arXiv subjects
Publications and source records attributed to P. G. Walsh.
A new method for solving quartic equations due to Luo and Lin is investigated both computationally and theoretically. As a result, a completely straightforward elementary method is given for solving Bumby's equation $3X^4-2Y^2=1$, along with a conjecture, which if resolved, would enable a similar proof for a possibly infinite family of similar equations.
We describe how the Mordell-Weil group of rational points on a certain family of elliptic curves give rise to solutions to a conjecture of Erdös on $3$-powerful numbers, and state a related conjecture which can be viewed as an elliptic curve analogue of the Ankeny-Artin-Chowla conjecture.
We examine the ranks of a subfamily of curves in a previous article, which are derived from the existence of solutions to certain Pell equations. We exhibit an abundance of curves of moderately large rank, and prove under mild conditions that these curves have rank at least three.
In this article we further develop methods for representing integers as a sum of three cubes. In particular, a barrier to solving the case $k=3$, which was outlined in a previous paper of the second author, is overcome. A very recent computation indicates that the method is quite favourable to other methods in terms of time estimates. A hybrid of the method presented here and those in a previous paper is currently underway for unsolved cases.