# American Mathematical Monthly -April 2006

## April 2006

Look-and-Say Biochemistry: Exponential RNA and Multistranded DNA
by Óscar Martín
[email protected]
1, 11, 21, 1211, 111221, 312211... Two decades ago, J. H. Conway studied this singular sequence of integers, the "look-and-say" sequence, and exposed its wonderful properties. He also showed some formal similarities between the behavior of the sequence and that of chemical elements. Since then, few contributions to the subject have appeared. We show how it is possible to bring the metaphor a step forward to produce infinite, self-descriptive "molecules" that in some sense formally resemble the macromolecules of real-world biochemistry. Our list of such includes RNA, DNA, double and multiple helixes, self-replication...

Congruent Numbers and Elliptic Curves
by Jasbir S. Chahal
[email protected]
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
[email protected]
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
[email protected]
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.

Notes

Plane Intersections of Rotational Ellipsoids
by Nils Abramson, Jan Boman, and Björn Bonnevier
[email protected], [email protected], [email protected]

Semiregular Polygons
by Oleg Mushkarov and Nikolai Nikolov
[email protected], [email protected]

Fubinito (Immediately) Implies FTA
by R. B. Burckel
[email protected]

On the Fundamental Theorem of Algebra
by T. W. Körner
[email protected]

A Topological Menagerie
by Paul Melvin
[email protected]

Evolution ofÂ…

Mathesis Perennis: Mathematics in Ancient, Renaissance, and Modern Times
by Eberhard Knobloch

Problems and Solutions

Reviews

Topology: A Geometric Approach.
by Terry Lawson.
[email protected]

Alfred Tarski: Life and Logic.
by Anita Burdman Feferman and Solomon Feferman
Reviewed by Carol Wood
[email protected]

Gersgorin and His Circles.
by Richard S. Varga
Reviewed by L. Elsner
[email protected]