Congruent Numbers and Elliptic Curves
by Jasbir S. Chahal
A positive rational number is a congruent number if it is the area of a rational right triangle. How to determine whether or not a given positive rational number is a congruent number is still an open question. It is obvious that we need to study the problem only for square-free positive integers. There are many conjectures on this topic. For example, a conjecture attributed to folklore asserts that if a square-free positive integer a is congruent to 5, 6, or 7 modulo 8, then a is a congruent number. One of the goals of this article is to prove that the converse of this conjecture is as false as it can get.
Unbounded Spigot Algorithms for the Digits of Pi
by Jeremy Gibbons
Rabinowitz and Wagon (in the April 1995 issue of this MONTHLY) present a spigot algorithm for computing the digits of π. A spigot algorithm yields its outputs incrementally and does not reuse them after producing them. Rabinowitz and Wagon's algorithm is inherently bounded: it requires a commitment in advance to the number of digits to be computed. We propose some streaming algorithms based on the same and some similar characterizations of π, with the same incremental characteristics, but without requiring the prior bound.
Counting Ordered Trees by Permuting Their Parts
by Bennet Vance
Any extended binary tree with n internal nodes has 2n edges and n + 1 leaves, and the number of such trees (assuming no labels) is given by the nth Catalan number, which can be written as (2n)!/((n + 1)!n!). That being so, can the factorials in the Catalan formula be seen as counting permutations of tree parts? We show that they can, and that this approach to interpreting the Catalan numbers also applies to more general counts of ordered trees and forests.
NotesPlane Intersections of Rotational Ellipsoids
by Oleg Mushkarov and Nikolai Nikolov
Fubinito (Immediately) Implies FTA
by R. B. Burckel
On the Fundamental Theorem of Algebra
by T. W. Körner
A Topological Menagerie
by Paul Melvin
Mathesis Perennis: Mathematics in Ancient, Renaissance, and Modern Times
by Eberhard Knobloch
Problems and Solutions
Topology: A Geometric Approach.
by Terry Lawson.
Reviewed by Paul G. Goerss
Alfred Tarski: Life and Logic.
by Anita Burdman Feferman and Solomon Feferman
Reviewed by Carol Wood
Gersgorin and His Circles.
by Richard S. Varga
Reviewed by L. Elsner