On the equation $Y^2 = X^6 + k$

TL;DR

This paper explicitly finds all rational solutions to the equation Y^2 = X^6 + k for integers |k| ≤ 50, using various methods from elementary to advanced algebraic techniques, highlighting the complexity and challenges in solving such equations.

Contribution

The paper provides a comprehensive solution for most cases within a range of k, demonstrating the limitations of current methods for certain challenging values.

Findings

Explicit solutions for most k within the range.

Identification of six challenging cases resistant to current methods.

Demonstration that no universal solution method exists for all k.

Abstract

We find explicitly all rational solutions of the title equation for all integers $k$ in the range $|k|\leq 50$ except for $k=-47,-39$. For the solution, a variety of methods is applied, which, depending on $k$, may range from elementary, such as divisibility and congruence considerations, to elliptic Chabauty techniques and highly technical computations in algebraic number fields, or a combination thereof. For certain sets of values of $k$ we can propose a more or less uniform method of solution, which might be applied successfully for quite a number of cases of $k$, even beyond the above range. It turns out, however, that in the range considered, six really challenging cases have to be dealt with individually, namely $k = 15,43,-11,-15,-39,-47$. More than half of the paper is devoted to the solution of the title equation for the first four of these values. For the last two values the solution of the equation, at present, has resisted all our efforts. The case with these six values of $k$ shows that one cannot expect a general method of solution which could be applied, even in principle, for {\em every} value of $k$. A summary of our results is shown at the end of the paper.