Table of Contents
Notations ix
Preface xv
1 Plane figurate numbers 1
1.1 Definitions and formulas 1
1.2 Main properties of polygonal numbers 12
1.3 Square triangular numbers 22
1.4 Other highly polygonal numbers 35
1.5 Amount of a given number in all polygonal numbers 45
1.6 Centered polygonal numbers 47
1.7 Other plane figurate numbers 61
1.8 Generalized plane figurate numbers 76
2 Space figurate numbers 87
2.1 Pyramidal numbers 88
2.2 Cubic numbers 99
2.3 Octahedral numbers 105
2.4 Other regular polyhedral numbers 109
2.5 Some semiregular and star polyhedral numbers 115
2.6 Centered space figurate numbers 120
2.7 Other space figurate numbers 138
2.8 Generalized space figurate numbers 145
3 Multidimensional figurate numbers 161
3.1 Pentatope numbers and their multidimensional analogues 162
3.2 Biquadratic numbers and their multidimensional analogues 169
3.3 Other regular polytope numbers 182
3.4 Nexus numbers 204
3.5 Pyramidal numbers of the second order and their multidimensional analogues 210
3.6 Centered multidimensional figurate numbers 219
3.7 Generalized multidimensional figurate numbers 232
4 Areas of number theory including figurate numbers 243
4.1 Addition and multiplication tables 243
4.2 Pascal's triangle and binomial theorem 247
4.3 Pythagorean triples and other Diophantine equations 254
4.4 Perfect numbers 263
4.5 Mersenne and Fermat numbers 267
4.6 Fibonacci and Lucas numbers 273
4.7 Pahndromic numbers 277
4.8 Other special numbers 282
4.9 Prime numbers 286
4.10 Magic constructions 293
4.11 Unrestricted partitions 298
4.12 Waring's problem 303
5 Fermat's polygonal number theorem 313
5.1 History of the problem 313
5.2 Lagrange's four-square theorem 316
5.3 Gauss's three-triangular-number theorem; elementary considerations 319
5.4 Proof of the Gauss's three-triangular-number theorem 323
5.5 Sums of squares and Minkowski's convex body theorem 337
5.6 Cauchy's proof of the polygonal number theorem 349
5.7 Pepin's proof of the polygonal number theorem 360
5.8 Other results related to the problem 370
6 Zoo of figurate-related numbers 379
7 Exercises 389
Bibliography 443
Index 449
