PREFACE. xvii 



The first example, 30 4 +120 4 +272 4 +315 4 = 353 4 , was found by Norrie in 

 1911. In the meantime various writers gave examples of five or more 

 biquadrates whose sum is a biquadrate and cases of equal sums of bi- 

 quadrates. 



Chapter XXIII, on equations of degree >4, will doubtless be more 

 useful than any other chapter in the volume since it reports on the papers 

 which offer general methods of attacking Diophantine equations. Lagrange 

 showed how to use continued fractions to solve f=c, where / is a binary 

 form of any degree. Runge and Maillet obtained conditions for the 

 existence of infinitely many pairs of integral solutions of f(x, y) = 0, where 

 / is an irreducible polynomial with integral coefficients. Thue proved the 

 useful theorem that, if U(x, y) is an irreducible homogeneous polynomial 

 of degree >2 with integral coefficients and c is a given constant, U = c 

 has only a finite number of pairs of integral solutions. Maillet gave a 

 generalization (p. 675) to non-homogeneous polynomials U. 



Hilbert and Hurwitz, in their joint paper of 1890-1, proved that any 

 homogeneous equation with integral coefficients which represents a curve 

 of genus zero can be transformed birationally into a linear or quadratic 

 equation. Poincare in 1901 proved the same theorem and found when 

 a curve of genus unity can be transformed birationally into a curve of 

 order p. The related later papers are cited on p. 677. 



It is convenient to define at this point the product 



F(x, y, . . ., z} = U(x+ay+ . . . +a n ~ 1 z) J 



extended over all the roots a, ... of any irreducible equation of degree n 

 with integral coefficients, to be the norm of the general number x -\-ay-\- . . . 

 of the algebraic field determined by a. Dirichlet noted that F 1 has 

 infinitude of integral solutions except when the field is an imaginary quad- 

 ratic field. If the field is real and if F can take a given value, it takes that 

 value for an infinitude of sets of integers x, . . ., z. Also Poincare (p. 678) 

 discussed this problem F = g. Lagrange (pp. 570, 691) proved in effect 

 that the norm of a product equals the product of the norms of the factors 

 and hence solved F(X, Y, . . ., Z] = V m , where V = F(x, y, . . .,2). This 

 method is of considerable power in seeking special solutions of various 

 types of equations. The particular case x 3 -\-ny z -\-n 2 z 3 3nxyz occurs in 

 the papers on pp. 593-5. This case is also a special case of another type 

 of equations of general degree obtained by Maillet from the theory of 

 recurring series (p. 695). A. Hurwitz's complete discussion (p. 697) of 

 the positive integral solutions of x\ + . . . -\-x z n = xx\ . . . x n furnishes a 

 model for thoroughness which may well be imitated by writers on Diophan- 

 tine equations, too many of whom seem to be content with a special solution 

 of their problems. 



Chapter XXIV deals with sets of integers with equal sums of like 

 powers. For example, a, b, c and a+b+c have the same sum and same 

 sum of squares as a +6, a+c, b+c. Of the seventy papers on this topic, 

 only five are prior to 1878. On pp. 714-6 is noted the connection of this 



