A note on class number one criteria of Sirola for real quadratic fields

In a recent paper, Sirola gives two necessary and sufficient conditions for the class number of a real quadratic field to be equal to one. The purpose of this note is to remark that the equivalence of these conditions can be proved by using an elementary result of Nagell, which itself is a simple consequence of the fact that the Pell equation X2 - dY2 = 1 always has solutions in positive integers when d > 1 is squarefree

A question of Erd\"{o}s on 33-powerful numbers and an elliptic curve analogue of the Ankeny-Artin-Chowla conjecture

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 Erd\"{o}s on $3$-powerful numbers, and state a related conjecture which can be viewed as an elliptic curve analogue of the Ankeny-Artin-Chowla conjecture

New tools in comparative political economy: The database of political institutions.

[Dataset available: http://hdl.handle.net/10411/15987]

Lower bounds for ranks using Pell equations

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.Comment: 4 page

On a family of quatric equations and a Diophantine problem of Martin Gardner

Wilhelm Ljunggren proved many fundamental theorems on equations of the form aX^2 - bY^4 = δ, where δ ∈ {±1, 2, ±4}. Recently, these results have been improved using a number of methods. Remarkably, the equation aX^2 - bY^4 = -2 remains elusive, as there have been no results in the literature which are comparable to results proved for the other values of δ. In this paper we give a sharp estimate for the number of integer solutions in the particular case that a = 1 and b is of a certain form. As a consequence of this result, we give an elementary solution to a Diophantine problem due to Martin Gardner which was previously solved by Charles Grinstead using Baker\u27s theory

Representing integers as a sum of three cubes

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.Comment: 4 page

Toward autonomous spacecraft

Ways in which autonomous behavior of spacecraft can be extended to treat situations wherein a closed loop control by a human may not be appropriate or even possible are explored. Predictive models that minimize mean least squared error and arbitrary cost functions are discussed. A methodology for extracting cyclic components for an arbitrary environment with respect to usual and arbitrary criteria is developed. An approach to prediction and control based on evolutionary programming is outlined. A computer program capable of predicting time series is presented. A design of a control system for a robotic dense with partially unknown physical properties is presented

The 2s atomic level in muonic 208-Pb

Relative intensities and energy measurements of 2s level in muonic Pb-20