October 2012 Contents
October’s Monthly opens with our annual look at the William Lowell Putnam Mathematics Competition. Our articles highlight Beatty sequences, ancient Indian square roots, means motivated by the Mean Value Theorem, and finiteness theorems for perfect numbers. Our notes feature a geometric proof of Erdös’ inequality for triangles, volume calculations using Cavalieri's Principle, evaluations of Legendre symbols, arithmetic polygons, and an elementary example in the compact-open topology. Mark Schwartz reviews I. Martin Isaacs’ Geometry for College Students, and our Problem Section will keep you busy as usual.
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The Seventy-Second William Lowell Putnam Mathematical CompetitionLeonard F. Klosinski, Gerald L. Alexanderson, and Mark Krusemeyer
A Model for Pairs of Beatty Sequences
Yuval Ginosar and Ilan Yona
Beatty sequences are described by two athletes running in opposite directions on a round track. This approach suggests an interesting interpretation for well-known partitioning criteria; such sequences (eventually) partition the integers essentially when the athletes have the same starting point.
Ancient Indian Square Roots: An Exercise in Forensic Paleo-Mathematics
David H. Bailey and Jonathan M. Borwein
This article examines the computation of square roots in ancient India in the context of the discovery of positional decimal arithmetic.
Looking for a Few Good MeansBruce Ebanks
The mean value theorem of integral calculus states that for any continuous real-valued function f on an interval I , and for any two distinct real numbers a, b I , there exists a value V(a, b) in the open interval between a and b for which . If in addition f is strictly monotonic, then the function V f defined by can be viewed as a two-variable mean on the interval I . We determine which of these means are homogeneous. The preparatory remarks preceding the proof provide a brief introduction to the theory of functional equations. Several questions for further exploration that the theorem suggests are indicated in the final sections.
Finiteness Theorems for Perfect Numbers and Their KinPaul Pollack
Since ancient times, a natural number has been called perfect if it equals the sum of its proper divisors; e.g., 6 = 1 + 2 + 3 is a perfect number. In 1913, Dickson showed that for each fixed k, there are only finitely many odd perfect numbers with at most k distinct prime factors. We show how this result, and many like it, follow from embedding the natural numbers in the supernatural numbers and imposing an appropriate topology on the latter; the notion of sequential compactness plays a starring role.
Vector Analysis Proof of Erdös Inequality for TrianglesAkira Sakurai
We define a new concept, the “1/2-power of plane vectors,” and use it to provide another proof of Erdös inequality for triangles.
Calculation of the Volume of Multidimensional Cones and Balls Using Cavalieri’s PrincipleBoris M. Makarov
Formulas for the volume of multidimensional cones and balls are deduced based on Cavalieri’s principle and basic properties of the Lebesgue measure (without using integration).
Evaluation of Certain Legendre SymbolsDavid Angell
We state and prove an apparently hitherto unrecorded evaluation of certain Legendre symbols: if p is prime, , and , then the Legendre symbol is given by
An Elementary Counterexample in the Compact-Open TopologyJonathan Groves
We give a short proof that the space of continuous functions from [0, 1] to [0, 1] is not compact in the compact-open topology.
Arithmetic PolygonsRobert Dawson
We consider the question of the existence of equiangular polygons with edge lengths in arithmetic progression, and show that they do not exist when the number of sides is a power of two and do exist if it is any other even number. A few results for small odd numbers are given.
PROBLEMS AND SOLUTIONS
REVIEWSGeometry for College Students. By I. Martin Isaacs. American Mathematical Society, The Sally Series, Pure and Applied Undergraduate Texts No. 8, Providence, RI, 2001. vii + 222 pp., ISBN 978-0-8218-4794-7. $62 ($49.60 to AMS members).
Reviewed by Mark Schwartz