xii PREFACE. 



Such an equation x z Ay 2 =l has long borne the name Pellian equation, 

 after John Pell, due to a confusion on the part of Euler; it would have been 

 more appropriately named after Fermat, who stated in 1657 that it has 

 an infinitude of integral solutions if A is any positive integer not a square, 

 and who stated in 1659 that he possessed a proof by indefinite descent. 

 He proposed it as a challenge problem to the English mathematicians 

 Lord Brouncker and John Wallis, who finally succeeded in discovering a 

 tentative method of solution, without giving a proof of the existence of 

 an infinitude of solutions. This theorem is really only the simplest and 

 first known case of Dirichlet 's elegant and very general theorem on the 

 existence of units in any algebraic field or domain. The former theorem 

 is also of great importance in the theory of binary quadratic forms. More- 

 over, the problem to find all the rational solutions of the most general 

 equation of the second degree in two unknowns reduces readily to that for 

 x 2 Ay- = B, all of whose solutions follow from one solution and the solutions 

 of x*-Ay 2 =l. 



In 1765 Euler exhibited the method of solving a Pellian equation due 

 to Brouncker and Wallis in a more convenient form by use of the continued 

 fraction for VZ and found various important facts, but gave no proof that 

 the process leads always to a solution in positive integers. This funda- 

 mental fact of the existence of solutions was first proved by Lagrange a 

 year or two later; while in 1769 and 1770 he brought out his classic 

 memoirs which give a direct method to find all integral solutions of 

 x 2 Ay 2 = B, as well as of an equation of degree n, by developing its real 

 roots into continued fractions. 



Of the further extensive literature on the Pellian equation, the most 

 notable papers are those by Legendre, Gauss, Dirichlet, Jacobi, and Perott; 

 limits for the least positive solution were obtained by Tchebychef in 1851 

 and by Remak, Perron, Schmitz, and Schur in 1913-18. Useful tables have 

 been given by Euler, Legendre, Degen, Tenner, Koenig, Arndt, Cayley, 

 Stern, Seeling, Roberts, Bickmore, Cunningham, and Whitford. 



Chapter XIII treats of further single equations of the second degree, 

 including axy -\-bx-\- cy+d = Q, x 2 y 2 = g, ax 2 -\-bxy +cy z = dz 2 or d, the most 

 general equation of the second degree in x, y, and its homogeneous form 

 aX 2 +bY 2 +cZ z +dXY+eXZ+fYZ = Q. Criteria for integral solutions of 

 the latter were stated by H. J. S. Smith (p. 431) and proved by Meyer for 

 the case of an odd determinant, while its complete solution was given by 

 Desboves (p. 432) when one solution is known. Lagrange's method for 

 x 2 Ay 2 = B, cited above, was employed by Legendre in 1785 to prove the 

 important theorem that, if no two of the integers a, b, c have a common 

 factor and if each is neither zero nor divisible by a square, then 

 ax 2 +&?/ 2 +cz 2 = has integral solutions not all zero if, and only if, be, ac, 

 ab are quadratic residues of a, b, c, respectively, and a, b, c are not all 

 of the same sign. Gauss gave a proof by means of ternary quadratic forms, 

 while a generalization was made by Dirichlet (p. 423) and Goldscheider 

 (p. 426). Meyer gave criteria (pp. 432-3) for integral solutions of /=0, 



