TABLE OP CONTENTS. xxv 



CHAPTER. PAGE. 



Binary cubic form made a square 569 



Numbers the sum of two rational cubes: x 3 +y 3 = Az 3 572 



Sum or difference of two cubes a square 578 



Sum of cubes of numbers in arithmetical progression a cube 582 



Sum of cubes of numbers in arithmetical progression a square 585 



Homogeneous cubic equation F(x, y, z) =0 588 



Ternary cubic form made a constant 593 



Miscellaneous single equations of degree three 595 



Systems of equations of degree three in two unknowns 599 



Systems of equations of degree three in three unknowns 602 



To find n numbers the cube of whose sum increased (or decreased) by any 



one of them gives a cube 607 



Systems of equations of degree 3 in n^.4 unknowns 612 



XXII. EQUATIONS OF DEGREE FOUR 615 



Sum or difference of two biquadrates never a square; area of a rational right 



triangle never a square 615 



2x*y*= D; right triangles whose hypotenuse and sum of legs are squares; 



x 2 +y 2 =B\ x+y=A\ Also, x 4 -2y 4 = D, 2 4 +8w 4 = D 620 



ax*+by* = cz* 627 



ax*+by*+dx 2 y 2 made a square 634 



Quartic function made a square 639 



A*+B* = C*+D* 644 



A*+hB* = C*+hD* 647 



Sum of three biquadrates never a biquadrate 648 



Sum of four or more biquadrates a biquadrate 648 



Equal sums of biquadrates 653 



Relations involving both biquadrates and squares 657 



Miscellaneous single equations of degree four 660 



To find n numbers whose sum is a square and sum of squares is a biquadrate . 665 



Miscellaneous systems of equations of degree four 667 



XXIII. EQUATIONS OF DEGREE n 673 



Solution of f=c, where / is a binary form 673 



Conditions for an infinitude of solutions of f(x, y) = 674 



Rational points on the plane curve f(x, y, z) = 675 



Equations formed from linear functions 677 



Product of consecutive integers not an exact power 679 



Further properties of products of consecutive integers 680 



Sum of nth powers an nth power 682 



Two equal sums of nth powers 684 



Miscellaneous results on sums of like powers 684 



Rational solutions of x v =y x 687 



Product of factors (x+l)/x equal to such a fraction 687 



Optic formula, l/Zi + . . . +l/x n = I/a 688 



Miscellaneous single equations of degree n >4 691 



Miscellaneous systems of equations of degree n>4 700 



XXIV. SETS OF INTEGERS WITH EQUAL SUMS OF LIKE POWERS 705 



An equivalent problem in the theory of logarithms 714 



XXV. WARING'S PROBLEM AND RELATED RESULTS 717 



Waring's problem 717 



Numbers expressible as sums of unlike powers 725 



Every number a sum of three rational cubes 726 



Every positive number a sum of 4 positive rational cubes 727 



XXVI. FERMAT'S LAST THEOREM, a.c r +fy/ s =cz', AND THE CONGRUENCE x n +y n =z n (modp) 731 



AUTHOR INDEX 777 



SUBJECT INDEX . 799 



