The 10 most-cited articles in American Mathematical Monthly, 2017

How can the deeply human themes that drive us to do mathematics be channeled to build a more beautiful and just world in which all can truly flourish?

A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is called the genus of the semigroup. How many numerical semigroups have genus equal to g? We outline Zhai's proof of a conjecture of Bras-Amorós that this sequence has Fibonacci-like growth. We now know that this sequence asymptotically grows as fast as the Fibonacci numbers, but it is still not known whether it is nondecreasing. We discuss this and other open problems. We highlight the many contributions made by undergraduates to problems in this area.

This survey concerns regular graphs that are extremal with respect to the number of independent sets and, more generally, graph homomorphisms. More precisely, in the family of of d-regular graphs, which graph G maximizes/minimizes the quantity i(G)1/v(G), the number of independent sets in G normalized exponentially by the size of G? What if i (G) is replaced by some other graph parameter? We review existing techniques, highlight some exciting recent developments, and discuss open problems and conjectures for future research.

The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence α, 2α,.…., Nα, for any integer N and real number α. This statement was proved in the 1950s independently by various authors. Here we present a different approach using the space of two-dimensional Euclidean lattices.

Recently, William Y.C. Chen, Qing-Hu Hou, and Doron Zeilberger developed an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequences, namely those (like the Catalan and Motzkin sequences) that are expressible in terms of constant terms of powers of Laurent polynomials. We first give a leisurely exposition of their approach and then extend it in two directions. The Laurent polynomials may be of several variables, and instead of single sums we have multiple sums. In fact, we even combine these two generalizations. We conclude with some super-challenges.

We recast Euclid's proof of the infinitude of prime numbers as a Euclidean Criterion for a domain to have infinitely many atoms. We make connections with Furstenberg's "topological" proof of the infinitude of prime numbers and show that our criterion applies even in certain domains in which not all nonzero nonunits factor into products of irreducibles.

The purpose of this paper is to give the flavor of the subject of self-similar tilings in a relatively elementary setting and to provide a novel method for the construction of such polygonal tilings.

The power sum 1n + 2n +… + xn has been of interest to mathematicians since classical times. Johann Faulhaber, Jacob Bernoulli, and others who followed expressed power sums as polynomials in x of degree n + 1 with rational coefficients. Here, we consider the denominators of these polynomials and prove some of their properties. A remarkable one is that such a denominator equals n + 1 times the squarefree product of certain primes p obeying the condition that the sum of the base-p digits of n + 1 is at least p. As an application, we derive a squarefree product formula for the denominators of the Bernoulli polynomials.

We present geometric proofs of Menger's results on isometrically embedding metric spaces in Euclidean space.

In recent years, several papers appeared on generalizations of Fermat's little theorem and the Gauss congruence to integer square matrices. We will give simple and intuitive proofs deriving these matrix versions from the corresponding number-theoretic results.