The American Mathematical Monthly

Wildberger, N. J.; Rubine, Dean

doi: 10.1080/00029890.2025.2460966pmid: N/A

Abstract The Catalan numbers C m count the number of subdivisions of a polygon into m triangles, and it is well known that their generating series is a solution to a particular quadratic equation. Analogously, the hyper-Catalan numbers C m count the number of subdivisions of a polygon into a given number of triangles, quadrilaterals, pentagons, etc. (its type m ), and we show that their generating series solves a polynomial equation of a particular geometric form. This solution is straightforwardly extended to solve the general univariate polynomial equation. A layering of this series by numbers of faces yields a remarkable factorization that reveals the Geode, a mysterious array that appears to underlie Catalan numerics.

Adamczewski, B.; Delaygue, É.

doi: 10.1080/00029890.2025.2459043pmid: N/A

Abstract The aim of this article is to show that any algebraic relation over Q ¯ between the values of the trigonometric functions sine and cosine at algebraic points can be derived from the Pythagorean identity and the angle addition formulas. This result is obtained as a consequence of the Lindemann–Weierstrass theorem.

Cerf, Raphaël

doi: 10.1080/00029890.2025.2459044pmid: N/A

Abstract We define the pivotal set of a Boolean function and we prove a fundamental inequality on its expected size, when the inputs are independent random coins of parameter p. We give two complete proofs of this inequality. Along the way, we obtain the classical Margulis–Russo formula. We give a short proof of the classical Hoeffding inequality for i.i.d. Bernoulli random variables, and we use it to derive more complex deviations inequalities associated to the pivotal set. We finally follow Talagrand’s footsteps and we discuss a beautiful inequality that he proved in the uniform case.

Prince, Thomas

doi: 10.1080/00029890.2025.2461447pmid: N/A

Abstract In this article we study Ceva’s theorem and its higher-dimensional extensions from the perspective of algebraic and projective geometry. First, we situate the theorem within the study of algebraic surfaces by relating it to the defining equation of a del Pezzo surface of degree six inside the product of three projective lines. Second, by interpreting (higher-dimensional analogues of) Ceva’s theorem in terms of projections from projective spaces, we recast these results as matrix completion problems. We use these ideas to offer proofs of some higher-dimensional analogues of Ceva’s theorem. This article is written with a nonspecialist audience in mind and we hope that some useful context is provided in the form of remarks in the sections on surfaces for students of algebraic geometry.

Kaneko, Yuto; Nakai, Hirofumi

doi: 10.1080/00029890.2025.2459045pmid: N/A

Abstract In 1893, Schatunowsky showed that the set of numbers, each of whose totatives except to 1 are primes, is finite. In this paper we examine a generalization of Schatunowsky’s theorem and prove that the associated sets are all finite.

Kim, Jaehyeon

doi: 10.1080/00029890.2025.2460413pmid: N/A

Abstract Various proofs of Jensen’s inequality exist. This note introduces a novel proof using water and potential energy concepts. By modeling the inequality using water, we establish a physical analogy that demonstrates the theorem visually and intuitively. This work is a generalization of [1] and [2].

Hardy, Michael

doi: 10.1080/00029890.2025.2459048pmid: N/A

Abstract It has been known at least since 1976 that the family of Cauchy-distributed random variables is closed under linear-fractional transformations with real coefficients [1]. We reduce the proof of that fact to the application of a simple trigonometric identity.

Olivieri, Julia

doi: 10.1080/00029890.2025.2460411pmid: N/A