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Visualising connections between types of polygonal number

Published online by Cambridge University Press:  16 February 2023

Umit Sarp
Umit Sarp*
Affiliation:
Distance Education, Application and Research Center, İzmir Kâtip Çelebi University, İzmir, Turkey e-mail: umit.sarp@ikcu.edu.tr & umitsarp@ymail.com
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Extract

Figurate numbers are numbers that can be represented by a regular and discrete geometric pattern of evenly-spaced points. Their study has attracted the attention of many mathematicians and scientists since the dawn of mathematical history, including Pythagoras of Samos (582 BC-507 BC), Diophantus of Alexandria (200/214-284/298), Fibonacci (1170-1250), Pierre de Fermat (1601-1665), Leonhard Euler (1707-1783), Waclaw Franciszek Sierpińshi (1882-1969) [1]. Although it is not one of the basic topics, it serves many fields of mathematics, such as Number Theory and Geometry. Many special numbers are related to figurate numbers. Polynomial values, some theorems and solutions of Diophantine equations are expressed in figurate numbers and studied [2, 3, 4]. Two-dimensional figurate numbers are known as polygonal numbers. In the early 1990's, polygonal numbers were expressed and visualised with the help of computers [5]. Richard K. Guy asked the question ‘Every number is expressible as the sum of how many polygonal numbers?’ [6]. As can be understood from his study, it is seen that most of the studies conducted are about ordinary polygonal numbers. But Euler proved the ‘generalized pentagonal number theorem’ about partitions [7].

Type
Articles
Information
The Mathematical Gazette , Volume 107 , Issue 568 , March 2023 , pp. 56 - 64
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Copyright
© The Authors, 2023. Published by Cambridge University Press on behalf of The Mathematical Association

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References

Deza, E. and Deza, M. M., Figurate numbers, World Scientific Publishing Co., Singapore (2012).CrossRefGoogle Scholar
Hajdu, L. and Varga, N., Polynomial values of figurate numbers, Journal of Number Theory, 214 (2020) pp. 79–99.CrossRefGoogle Scholar
Banerjee, S., Batavia, M., Kane, B., Kyranbay, M., Park, D., Saha, S., So, H.C. and Varyani, P., Fermat’s polygonal number theorem for repeated generalized polygonal numbers, Journal of Number Theory, 220 (2021) pp. 163–181.CrossRefGoogle Scholar
Oh, B. K., Ternary universal sums of generalized pentagonal numbers, Journal of the Korean Mathematical Society, 48(4) (2011) pp. 837–847.CrossRefGoogle Scholar
Hitt, F., Visualization, anchorage, availability and natural image: polygonal numbers in computer environments, International Journal of Mathematical Education in Science and Technology, 25(3) (1994) pp. 447–455.CrossRefGoogle Scholar
Guy, R. K., Every number is expressible as the sum of how many polygonal numbers? Amer. Math. Monthly, 101(2) (1994) pp. 169–172.CrossRefGoogle Scholar
Andrews, G. E., Euler’s pentagonal number theorem, Mathematics Magazine, 56(5) (1983) pp. 279–284.CrossRefGoogle Scholar
The on-line encyclopedia of integer sequences, OEIS Foundation, accessed August 2022 at: http://oeis.org Google Scholar
Caglayan, G., Covering a triangular number with pentagonal numbers, Mathematical Intelligencer 42(55) (2020).CrossRefGoogle Scholar