Geometry of Figurate Numbers and Sums of Powers of Consecutive Natural Numbers

Abstract

First, we give a geometric proof of Fermat’s fundamental formula for figurate numbers. Then we use geometrical reasoning to derive weighted identities with figurate numbers and observe some of their applications. Next, we utilize figurate numbers to provide a matrix formulation for the closed forms of the sums ${\sum }_{i=1}^n{i}^p,p=1,2,\dots ,$ thus generating Bernoulli numbers. Finally, we present a formula—motivated by the inclusion-exclusion principle—for ${\sum }_{i=1}^n{i}^p$ as a linear combination of figurate numbers.

• MSC: Primary 11B68
• Secondary 11B83
• 52C22

Acknowledgments

We would like to thank the anonymous referee for careful reading of the manuscript and the suggestions that helped us significantly improve its presentation.

Additional information

Notes on contributors

František Marko

FRANTIŠEK MARKO received his Ph.D. in number theory from the Slovak Academy of Sciences in Bratislava and his second Ph.D. in algebra from Carleton University in Ottawa. He held brief positions at Syracuse University and the University of Minnesota at Duluth before joining the faculty at Pennsylvania State University Hazleton. His research interests are in the areas of number theory, algebra, and representation theory.

Semyon Litvinov

Semyon Litvinov received a Ph.D. in operator algebras in 1987 from the Romanovsky Institute of Mathematics of the Academy of Sciences of Uzbekistan and a Ph.D. in noncommutative ergodic theory in 1999 from North Dakota State University. Before joining the faculty at Pennsylvania State University Hazleton he had taught mathematics at Tashkent State University, North Dakota State University, and Saint Cloud State University. His main research interests lie in the area of functional analysis with emphases in operator algebras and ergodic theory.