*The author presents three solutions to a problem concerning...*

We start the New Year out with a bang at the *Monthly*! In our lead articles for January, you will learn the connection between solutions of Diophantine equations and the decompositions of modules, how cake cutting may not be perfect among three or more persons, why Euler’s constant is most likely transcendental, and an interesting containment relation involving $$L^{p}$$ spaces. Our Notes section features a look at an integral identity found by M. L. Glasser, a proof that Zeilberger missed, what happens when Mamikon meets Kepler, and a simple characterization of differentiation. Fernando Gouvêa reviews *Elliptic Curves, Modular Forms, and Their L-functions* by Álvaro Lozano-Robledo. Finally, don’t forget our world-famous Problem Section. —*Scott Chapman*

Vol. 120, No. 1, pp.3-98.

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Nicholas R. Baeth and Roger Wiegand

Let $$R$$ be a commutative ring with identity. It often happens that $$M{1}\bigoplus\cdots\bigoplus M_{s}\cong N_{1}\bigoplus\cdots\bigoplus N_{t}$$ for indecomposable $$R$$-modules $$M_{1},\dots,M_{s}$$ and $$N_{1},\dots,N_{t}$$ with $$s\neq t$$. This behavior can be captured by studying the commutative monoid {[$$M$$]|$$M$$ is an $$R$$-module} of isomorphism classes of $$R$$-modules with operation given by $$[M]+[N]=[M\bigoplus N]$$. In this mostly self-contained exposition, we introduce the reader to the interplay between the study of direct-sum decompositions of modules and the study of factorizations in integral domains.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.01.003

Steven J. Brams, Michael A. Jones, and Christian Klamler

A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envy-free, and equitable. We give an example of a cake that is impossible to divide among three players, so that these three properties are satisfied, however many (finite) cuts are made. It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which is not envy-free, is more desirable (a 2-cut division can at best satisfy either envy-freeness or equitability, but not both). We prove that no perfect division exists for more than 4 cuts and for an extension of this example to more than three players.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.01.035

M. Ram Murty and Anastasia Zaytseva

We consider a class of analogues of Euler’s constant $$\gamma$$ and use Baker’s theory of linear forms in logarithms to study its arithmetic properties. In particular, we show that with at most one exception, all of these analogues are transcendental.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.01.048

Jean-Baptiste Hiriart-Urruty and Patrice Lassère

We prove a necessary and sufficient condition on the exponents $$p$$, $$q$$, $$r\geq1$$ such that $$L^{r}(\mathbb{R})\subset L^{p}(\mathbb{R})+L^{q}(\mathbb{R})$$. In doing so, we explore the structure of $$L^{p}(\mathbb{R})+L^{q}(\mathbb{R})$$ as a normed vector space.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.01.055

Vinicius Nicolae Petre Anghel

The integral identity found by M. L. Glasser [3] is generalized using the permutation symmetry of coordinates of an $$n$$-spherical surface simplex. The first calculation technique is simple to apply, but the second technique allows further generalization of M. L. Glasser’s identity. Analogous results are discussed for the $$n$$-hemispherical surface of the unit $$n$$-sphere and for the entire surface of the $$n$$-sphere. The $$n$$-sphere surface result is used to generalize M. L. Glasser’s solution to a problem proposed by J. R. Bottiger [2].

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.01.062

Yi Jun Chen

In this paper, a succinct new proof of an identity by Chaundy and Bullard is given, based on the Wilf-Zeilberger theory.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.01.069

L. P. Withers, Jr.

A new areal proof of Kepler’s second law of planetary motion is presented, based on Mamikon’s sweeping-tangent theorem.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.01.071

Wlodzimierz Bak

In this short note we will give a result that can be treated as a characterization of differentiation.

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Problems 11684-11690

Solutions 11540, 11558, 11559, 11560, 11561, 11564, 11565, 11571

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.01.076

*Elliptic Curves, Modular Forms, and Their L-functions* (Student Mathematical Library IAS/Park City Subseries, vol. 58), by Alvaro Lozano-Robledo. American Mathematical Society, Providence, RI, 2011, 195 pp., ISBN 978-0-8218-5242-2, $29.60. Reviewed by Fernando Q. Gouvêa

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.01.084