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American Mathematical Monthly - October 2005

A Pascal-Type Triangle Characterizing Twin Primes
by Karl Dilcher and Kenneth B. Stolarsky
dilcher@mathstat.dal.ca, stolarsk@math.uiuc.edu
It is a well-known property of Pascal's triangle that the entries of the kth row, without the initial and final entries 1, are all divisible by k if and only if k is prime. This paper presents a triangular array, analogous to Pascal's, that characterizes twin prime pairs in a similar fashion. The proof involves generating function techniques. Connections with orthogonal polynomials, in particular Chebyshev and ultraspherical polynomials, are also discussed.

 

Geometry and Number Theory on Clovers
by David A. Cox and Jerry Shurman
dac@cs.amherst.edu, jerry@reed.edu
Gauss characterized the positive integers n for which the circle can be divided into n arcs of equal length by straightedge and compass, Abel solved the corresponding problem for the lemniscate, and Pierpont solved the circle problem for origami constructions. This paper places these problems in the context of a family of curves called m-clovers. The first four m-clovers are the cardioid, the circle, a three-leaf clover that may be new, and the lemniscate. Arc length on m-clovers is given by a family of integrals, the first two elementary, the next two elliptic, and the rest hyperelliptic. Basic methods show that the cardioid can be divided into any number n of equal arcs by straightedge and compass. On the other hand, our solution of the origami division problem on the three-clover and lemniscate uses Galois theory, elliptic functions, complex multiplication, and class field theory. The resulting values of n are characterized in terms of Pierpont primes, namely, those primes greater than three of the form 1 plus a power of 2 times a power of 3.

 

A Nonstandard Proof of the Fundamental Theorem of Algebra
by George Leibman
gleibman@acedsl.com
The "Fundamental Theorem of Algebra" states that the field of complex numbers is algebraically closed—every polynomial of a positive degree with complex coefficients has a complex zero. There are many proofs of this result that use (alone and in various combinations) Galois theory, topology, complex analysis, linear algebra, and nonstandard analysis. This article presents a proof that relies upon a combination of topology and nonstandard analysis. The article is self-contained, including a brief introduction to nonstandard analysis and a review of the necessary concepts from topology that are used in the proof. The article demonstrates, by sketching a related standard proof, how nonstandard analysis helps to expose new "standard" proofs of classical results, often containing fresh intuitive insights about old ideas.

 

The Sixty-Fifth William Lowell Putnam Mathematical Competition
by Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson

 

Notes

The Stone-Weierstrass Theorem Revisited
by N. V. Rao
rnagise@math.utoledo.edu

A Faster Product for π and a New Integral for ln π/2
by Jonathan Sondow
jsondow@alumni.princeton.edu

Generalizations of Fermat’s Little Theorem via Group Theory
by I. M. Isaacs and M. R. Pournaki
isaacs@math.wisc.edu, pournaki@ipm.ir

Convexity and Minkowski’s Inequality
by Geoffrey Brown
gwbrown@mast.queensu.ca

Evolution ofÂ…
The Analytic Principle of Continuity

by B. A. Rosenfeld

Problems and Solutions

Reviews

From Newton to Hawking: A History of Cambridge University’s Lucasian Professors of Mathematics Edited by Kevin C. Knox and Richard Noakes Reviewed by Judith Grabiner jgrabine@pitzer.edu

 

Dummy View - NOT TO BE DELETED