The 10 most-cited articles in American Mathematical Monthly, 2022

We give a short proof of a recently established Hardy-type inequality due to Keller, Pinchover, and Pogorzelski together with its optimality. Moreover, we identify the remainder term which makes it into an identity.

Euclid pioneered the concept of a mathematical theory developed from axioms by a series of justified proof steps. From the outset there were critics and improvers. In this century the use of computers to check proofs for correctness sets a new standard of rigor. How does Euclid stand up under such an examination? And what does the exercise have to teach us about geometry, mathematical foundations, and the relation of logic to truth?

In this article, using only elementary knowledge of complex numbers, we sketch a proof of the celebrated Abel–Ruffini theorem, which states that the general solution to an algebraic equation of degree five or more cannot be written using radicals, that is, using its coefficients and arithmetic operations +,−,×,÷, and . The present article is written purposely with concise and pedagogical terms and dedicated to students and researchers not familiar with Galois theory, or even group theory in general, which are the usual tools used to prove this remarkable theorem. In particular, the proof is self-contained and gives some insight as to why formulae exist for equations of degree less than five (and how they are constructed), and why they do not for degree five or more.

In simple card games, cards are dealt one at a time and the player guesses each card sequentially. We study problems where feedback (e.g., correct/incorrect) is given after each guess. For decks with repeated values (as in blackjack where suits do not matter), the optimal strategy differs from the “greedy strategy” (of guessing a most likely card each round). Further, both optimal and greedy strategies are far too complicated for real-time use by human players. Our main results show that simple heuristics perform close to optimal.

We provide lower bounds on the number of subgroups of a group G as a function of the primes and exponents appearing in the prime factorization of |G|. Using these bounds, we classify all abelian groups with 22 or fewer subgroups, and all nonabelian groups with 19 or fewer subgroups. This allows us to extend the integer sequence A274847 in the On-Line Encyclopedia of Integer Sequences introduced by Slattery.

Piers Bohl (1865–1921) was an outstanding Latvian mathematician whose name is particularly connected with several crucial achievements that were ahead of their time. Unfortunately, the contemporary mathematical community did not recognize their significance. This paper discusses one of Bohl’s results from 1908 that remained nearly unnoticed, even up until the present. It concerns a general complex trinomial and answers the problem of the distribution of its roots with respect to a given modulus. During the last few decades, this and other related problems have been extensively studied in numerous particular cases, without knowledge of this existing answer. This paper recalls the aforementioned Bohl’s result and illustrates how easily it can imply conclusions of recent as well as older works regarding this topic.

How should we place n great circles on a sphere to minimize the furthest distance between a point on the sphere and its nearest great circle? Fejes Tóth conjectured that the optimum is attained by placing n circles evenly spaced all passing through the north and south poles. This conjecture was recently proved by Jiang and Polyanskii. We present a short simplification of Ortega-Moreno’s alternate proof of this conjecture.

J. Marshall Ash [1] gave a simple and elegant proof to the fact, first noticed by N.H. Abel in 1828, that for each divergent positive series there is always another one which diverges slower. We sh...

The classical D’Alembert’s Ratio Test is a powerful test that we learn from calculus to determine convergence for a series of positive terms. Its range of applicability and ease of computation make this test extremely appealing. However, when the limiting ratio of the terms equals 1, then the test is inconclusive. Several series tests like Raabe’s and Gauss’ Tests have been proposed in order to address this case. These tests were also generalized by Kummer through Kummer’s Test. More recently, a Second Ratio Test was constructed that also possessed an inconclusive case. This article presents a survey of existing series tests, introduces an extension of Raabe’s Test to the Second Ratio Test, and proposes extensions of classical tests such as Gauss’s Test and Kummer’s Test. It also offers proofs of the aforementioned tests and a brief application of the Second Raabe’s Test.