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Euler’s Inequality for Circumradius and Inradius Am. Math. Monthly (IF 0.5) Pub Date : 2024-03-04 Wagner Oliveira Costa Filho
Published in The American Mathematical Monthly (Ahead of Print, 2024)
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Reviews Am. Math. Monthly (IF 0.5) Pub Date : 2024-02-22 Reviewed by Daniel E. Otero
Published in The American Mathematical Monthly (Ahead of Print, 2024)
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On Nonnegativity of the Generalized Discriminants of Quadratics Am. Math. Monthly (IF 0.5) Pub Date : 2024-02-23 Yusuf Z. Gürtaş
Discriminants of quadratics have been recently generalized as Δk=k(n−k)cn−k2−(k+1)(n−k+1)cn−k−1cn−k+1 for a polynomial f(x)=cnxn+⋯+c1x+c0 of degree n≥2 for 1≤k≤n−1 and it has been shown that Δ1≥0 i...
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Dubious Identities: A Visit to the Borwein Zoo Am. Math. Monthly (IF 0.5) Pub Date : 2024-02-22 Zachary P. Bradshaw, Christophe Vignat
We contribute to the zoo of dubious identities established by J.M. and P.B. Borwein in their 1992 paper, “Strange Series and High Precision Fraud” with five new entries, each of a different variety...
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On the Primorial Counting Function Am. Math. Monthly (IF 0.5) Pub Date : 2024-02-14 Christian Axler
Let Nk denote the kth primorial and let K(x) be the primorial counting function, the function that counts the number of primorials not exceeding x. In this note, we examine the sign of the function...
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100 Years Ago This Month in The American Mathematical Monthly Am. Math. Monthly (IF 0.5) Pub Date : 2024-02-14 Edited by Vadim Ponomarenko
Published in The American Mathematical Monthly (Ahead of Print, 2024)
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Counting Parity Palindrome Compositions Am. Math. Monthly (IF 0.5) Pub Date : 2024-02-14 Vincent Vatter
Published in The American Mathematical Monthly (Ahead of Print, 2024)
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Yet Another Integral Representation of Catalan’s Constant Am. Math. Monthly (IF 0.5) Pub Date : 2024-01-29 Raymond Mortini
Published in The American Mathematical Monthly (Vol. 131, No. 2, 2024)
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A Sufficient Condition for a Prime to be a Sophie Germain Prime Am. Math. Monthly (IF 0.5) Pub Date : 2024-01-29 Katsumasa Ishii
Published in The American Mathematical Monthly (Vol. 131, No. 2, 2024)
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On Random Walks with Geometric Lifetimes Am. Math. Monthly (IF 0.5) Pub Date : 2024-01-29 Elcio Lebensztayn, Vicenzo Pereira
We consider the following stochastic process. Initially, there is a single particle placed at the origin of Z. This particle moves as a discrete-time simple random walk on Z, which takes one step t...
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Another Proof of Euler’s Circuit Theorem Am. Math. Monthly (IF 0.5) Pub Date : 2024-01-29 Submitted by Carl Feghali
Published in The American Mathematical Monthly (Vol. 131, No. 2, 2024)
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100 Years Ago This Month in The American Mathematical Monthly Am. Math. Monthly (IF 0.5) Pub Date : 2024-01-29 Edited by Vadim Ponomarenko
Published in The American Mathematical Monthly (Vol. 131, No. 2, 2024)
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On Some Localizations and Their Relation to Quotients Am. Math. Monthly (IF 0.5) Pub Date : 2024-01-25 Nawal Kishor Hazarika, Pijush Pratim Sarmah
We describe the ring of fractions of Z/nZ with respect to the multiplicative set S={1,x,x2,x3,...}, where x is any non-nilpotent element and we observe how it identifies with a quotient of the ring...
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Divisibility Graphs and Modular Multiplication Tables Am. Math. Monthly (IF 0.5) Pub Date : 2024-01-09 Antonella Perucca, Tim Seuré, Vincent Wolff
Consider the graph with the residue classes modulo n as vertices, and the following edges: “additive” edges from a to a + 1; “multiplicative” edges from a to ab for some fixed b. This graph illustr...
