Edray Herber Goins

On the diophantine equation x 2 + 2 α 5 β 13 γ = y n

In this paper, we find all the solutions of the Diophantineequation x2 + 2α 5β13γ = yn in nonnegative integers x, y, α, β, γ, n ≥ 3with x and y coprime. In fact, for n = 3, 4, 6, 8, 12, we transform the aboveequation into several elliptic equations written in cubic or quartic modelsfor which we determine all their {2, 5, 13}-integer points. For n ≥ 5, weapply a method that uses primitive divisors of Lucas sequences. Againwe are able to obtain several elliptic equations written in cubic modelsfor which we find all their {2, 5, 13}-integer points. All the computationsare done with MAGMA [12].A coin sorting machine in which coins are fed onto the center of a rotating disc having a flexible surface. An annular guide plate is positioned over the disc, being open in the center for receiving coins, and extending outward to the peripheral edge of the disc. The underside of the guide plate is formed with a peripheral stop extending around approximately half of the disc, and a series of discrete radiused guides extend over portions of the other half of the disc. The first of the series of guides would have an outer edge differing in radius from the peripheral stop by the diameter of the smallest coin to be sorted, and each succeeding guide would have a progressively smaller radiused outside edge defined by the difference between coins to be sorted by succeeding guides.

Palindromes in different bases: a conjecture of J. Ernest Wilkins.

Abstract We show that there exist exactly 203 positive integers N such that for some integer d ≥ 2 this number is a d-digit palindrome base 10 as well as a d-digit palindrome for some base b different from 10. To be more precise, such N range from 22 to 9986831781362631871386899.

Lattice Point Visibility on Generalized Lines of Sight

Abstract For a fixed we say that a point (r, s) in the integer lattice is b-visible from the origin if it lies on the graph of a power function f(x) = axb with and no other integer lattice point lies on this curve (i.e., line of sight) between (0, 0) and (r, s). We prove that the proportion of b-visible integer lattice points is given by 1/ζ(b + 1), where ζ(s) denotes the Riemann zeta function. We also show that even though the proportion of b-visible lattice points approaches 1 as b approaches infinity, there exist arbitrarily large rectangular arrays of b-invisible lattice points for any fixed b. This work specialized to b = 1 recovers original results from the classical lattice point visibility setting where the lines of sight are given by linear functions with rational slope through the origin.

Branch-decomposition heuristics for linear matroids

Abstract This paper presents three new heuristics which utilize classification, max-flow, and matroid intersection algorithms respectively to derive near-optimal branch decompositions for linear matroids. In the literature, there are already excellent heuristics for graphs, however, no practical branch decomposition methods for general linear matroids have been addressed yet. Introducing a “measure” which compares the “similarity” of elements of a linear matroid, this work reforms the linear matroid into a similarity graph. Then, the classification method, the max-flow method, and the mat-flow method, all based on the similarity graph, are utilized on the similarity graph to derive separations for a near-optimal branch decomposition. Computational results using the methods on linear matroid instances are shown respectively.

ARITHMETIC PROGRESSIONS ON CONIC SECTIONS

The set {1, 25, 49} is a 3-term collection of integers which forms an arithmetic progression of perfect squares. We view the set {(1, 1), (5, 25), (7, 49)} as a 3-term collection of rational points on the parabola y = x2 whose y-coordinates form an arithmetic progression. In this exposition, we provide a generalization to 3-term arithmetic progressions on arbitrary conic sections with respect to a linear rational map . We explain how this construction is related to rational points on the universal elliptic curve Y2 + 4XY + 4kY = X3 + kX2 classifying those curves possessing a rational 4-torsion point.

The Area of a Surface Generated by Revolving a Graph About any Line

Abstract We discuss a general formula for the area of the surface that is generated by a graph sending revolved around a general line . As a corollary, we obtain a formula for the area of the surface formed by revolving y = f(x) around the line y = mx + k.

POINTS ON HYPERBOLAS AT RATIONAL DISTANCE

Richard Guy asked for the largest set of points which can be placed in the plane so that their pairwise distances are rational numbers. In this article, we consider such a set of rational points restricted to a given hyperbola. To be precise, for rational numbers

Heron Triangles via Elliptic Curves

a

Explicit Descent via 4-Isogeny on an Elliptic Curve

,

Icosahedral Q-curve extensions

b