Author(s)

Fuju Wu ¹, Deyi Chen ²

Affiliation(s)

¹ Wuhan Britain-China School, Wuhan, Hubei, 430030, China
² Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, 310027, China

Corresponding Author

Fuju Wu

ABSTRACT

In this paper, we have demonstrated that there are infinitely many pentagonal numbers which have two different ways to be decomposed as the product of two non-1 pentagonal numbers, with the domain being in positive rational numbers. This was achieved by transforming an equation into an elliptic curve, identifying a rational point on this curve, and subsequently employing the Nagell-Lutz Theorem to establish the existence of infinitely many rational points on the elliptic curve. Finally, we conjecture that if the domain is restricted to positive integers, then there do not exist such two different decompositions.

KEYWORDS

Pentagonal Numbers; Elliptic curves; Pell's equation

CITE THIS PAPER

Fuju Wu, Deyi Chen, The Decomposition of Pentagonal Numbers. Transactions on Computational and Applied Mathematics (2024) Vol. 4: 113-118. DOI: http://dx.doi.org/10.23977/tracam.2024.040115.

REFERENCES

[1] Euler, On the remarkable properties of the pentagonal numbers. Acta Academia of the Science of the Empire of Petropolitan, 1783, 4(1):56-75. 
[2] Heath, Sir Thomas Little, Diophantus of Alexandria: A study in the history of Greek algebra, Cambridge University Press, 1910:188.
[3] Chen D, Cai T. The decomposition of triangular numbers [J]. The Rocky Mountain Journal of Mathematics, 2013: 75-81.
[4] Mordell L J. Diophantine equations: Diophantine Equations [M]. Academic press, 1969. 
[5] Silverman J H, Tate J T. Rational points on elliptic curves [M]. New York: Springer-Verlag, 1992.

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