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DYNAMICAL SYSTEMS FOR FIGURATE NUMBERS AND ELLIPTIC SURFACES
Keywords:
figurate number, recurrence, quadrangular number, elliptic surface, triangular number, cubic surfaceDOI:
https://doi.org/10.17654/0972555525004Abstract
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 $g\left(x_0\right)$ obtained by putting $x=x_0$ in a quadratic $g(x)=c x^2+$ $d x+e$. 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.
Received: August 6, 2024
Accepted: November 9, 2024
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