TABLE OF CONTENTS. 



CHAPTER. PAGE. 



I. POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS 1 



II. LINEAR DIOPHANTINE EQUATIONS AND CONGRUENCES 41 



Solution of ax-\-by = c 41 



Solution of ax = b (mod w) without Fermat's theorem 54 



Solution of ax =b (mod m) by Fermat's or Wilson's theorem 55 



Chinese problem of remainders 57 



Number of positive integral solutions of ax-\-by = n 64 



One linear equation in three unknowns 71 



One linear equation in n>3 unknowns 73 



System of linear equations 77 



One linear congruence in two or more unknowns 86 



System of linear congruences 88 



Linear forms, approximation 93 



III. PARTITIONS 101 



IV. RATIONAL RIGHT TRIANGLES 165 



Methods of solving x 2 +y 2 = z 2 165 



Sides divisible by 3, 4, or 5 171 



Number of right triangles with a given side 172 



Right triangles of equal area 172 



Right triangles whose areas have a given ratio 174 



Other problems involving only area 175 



Problems involving area and other elements 176 



Right triangles whose legs differ by unity 181 



Difference or sum of legs given 183 



Two right triangles with equal differences of legs, and larger leg of one equal 



to hypotenuse of the other 184 



Problems involving the sides, but not the area 186 



Right triangles with a rational angle-bisector 188 



Tables of right triangles with integral sides 189 



V. TRIANGLES, QUADRILATERALS, AND TETRAHEDRA 191 



Rational or Heron triangles 191 



Pairs of rational triangles 201 



Triangles whofee sides and medians are all rational 202 



Triangles with a rational median and rational sides, parallelograms with 



rational sides and diagonals 205 



Heron triangles with a rational median, Heron parallelograms 207 



Triangles with one or more rational angle-bisectors 209 



Triangles with a linear relation between the angles 213 



Triangles whose area need not be rational 214 



Rational quadrilaterals 216 



Rational inscribed polygons 221 



Rational pyramids and trihedral angles 221 



VI. SUM OF Two SQUARES 225 



VII. SUM OF THREE SQUARES 259 



VIII. SUM OF FOUR SQUARES 275 



IX. SUM OF n SQUARES 305 



Representation as a sum of 5 or more squares 305 



Relations between squares 318 



X. NUMBER OF SOLUTIONS OF QUADRATIC CONGRUENCES IN n UNKNOWNS 325 



XI. LIOUVILLE'S SERIES OF EIGHTEEN ARTICLES 329 



XII. PELL EQUATION; ax 2 +bx+c MADE A SQUARE 341 



XIII. FURTHER SINGLE EQUATIONS OF THE SECOND DEGREE 401 



Equations linear in one unknown 401 



Solution of x 2 -y 2 = g 402 



Solution of ax 2 +bxy+cy 2 =dz 2 404 



Solution of ax 2 +by 2 = c 407 



Solution of ax z +bxy+cy 2 = k 408 



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