(2020). The Eightieth William Lowell Putnam Mathematical Competition. The American Mathematical Monthly: Vol. 127, No. 8, pp. 675-687.

A beautiful theorem of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse generalizing a classic problem about circles. We give a brief history of the circle problem an account of Price’s ellipse proof and a reorganized proof with some new ideas designed to situate the result within a dense web of connections to classical mathematics. It is inspired

In Conway’s game, Sylver Coinage, the set of legal plays forms the complement of a numerical semigroup after a finite number of turns. Our goal is to show how the tools and techniques of numerical semigroups can be brought to bear on questions related to Sylver Coinage. We begin by formally connecting the definitions and concepts related to the game of Sylver Coinage with those of numerical semigroups

(2020). Wald’s Identity and Geometric Expectation. The American Mathematical Monthly: Vol. 127, No. 8, pp. 716-716.

Zero divided by zero is arguably the single most important concept underlying calculus. For functions of more than one variable, methods of proof for indeterminate limits are not as familiar as for functions of a single variable. We present a l’Hôpital’s rule that provides a way to simplify and resolve a wide variety of zero-over-zero limits in terms of quotients of their derivatives.

We solve a very classical problem: providing a description of the geometry of a Euclidean tetrahedron from the initial data of the areas of the faces and the areas of the medial parallelograms of Yetter, or equivalently of the pseudofaces of McConnell. In particular, we derive expressions for the dihedral angles, face angles, and (an) edge length, the remaining parts being derivable by symmetry or

We give a simple proof of the Siebeck–Marden theorem. We use only basic facts of geometry and complex numbers.

We provide an elementary derivation of the Green function for Poisson’s equation with Neumann boundary data on balls of arbitrary dimension. Surprisingly, until very recently this Green function was only known in dimensions up to three, and an explicit construction (even in low dimensions) on the level of an undergraduate PDE class was lacking. This changes if one derives the Green function for Poisson’s

(2020). Null Subsets of All Sizes Inside Vitali Sets. The American Mathematical Monthly: Vol. 127, No. 8, pp. 743-743.

A numerical semigroup S is an additive subsemigroup of the nonnegative integers, containing zero, with finite complement. Its multiplicity m is its smallest nonzero element. The Apéry set of S is the set of elements Ap(S)={n∈S:n−m∉S}. Fixing a numerical semigroup, we ask how many elements of its Apéry set have nonunique factorization and define several new invariants.

Using the group consisting of the eight Möbius transformations x, – x, 1/x,−1/x, (x−1)/(x+1),(x+1)/(1−x), (x+1)/(x−1), and (1−x)/(x+1), we present an enumerative proof of the classical result for when the element 2 is a quadratic residue in the finite field Fq .

Proposed problems should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed solutions to the problems below should be submitted by February 28, 2021, via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions

(2020). A Formula for the Stirling Numbers of the Second Kind. The American Mathematical Monthly: Vol. 127, No. 8, pp. 762-762.

(2020). Discrete Morse Theory by Nicholas Scoville. The American Mathematical Monthly: Vol. 127, No. 8, pp. 763-768.

We say that a simple, closed curve γ in the plane has bounded convex curvature if for every point x on γ, there is an open unit disk Ux and εx>0 such that x∈∂Ux and Bεx(x)∩Ux⊂Int γ. We prove that the interior of every curve of bounded convex curvature contains an open unit disk.

In the chapter “Magic with a Matrix” in Hexaflexagons and Other Mathematical Diversions (1988); Martin Gardner describes a delightful “party trick” to fill the squares of a d-by-d chessboard with nonnegative integers such that the sum of the numbers covered by any placement of d nonthreatening rooks is a given number N. We consider such chessboards from a geometric perspective which gives rise to a

A proof is given, for a large class of polygons, of a conjecture of Shier concerning his algorithm for the disjoint but otherwise random placement of successively smaller copies of an arbitrary shape within a bounded region whose area is the infinite sum of all the shape areas. If the shapes decrease successively in size according to a power law with an exponent falling in a specified range of values

We examine the aggregate behavior of one-dimensional random walks in a model known as (one-dimensional) Internal Diffusion Limited Aggregation. In this model, a sequence of n particles perform random walks on the integers, beginning at the origin. Each particle walks until it reaches an unoccupied site, at which point it occupies that site and the next particle begins its walk. After all walks are

In 1934, B. Berggren first discovered the surprising result that every Pythagorean triangle is the pre-product of the triangle (3, 4, 5) given as a column vector by a product of three matrices, and that every triangle is obtained in this manner exactly once and in primitive form. In this article, we show a similar result for integer triangles with an angle of 60 and 120 degrees (also known as Eisensteinian

(2020). A Short Proof of Fermat’s Two-square Theorem. The American Mathematical Monthly: Vol. 127, No. 7, pp. 638-638.

