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April 2006

**Look-and-Say Biochemistry: Exponential RNA and Multistranded DNA**

by Óscar Martín

oscar.martin@uam.es

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

jasbir@math.byu.edu

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

Jeremy.Gibbons@comlab.ox.ac.uk

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

bennet.vance@dartmouth.edu

Any extended binary tree with n internal nodes has 2*n* edges and *n* + 1 leaves, and the number of such trees (assuming no labels) is given by the *n*th *Catalan number*, which can be written as (2*n*)!/((*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

nilsa@iip.kth.se, jabo@math.su.se, bjornbo@kth.se

**Semiregular Polygons**

by Oleg Mushkarov and Nikolai Nikolov

muskarov@math.bas.bg, nik@math.bas.bg

**Fubinito (Immediately) Implies FTA **

by R. B. Burckel

burckel@math.ksu.edu

**On the Fundamental Theorem of Algebra **

by T. W. Körner

twk@dpmms.cam.ac.uk

**A Topological Menagerie**

by Paul Melvin

pmelvin@brynmawr.edu

**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.

Reviewed by Paul G. Goerss

pgoerss@math.northwestern.edu

**Alfred Tarski: Life and Logic.**

by Anita Burdman Feferman and Solomon Feferman

Reviewed by Carol Wood

cwood@wesleyan.edu

**Gersgorin and His Circles.**

by Richard S. Varga

Reviewed by L. Elsner

elsner@math.uni-bielefeld.de