We consider the problem how long a linear iteration continues to produce figurate numbers. We reduce the problem to considering rational points on a certain elliptic surface. Next, we consider a similar problem given by replacing a quadrangular number with a value g(x₀) obtained by putting x = x₀ in a quadratic   It is reduced to investigate rational points on an analogous algebraic variety. As a result, we show that there are infinitely many triangular numbers that satisfy this condition.

figurate number, recurrence, quadrangular number, elliptic surface, triangular number, cubic surface.

Received: August 6, 2024; Accepted: November 9, 2024; Published: December 5, 2024

How to cite this article: Hizuru Yamagishi, Dynamical systems for figurate numbers and elliptic surfaces, JP Journal of Algebra, Number Theory and Applications 64(1) (2025), 47-79. https://doi.org/10.17654/0972555525004

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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