Mills proved that there exists a real constant A > 1 such that for all n∈N the values ⌊A3n⌋ are prime numbers. No explicit value of A is known, but assuming the Riemann hypothesis one can choose A=1.3063778838…. Here we give a first unconditional variant: ⌊A1010n⌋ is prime, where A=1.00536773279814724017… can be computed to millions of digits. Similarly, ⌊A313n⌋ is prime, with A=3.8249998073439146171615551375…

This note gives a short, direct proof of the uniform smoothness of Lp spaces with 2

We offer an alternative to the so-called Weierstrass substitution for a class of trigonometric integrals, with references to Euler, Hardy, and others.

After reviewing the history of the crank of an integer partition, involving work of Freeman Dyson from 1944 and Frank Garvan and George Andrews in 1988, we connect this concept with more recent work of Andrews based on the smallest missing part of a partition. This alternative view of partitions having certain crank values allows us to strengthen a recent result of Yuefei Shen with a greatly simplified

(2020). The Paul R. Halmos–Lester R. Ford Awards for 2019. The American Mathematical Monthly: Vol. 127, No. 7, pp. 657-657.

(2020). Components Used in the Stone–Weierstrass Theorem. The American Mathematical Monthly: Vol. 127, No. 7, pp. 658-658.

Proposed problems should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed solutions to the problems below should be submitted by December 31, 2020, via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions

(2020). Reviews. The American Mathematical Monthly: Vol. 127, No. 7, pp. 668-671.

(2020). The Mathematical Association of America Prime. The American Mathematical Monthly: Vol. 127, No. 7, pp. 672-672.

We revisit the nineteenth-century version of the bounded convergence theorem formulated by C. Arzelà in 1885 for Riemann integrable functions and, independently, by W. F. Osgood in 1897 for continuous functions.

We review some basic results of convex analysis and geometry in Rn in the context of formulating a differential equation to track the distance between an observer flying outside a convex set K and K itself.

(2020). Portrait of the Mathematician as a First-Year Undergrad. The American Mathematical Monthly: Vol. 127, No. 6, pp. 518-518.

In this article, we study several probabilistic properties of polynomials defined over the ring of p-adic integers under the Haar measure. First, we calculate the probability that a monic polynomial is separable, generalizing a result of Polak. Second, we introduce the notion of two polynomials being strongly coprime and calculate the probability of two monic polynomials being strongly coprime. Finally

We show how the determinant of a matrix can be calculated using complex analysis of one variable. We also describe the connection between unitary matrices and the problem of holomorphic isometries in complex geometry.

To make a marked point on the boundary of a circular wheel of radius 1 trace the curve known as a cycloid, let this wheel roll along an arbitrary fixed curve called the base curve. There are two cycloids, each of which is drawn on one side of the base curve and share cusps with each other. Consider the two arcs joining the same pair of adjacent cusps, one arc from each cycloid. T. M. Apostol and M

(2020). Paying Our Way Out of a Crisis. The American Mathematical Monthly: Vol. 127, No. 6, pp. 544-544.

Many proofs are known for the inequality between the arithmetic mean and the geometric mean. This note gives a new derivation, interpreting the means as final and initial values of entropy, and the inequality as the second law of thermodynamics.

Homeomorphisms are one of the main objects of study in topology and, hence, it is interesting to provide conditions that ensure that a map between topological spaces is a homeomorphism. Some examples of these kinds of results are the invariance of domain theorem (IDT) or the well-known lemma that states that a continuous bijection from a compact topological space to a Hausdorff topological space is

(2020). Portrait of the Mathematician as a New Mother. The American Mathematical Monthly: Vol. 127, No. 6, pp. 553-553.

Although no known closed-form expression exists for the Nth partial sum of the p-series, we show that a closed-form expression can at least be found for the integer part of these partial sums, where 0

The elementary three distance theorem (Steinhaus conjecture) is specified so that the two or at most three distances are concretely represented by the marginal gaps and the gap that covers 1−α where α is the fractional part of a given irrational number. It is interesting to see that the orderly structure of the fractional parts of the first n multiples of an irrational number is completely represented

Proposed problems should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed solutions to the problems below should be submitted by October 31, 2020, via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions

(2020). A Water-Based Proof of the Cauchy–Schwarz Inequality. The American Mathematical Monthly: Vol. 127, No. 6, pp. 572-572.

(2020). Reviews. The American Mathematical Monthly: Vol. 127, No. 6, pp. 573-576.

This note is devoted to a short, but elementary, proof of Hadamard's determinant inequality.

In a recent issue of this journal, Mordukhovich, Nam, and Salinas pose and solve an interesting non-differentiable generalization of the Heron problem in the framework of modern convex analysis. In the generalized Heron problem, one is given k + 1 closed convex sets in ℝ d equipped with its Euclidean norm and asked to find the point in the last set such that the sum of the distances to the first k