@article{Booker2021OnAQ,
title={On a question of Mordell},
author={Andrew R. Booker and Andrew V. Sutherland},
journal={Proceedings of the National Academy of Sciences of the United States of America},
year={2021},
volume={118}
}

Proceedings of the National Academy of Sciences of the United States of America

Significance A Diophantine equation is a polynomial equation to which one seeks solutions in integers. There is a notable disparity between the difficulty of stating Diophantine equations and that of solving them. This feature was formalized in the 20th century by Matiyasevich’s negative answer to Hilbert’s tenth problem: It is impossible to tell whether some Diophantine equations have solutions or not. One need not look very far to find examples whose status is unknown. A striking example was… Expand

The search of solutions of the Diophantine equation $x^3 + y^3 + z^3 = k$ for $k<1000$ has been extended with bounds of $|x|$, $|y|$ and $|z|$ up to $10^{15}$. The first solution for $k=74$ is… Expand

Inspired by the Numberphile video "The uncracked problem with 33" by Tim Browning and Brady Haran, we investigate solutions to $x^3+y^3+z^3=k$ for a few small values of $k$. We find the first known… Expand

We determine the shapes of pure cubic fields and show that they fall into two families based on whether the field is wildly or tamely ramified (of Type I or Type II in the sense of Dedekind). We show… Expand

In this paper we point out that the proof of Kable's extension of the Wiener-Ikehara Tauberian theorem can be applied to the case where the Dirichlet series has a pole of order "$l / m$" without much… Expand