MODERN MATHEMATICAL RESEARCH— MILLER. 191 



He added that lie had discovered a wonderful proof of this theorem, 

 but that the margm of the page did not afford enough room to add 

 it.* This theorem has since become known as Fermat's greater tlieo- 

 rem and has a most interesting and important history, which we pro- 

 ceed to sketch. 



About a century after Fermat had noted this theorem, Euler 

 (1707-1783) proved it for all the cases when n is a multiple of either 

 3 or 4, and during the following century Dirichlet (1805-1859) and 

 Legendre (1752-1833) proved it for all the cases when n is a multiple 

 of 5. The most important step toward a general proof was taken 

 by Kummer (1810-1893), who applied to this problem the modern 

 theory of algebraic numbers and was thus able to })rove its truth for 

 all multiples of prunes which do not exceed 100 and also for all the 

 multiples of many larger primes. 



The fact that such eminent mathematicians as Fermat, Euler, 

 Duichlet, Legendre, and Kummer were greatly interested in this 

 problem was sufficient to secure for it considerable prominence in 

 mathematical literature, and several mathematicians, including 

 Dickson, of Chicago, succeeded in extending materially some of the 

 results indicated above. The cii'cle of those taldng an active interest 

 in the problem was suddenly greatly enlarged, a few years ago, when 

 it became known that a prize of 100,000 marks (about $25,000) was 

 awaiting the one who could present the first complete solution. 

 This amount was put in trust of the Gottingen Gesellschaft der Wissen- 

 schaften by the will of a deceased German mathematician named 

 Wolfskehl, and it is to remain open for about a centur}^, until 2007, 

 unless some one should successfully solve the problem at an earlier 

 date. 



It is too early to determine whether the balance of the effects of 

 this prize will tend toward real progi'ess. One desu-able feature is the 

 fact that the interest on the money is being used from year to year to 

 further important mathematical enterprises. A certain amount of 

 this has aheady been given to A. Wieferich for results of importance 

 toward the solution of Fermat's problem, and other amounts were 

 employed to secure at Gottingen courses of lectures by Poincar6 and 

 Lorentz. 



What appears as a bad effect of this offered prize is the fact that 

 many people with very meager mathematical training and still less 

 ability are wasting their tune and money by working out and publish- 

 ing supposed proofs. The number of these is aheady nmch beyond 

 1,000, and no one can foresee the extent to which this kind of literatm'e 

 will grow, especially if the complete solution will not be attained 



' Fermat's words are as follows: "Cujus rei demonstratiouein inirabilem saue detexi. Ilanc niarginis 

 exiguitas non caperet." 



