Why Dickson Left Quadratic Reciprocity out of his History of the Theory of Numbers
by Della Dumbaugh Fenster
In the closing years of the nineteenth century, Leonard Dickson began what would ultimately grow into a distinguished forty-two-year long career in mathematics. He spent all but the first two of these years on the faculty at the University of Chicago, where he pursued research interests in group theory, finite field theory, invariant theory, the theory of algebras, and number theory. He, however, interrupted his thriving pure mathematical investigations for over a decade to write his monumental History of the Theory of Numbers. And, yet, in this widely recognized compendium of the subject, he omitted almost any mention of quadratic reciprocity, the "crown jewel of elementary number theory". This paper explores this perplexing omission.
Uniform Calculus and the Law of Bounded Change
by Mark Bridger and Gabriel Stolzenberg
In this article, we show the simplicity and power of a theory of the calculus based on uniform continuity and differentiability on compact intervals. We employ Caratheodory's elegant definition of the derivative, which leads to an interestingly different proof of the fundamental theorem of the calculus.
In the first part of the paper, we show that, for the functions of the calculus, uniform continuity and differentiability on compact intervals are as easy to verify as their pointwise counterparts. We work out their laws of arithmetic (sum, product, etc.) on an arbitrary domain, subject only to certain boundedness assumptions that are always satisfied on a compact interval.
In the second half of the article, we focus on the law of bounded change, our choice for the role that is traditionally played by the mean value theorem. It reads, "Bounds for the derivative are bounds for the difference quotient." (Have your students chant this.) Our simple, elementary proof (no appeal to completeness) is adapted from a familiar argument in which the uniform continuity of the derivative of f is used to approximate f(b) - f(a) by Riemann sums for the integral of this derivative.
No doubt the most important application of the law of bounded change (or the mean value theorem) for the calculus is that, on an interval, a function with 0 derivative is constant. Is there any simpler way of proving this deceptively obvious-looking assertion?
We show that the law of bounded change serves well in place of the mean value theorem in proofs of the two cases of L'Hopital's Rule and differentiation under the integral sign.
Finally, although the mean value theorem fails even for functions from an interval to the plane, our proof of the law of bounded change generalizes to higher dimensions almost without change, provided only that we take convex sets to be our higher-dimensional counterparts to intervals.
The Answer is 2n n! What's the Question?
by Gary Gordon
How many different questions can you find that have 2n n! as the answer? Working backwards this way (motivated by the game show Jeopardy!), four fictitious mathematicians find their own questions. Allison finds an algebraic approach, Cory likes combinatorics, Heidi uses hyperplanes, and Zach knows about zonotopes. Along the way, we meet the group of symmetries of a hypercube and get a combinatorial feel for its structure.
Conditional Convergence of Infinite Products
by William F. Trench
There are a few results in the textbook literature that imply conditional convergence of infinite products of complex numbers. The paper presents a hierarchy of tests for conditional convergence, with examples showing that they have nontrivial applications.
H. J. S. Smith and the Fermat Two Squares Theorem
by F. W. Clarke, W. N. Everitt, L. L. Littlejohn, and S. J. R. Vorster
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The two squares theorem of Fermat gives a representation of a prime congruent to 1 modulo 4, as the sum of two integer squares. Fermat (1659) is credited with the first proof of this result, but the first recorded proof is due to Euler (1749). Gauss (1801) showed that the two squares representation is essentially unique. In 1855 the Oxford mathematician Henry Smith gave an elementary proof involving the use of continuants. This paper discusses the Smith proof and shows how his method can be extended to give uniqueness. There is a brief account of the life and achievements of Henry Smith, recently called "the mathematician the world forgot".
Number Theory and Semigroups of Intermediate Growth
by Melvyn B. Nathanson
The Gottschalk-Hedlund Theorem
by Randall McCutcheon
A Mean Value Theorem
by Tadashi F. Tokieda
On an Example of Jacobson
by B. Sury
THE EVOLUTION OF...
Field Theory: From Equations to Axiomatization. Part I
by Israel Kleiner
PROBLEMS AND SOLUTIONS
My Brain Is Open: The Mathematical Journeys of Paul Erdös
By Bruce Schechter
The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth
By Paul Hoffman
Reviewed by Albert A. Mullin