Rad HAZU, Matematičke znanosti, Vol. 27 (2023), 81-85.

A NOTE ON LOWER BOUNDS FOR RANKS USING PELL EQUATIONS

P. G. Walsh

Abstract.   In this short note, we examine the ranks of a subfamily of curves from a previous paper derived from the existence of solutions to certain Pell equations. We exhibit an abundance of curves of moderately large rank, and using certain well known results from Diophantine analysis, we prove under mild conditions that these curves have rank at least three provided that the constant coefficient of the cubic polynomial defining the curve is sufficiently large.

2020 Mathematics Subject Classification.   11G05, 11D09.

Key words and phrases.   Elliptic curve, rank, Pell equation.

DOI: https://doi.org/10.21857/90836czpwy

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