xviii PREFACE. 



problem with the older one of rapidly converging series convenient for the 

 computation of logarithms, in which we desire two polynomials in x which 

 differ only in their constant terms and have exclusively integers as their 

 roots. 



Chapter XXV furnishes a typical example in the theory of numbers of 

 the contrast between the ease with which empirical theorems are discovered 

 and the difficulty attending a complete mathematical proof. On the basis 

 of numerical experiments, Waring announced hi 1770 the empirical theorem 

 that every positive integer is a sum of at most 9 positive cubes, a sum of 

 at most 19 biquadrates, and in general a sum of a limited number of positive 

 mih powers. The last fact was first proved in 1909 by Hilbert, although 

 his investigation does not determine the precise value of the number N m 

 such that every positive integer is a sum of at most N m positive mth powers. 

 About the year 1772, J. A. Euler stated that N m ^v+2 m 2, where v is 

 the largest integer < (3/2) m . Just before 1859, Liouville proved that 

 Ni^=53 by means of an identity equivalent to 



and the fact that any positive integer n is expressible in the form x\-\-x\ 

 +#3 +#4, so that 6n 2 is a sum of 12 biquadrates. But any positive integer 

 is of one of the six forms 6p, 6p+l, . . ., 6p+5, while p = nl+nl-\-n 2 3 +nl. 

 Thus Qp is a sum of 4X12 biquadrates. Since 1, . . .,5 are sums of as 

 many units, each a biquadrate, we have 7V 4 ^4Xl2+5. Maillet was the 

 first to prove, in 1895, that N 3 is finite, in fact ^21. Later writers suc- 

 ceeded in proving that A^ 3 = 9. In his proof that N m is finite, Hilbert 

 employed a five-fold integral, while later writers have given an algebraic 

 proof. Quite recently, Hardy and Littlewood gave a proof by use of the 

 theory of analytic functions and showed that N m ^=(m 2)2 TO ~ 1 +5, which 

 gives 9 cubes, 21 biquadrates, 53 fifth powers, etc. Earlier papers (pp. 

 726-9) gave elementary proofs that every positive rational number is a 

 sum of three rational cubes and a sum of four positive rational cubes. 



The final chapter devotes 46 pages to reports on more than 300 papers 

 on Fermat's last theorem, which states that it is impossible to separate 

 any power higher than the second into two powers of like degree, and the 

 more general trinomial equation ax r -{-by a = cz t , and congruence of the same 

 form. In letters and in annotations to his copy of Diophantus, Fermat 

 announced many interesting discoveries in the theory of numbers, usually 

 with the statement that he possessed a proof. All of these facts have since 

 been proved with the exception of his "last theorem" above, for which 

 he stated that he had found a truly remarkable proof. If there was an 

 oversight in his proof it was certainly not one of the foolish errors com- 

 mitted in the past decade in the thousands of efforts to secure a large cash 

 prize. Fermat proposed the cases of exponents 3 and 4 (p. 545, pp. 616-7) 

 as challenge problems to the mathematicians of his time. The general 

 case has remained a challenge problem to the mathematicians of the sub- 



