp. 739, 740) that the mathematical world became definitely 

 conscious of the fact that complex integers a +air+ . . . -fa^-ir"- 1 , where 

 the a's are ordinary integers and r is an imaginary nth root of unity, do 

 not in general decompose into complex primes in a single manner, do not 

 possess a greatest common divisor, and hence do not obey the laws of 

 elementary arithmetic. This historical fact came to light through dis- 

 cussions of lacunae in the attempted proof by Lame that, if n is an odd 

 prime, x n +y n = z n is not satisfied by such complex integers. Other errors 

 of the same nature were made in the same year by Wantzel and by so great 

 a mathematician as Cauchy. Curiously enough, Kummer himself made 

 the error, in a letter of about 1843 to Dirichlet, of assuming that factoriza- 

 tion is unique, so that his initial proof of Fermat's last theorem was 

 incomplete. But Kummer did not stop with the mere recognition of the 

 fact that algebraic numbers do not obey the laws of arithmetic; he suc- 

 ceeded in restoring those laws by the introduction of ideal elements, this 

 restoration of law in the midst of chaos being one of the chief scientific 

 triumphs of the past century. 



Although the theory of algebraic numbers appears to be a powerful 

 tool especially adapted to attack Fermat's last theorem, it has not yet led 

 to a complete proof of it. Numerous facts have been obtained by a variety 

 of more elementary methods. Until the theorem is actually proved, it will 

 obviously be unwise to attempt to weigh the importance of any particular 

 fact or method. Hence no further analysis will be given here of the con- 

 tents of the long Chapter XXVI which is itself a condensed history of 

 Fermat's last theorem. Moreover this subject is one of those for which 

 the subject index gives a rather minute classification of the subject matter. 



In the preceding summary mention was made of only the most im- 

 portant of the upwards of 5,000 writings upon which report has been made 