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100 Years Ago This Month in The American Mathematical Monthly Am. Math. Monthly (IF 0.5) Pub Date : 2023-12-21 Edited by Vadim Ponomarenko
Published in The American Mathematical Monthly (Vol. 131, No. 1, 2024)
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Lucky Cars: Expected Values and Generating Functions Am. Math. Monthly (IF 0.5) Pub Date : 2023-12-21 Antonín Slavík, Marie Vestenická
We count lucky cars in parking functions of a given length, and provide elementary derivations for the generating functions, expected values and variances. We consider not only classical parking fu...
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Collective Evolution Under Catastrophes Am. Math. Monthly (IF 0.5) Pub Date : 2023-12-21 Rinaldo B. Schinazi
We introduce the following discrete time model. Each site of N represents an ecological niche and is assigned a fitness in (0,1). All the sites are updated simultaneously at every discrete time. At...
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The Discrete Theorema Egregium Am. Math. Monthly (IF 0.5) Pub Date : 2023-12-21 Thomas F. Banchoff, Felix Günther
In 1827, Gauss proved that Gaussian curvature is actually an intrinsic quantity, meaning that it can be calculated just from measurements within the surface. Before, curvature of surfaces could onl...
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Problems and Solutions Am. Math. Monthly (IF 0.5) Pub Date : 2023-12-21 Edited by Daniel H. Ullman, Daniel J. Velleman, Stan Wagon, Douglas B. West
Published in The American Mathematical Monthly (Vol. 131, No. 1, 2024)
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A Short Simple Proof of Closedness of Convex Cones and Farkas’ Lemma Am. Math. Monthly (IF 0.5) Pub Date : 2023-12-21 Wouter Kager
Proving that a finitely generated convex cone is closed is often considered the most difficult part of geometric proofs of Farkas’ lemma. We provide a short simple proof of this fact and (for compl...
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A Derivation of the Infinitude of Primes Am. Math. Monthly (IF 0.5) Pub Date : 2023-12-21 Hector Pasten
The well-known analogy between polynomials and integers breaks down when it comes to considering the polynomial derivative. This is rather unfortunate since derivatives are a powerful tool for doin...
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Tiling with Monotone Polyominos Am. Math. Monthly (IF 0.5) Pub Date : 2023-12-21 István Tomon
A monotone polyomino is a set of grid cells pierced by a continuous monotone function f:[a,b]→R. We prove that the minimum number of monotone polyominos in a tiling of the n×n lattice square is n. ...
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Six Proofs of the Fáry–Milnor Theorem Am. Math. Monthly (IF 0.5) Pub Date : 2023-12-15 Anton Petrunin, Stephan Stadler
We survey known proofs of the Fáry–Milnor theorem; it states that any nontrivial knot makes at least two full turns.
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On Erdős’s Proof of the Existence of Cages Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-21 Vincent Vatter
Published in The American Mathematical Monthly (Vol. 130, No. 10, 2023)
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100 Years Ago This Month in The American Mathematical Monthly Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-21 Vadim Ponomarenko
Published in The American Mathematical Monthly (Vol. 130, No. 10, 2023)
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The Hidden Twin of Morley’s Five Circles Theorem Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-21 Lorenz Halbeisen, Norbert Hungerbühler, Vanessa Loureiro
We give an algebraic proof of a slightly extended version of Morley’s Five Circles Theorem. The theorem holds in all Miquelian Möbius planes obtained from a separable quadratic field extension, in ...
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The Importance of Being Prime⋆, a Nontrivial Generalization for Nonunique Factorizations Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-21 Nicholas R. Baeth, Scott T. Chapman
The notion of primeness is the key to the phenomenon of unique factorization. In particular, when unique factorization in a monoid fails, the arithmetic of that monoid is determined by the irreduci...
