We consider Catalan's equation $x^p-y^q=1$ (where all variables are integers and $p,q$ are greater than $1$), which has the obvious solution $9-8=1$. Are there others? Applying old and new theoretical results to a systematic computation, we were able to improve on recent work of Mignotte and show that Catalan's equation has only the obvious solutions when $\min\{p,q\}<10651$. Two crucial tools used are a new result of Laurent, Mignotte, and Nesterenko on linear forms of logarithms, and a criterion obtained by W. Schwarz in 1994.

The soap film problem is to minimize area, and its dual is to maximize the flux of a divergenceless bounded vector field. This paper discretizes the continuous problem and solves it numerically. This gives upper and lower bounds on the area of the globally minimizing film. In favorable cases, the method can be used to discover previously unknown films. No initial assumptions about the topology of the film are needed. The paired calibration or covering space model of soap films is used to enable representation of films with singularities.

We give a brief exposition of results of Bredon and others on passage to fixed points from stable $C_2$ equivariant homotopy (where $C_2$ is the group of order two) and its relation to Mahowald's root invariant. In particular we give Bredon's easy equivariant proof that the root invariant doubles the stem; the conjecture of the title is equivalent to the Mahowald--Ravenel conjecture that the root invariant never more than triples the stem. Our main result is to verify by computation that the algebraic analogue of this holds in an extensive range: this improves on results of [Mahowald and Shick 1983].

The generalized Igusa local zeta function $Z_{\Q _p}(s)$ associated to $(\SL_7,\rho)$, where $\rho$ is the Cartan product of the first, third and fifth fundamental representations of $\SL_7$, is explicitly computed and shown not to satisfy the expected functional equation $$Z_{\Q _p}(s)|_{p \mapsto p^{-1}}=p^{-7s}Z_{\Q _p}(s).$$

Circle packings are configurations of circles with specified patterns of tangency, and lend themselves naturally to computer experimentation and visualization. Maps between them display, with surprising faithfulness, many of the geometric properties associated with classical analytic functions. This paper introduces the fundamentals of an emerging "discrete analytic function theory'' and investigates connections with the classical theory. It then describes several experiments, ranging from investigation of a conjectured discrete Koebe $\sans{\quarter}$ theorem to a multigrid method for computing discrete approximations of classical analytic functions. These experiments were performed using CirclePack, a software package described in the paper and available free of charge.