# American Mathematical Monthly - May 2018

The May issue of the Monthly has all sorts of springtime treats in store. Explore maximally nontransitive dice; examine factorials and the gamma function through the pages of the Monthly; study a model for the moduli space of Lorentzian triangles; use symmetric polynomials to compute determinants of general Vandermode and Toeplitz matrices.

In the Notes, you can investigate sequences generated by powers of the kth-order Fibonacci recurrence relation; enumerate partitions of {1, 2,…, n} containing isolated elements; consider inequalities involving the nth derivative of cos(√x) and its analytic continuation, “slow” sequences in normed spaces, an irreducibility criterion for integer polynomials, and a short proof of the hairy ball theorem.

Enjoy the satisfaction of solving tantalizing Monthly problems and ponder one of the great mathematical mysteries as you read Avner Ash’s review of Mazur and Stein’s Prime Numbers and the Riemann Hypothesis.

— Susan Jane Colley, Editor

### Maximally Nontransitive Dice

p. 387.

Joe Buhler, Ron Graham & Al Hales

We construct arbitrarily large sets of dice with some remarkable nontransitivity properties. In a sense made precise later, each set exhibits all possible pairwise win/loss relationships by summing different numbers of rolls. The proof of this fact relies on an asymptotic formula for the difference between the median and mean of sums of multiple rolls of dice. This formula is a consequence of a suitable Edgeworth series (an asymptotic refinement of the central limit theorem), for which we give a detailed sketch of a proof in the final section.

### Gamma and Factorial in the Monthly

p. 400.

Jonathan M. Borwein & Robert M. Corless

Since its inception in 1894, the Monthly has printed 50 articles on the Γ function or Stirling's asymptotic formula, including the magisterial 1959 paper by Phillip J. Davis, which won the 1963 Chauvenet prize, and the eye-opening 2000 paper by the Fields medalist Manjul Bhargava. In this article, we look back and comment on what has been said, and why, and try to guess what will be said about the Γ function in future Monthly issues. We also identify some gaps, which surprised us: phase plots, Riemann surfaces, and the functional inverse of Γ make their first appearance in the Monthly here. We also give a new elementary treatment of the asymptotics of n! and the first few terms of a new asymptotic formula for invΓ.

### A Visual Model of the Lorentz Triangular Moduli Space

p. 426.

Daniel de la Fuente, Rafael Ramírez-Uclés & Juan F. Ruiz-Hidalgo

We present a schematic construction of the triangularly-shaped space in the Lorentz plane, i.e., the moduli space of all Lorentzian triangles up to similarity. This space turns out to be homeomorphic to a band where the different types of triangles may be located according to the length and the causal character of their sides. In particular, several kinds of equilateral and isosceles triangles are identified.

### On One Type of Generalized Vandermonde Determinants

p. 433.

João Lita da Silva

An expression including symmetric polynomials for the determinant of one type of generalized Vandermonde matrix is presented leading to a handy formula to compute determinants of general Toeplitz band matrices.

## NOTES

### Sequences Generated by Powers of the kth-order Fibonacci Recurrence Relation

p. 443.

Chris D. Lynd & James Wright Sharpe

Every positive sequence generated by the Fibonacci recurrence relation diverges to infinity. However, every positive sequence generated by the square root of the Fibonacci recurrence relation converges to 2. In this note, we investigate different powers of the Fibonacci recurrence relation and higher-order Fibonacci recurrence relations. We then prove a theorem that gives the limit of positive sequences generated by the pth-power of the kth-order Fibonacci recurrence relation.

### Set Partitions with Isolated Singletons

p. 447.

Augustine O. Munagi

We give results for partitions of the set of the first n natural numbers into k nonempty subsets S1, S1, …, Sk with respect to the number of isolated elements, that is, with respect to integers a such that if a belongs to Si, then neither a − 1 nor a + 1 belongs to Si.

### A Gronwall-type Trigonometric Inequality

p. 453.

A. G. Smirnov

We prove that the absolute value of the nth derivative of cos(√x) does not exceed n!/(2n)! for all x > 0 and n = 0, 1, … and obtain a natural generalization of this inequality involving the analytic continuation of cos(√x).

### Cluster Points of Slow Sequences

p. 457.

Youssef Azouzi

A sequence in a metric space is said to be slow if the distance between consecutive terms approaches zero. If a slow sequence lies in a compact metric space or in the real line, then the set of its cluster points is connected. The aim of the present note is to give a characterization of the set of all cluster points of a slow sequence in a normed space.

### Another Short Proof of the Hairy Ball Theorem

p. 462.

Eugene Curtin

We show how the assumption of the existence of a continuous unit tangent vector field on the sphere leads to an explicit formula for a homotopy between curves of winding number 1 and − 1 about the origin, thus proving the hairy ball theorem by contradiction.

### An Irreducibility Criterion for Polynomials with Integer Coefficients

p. 464.

Anuj Jakhar

We provide irreducibility conditions for polynomials f(x) with integer coefficients, with f written in φ(x)-expansion, where φ is monic and irreducible modulo some prime number p. Examples of some explicit families of such irreducible polynomials are also given.

p. 466.

## Book Review

p. 476.

### Prime Numbers and the Riemann Hypothesis by Barry Mazur and William Stein

Reviewed by Avner Ash

p. 425.

p. 446.

p. 463.