Generalized Pell-Fermat equations and Pascal triangle

TL;DR

This paper generalizes Pell-Fermat equations using Pascal's triangle to analyze distributions related to tetrahedral numbers, providing a novel approach to solving higher-order Pell-Fermat equations with computational tools.

Contribution

It introduces a new generalization of Pell-Fermat equations linked to Pascal's triangle and develops a computational method for finding integer solutions.

Findings

Reduction of median and quartiles to higher-order Pell-Fermat equations

Existence of integer sequences of best approximation for these equations

Use of Mathematica for formal calculations in the absence of a general theory

Abstract

Using Pascal triangle, we give a simple generalization to the so-called STRAND Puzzle solved by Srinivasa Ramanujan. Thus we are interested in computing the median, first and third quartiles of some integer valued distributions, arising naturally when extending partial sums of the arithmetic progression (triangular numbers) to tetrahedral numbers and beyond. We show this reduces to equations of Pell-Fermat type of higher order, which admit very few integer solutions, but for which, following Ramanujan's original idea, we can always find integer sequences of best approximation, in the Diophantine sense. In absence of a general theory on Pell-Fermat equation of higher order, our procedure relies much on formal Calculus with Mathematica.