# Book:Elena Deza/Figurate Numbers

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## Elena Deza and Michel Marie Deza: *Figurate Numbers*

Published $2012$, **World Scientific Publishing Co. Pte. Ltd.**

- ISBN 978-981-4355-48-3.

### Contents

- Notations
- Preface
- 1. Plane figurate numbers
- 1.1 Definitions and formulas
- 1.2 Main properties of polygonal numbers
- 1.3 Square triangular numbers
- 1.4 Other highly polygonal numbers
- 1.5 Amount of a given number in all polygonal numbers
- 1.6 Centered polygonal numbers
- 1.7 Other plane figurate numbers
- 1.8 Generalized plane figurate numbers

- 2. Space figurate numbers
- 2.1 Pyramidal numbers
- 2.2 Cubic numbers
- 2.3 Octahedral numbers
- 2.4 Other regular polyhedral numbers
- 2.5 Some semiregular and star polyhedral numbers
- 2.6 Centered space figurate numbers
- 2.7 Other space figurate numbers
- 2.8 Generalized space figurate numbers

- 3. Multidimensional figurate numbers
- 3.1 Pentatope numbers and their multidimensional analogues
- 3.2 Biquadratic numbers and their multidimensional analogues
- 3.3 Other regular polytope numbers
- 3.4 Nexus numbers
- 3.5 Pyramidal numbers of the second order and their multidimensional analogues
- 3.6 Centered multidimensional figurate numbers
- 3.7 Generalized multidimensional figurate numbers

- 4. Areas of number theory including figurate numbers
- 4.1 Addition and multiplication tables
- 4.2 Pascal's triangle and binomial theorem
- 4.3 Pythagorean triples and other Diophantine equations
- 4.4 Perfect numbers
- 4.5 Mersenne and Fermat numbers
- 4.6 Fibonacci and Lucas numbers
- 4.7 Palindromic numbers
- 4.8 Other special numbers
- 4.9 Prime numbers
- 4.10 Magic constructions
- 4.11 Unrestricted partitions
- 4.12 Waring's problem

- 5. Fermat's polygonal number theorem
- 5.1 History of the problem
- 5.2 Lagrange's four-square theorem
- 5.3 Gauss's three-triangular-number theorem; elementary considerations
- 5.4 Proof of the Gauss's three-triangular-number theorem
- 5.5 Sums of squares and Minkowski's convex body theorem
- 5.6 Cauchy's proof of the polygonal number theorem
- 5.7 Pepin's proof of the polygonal number theorem
- 5.8 Other results related to the problem

- 6. Zoo of figurate-related numbers
- 7. Exercises
- Bibliography
- Index