Figurate numbers - Deza, Elena (World Scientific, 2012)
Made available through World Scientific e-Books
Includes bibliographical references (p. 443-447) and index
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
Figurate numbers have a rich history with many applications. The main purpose of this book is to provide a thorough and complete presentation of the theory of figurate numbers, giving much of their properties, facts and theorems with full proofs. This book is the first of this topic written in unified systematic way. It also contains many exercises with solutions