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Packing Density of Combinatorial Settlement Planning Models Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-21 Mate Puljiz, Stjepan Šebek, Josip Žubrinić
We consider a combinatorial settlement model on a rectangular grid where each house must be exposed to sunlight from east, south, or west. We are interested in maximal configurations, where no addi...
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The Inverse of a Bad Primitive Root is Not Bad Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-21 V. P. Ramesh, R. Gowtham
Published in The American Mathematical Monthly (Vol. 130, No. 10, 2023)
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Odd-Period Cycles of the Logistic Map Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-21 David Calvis
We locate the onset of odd-period cycles of the logistic map using only elementary algebra and calculus.
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A Note on Minimal Prime Submodules Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-21 Ali Reza Naghipour
In this short note, we give a characterization of the modules with finitely many minimal prime submodules over an arbitrary submodule.
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Problems and Solutions Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-21 Edited by Daniel H. Ullman, Daniel J. Velleman, Stan Wagon, Douglas B. West
Published in The American Mathematical Monthly (Vol. 130, No. 10, 2023)
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Reviews Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-21 Reviewed by Pamela Richardson
Published in The American Mathematical Monthly (Vol. 130, No. 10, 2023)
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Editor’s Endnotes Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-21
Published in The American Mathematical Monthly (Vol. 130, No. 10, 2023)
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Monthly Referees for 2023 Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-21
Published in The American Mathematical Monthly (Vol. 130, No. 10, 2023)
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Reviews Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-17 Reviewed by Victor Piercey
Published in The American Mathematical Monthly (Ahead of Print, 2023)
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Abbott Dimension, Mathematics Inspired by Flatland Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-14 Jeremy Siegert
What is the “right way” to define dimension? Mathematicians working in the early and middle 20th-century formalized three intuitive definitions of dimension that all turned out to be equivalent on ...
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From Abel’s Binomial Theorem to Cayley’s Tree Formula Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-14 Marc Zucker
We derive Abel’s generalization of the binomial theorem and use it to present a short proof of Cayley’s theorem on the number of trees on n labeled vertices.
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Ten Points on a Cubic Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-14 Will Traves, David Wehlau
The 16-year old Blaise Pascal found an incidence relation that holds when six points lie on a conic. A century later, Braikenridge and Maclaurin extended Pascal’s result to a straightedge construct...
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Problems and Solutions Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-17 Edited by Daniel H. Ullman, Daniel J. Velleman, Stan Wagon, Douglas B. West
Published in The American Mathematical Monthly (Ahead of Print, 2023)
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Another Proof of Euler’s Circuit Theorem Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-17 Submitted by Carl Feghali
Published in The American Mathematical Monthly (Ahead of Print, 2023)
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On Random Walks with Geometric Lifetimes Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-17 Elcio Lebensztayn, Vicenzo Pereira
We consider the following stochastic process. Initially, there is a single particle placed at the origin of Z. This particle moves as a discrete-time simple random walk on Z, which takes one step t...
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The Altitudes of a Triangle Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-15 Mark Mandelkern
A long-standing, unanswered question regarding Euclid’s Elements concerns the absence of a theorem for the concurrence of the altitudes of a triangle, and the possible reasons for this omission. Th...
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Yet Another Integral Representation of Catalan’s Constant Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-06 Raymond Mortini
Published in The American Mathematical Monthly (Ahead of Print, 2023)
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100 Years Ago This Month in The American Mathematical Monthly Am. Math. Monthly (IF 0.5) Pub Date : 2023-11-06 Edited by Vadim Ponomarenko
Published in The American Mathematical Monthly (Ahead of Print, 2023)
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The Discrete Theorema Egregium Am. Math. Monthly (IF 0.5) Pub Date : 2023-10-20 Thomas F. Banchoff, Felix Günther
In 1827, Gauss proved that Gaussian curvature is actually an intrinsic quantity, meaning that it can be calculated just from measurements within the surface. Before, curvature of surfaces could onl...
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Collective Evolution Under Catastrophes Am. Math. Monthly (IF 0.5) Pub Date : 2023-10-17 Rinaldo B. Schinazi
We introduce the following discrete time model. Each site of N represents an ecological niche and is assigned a fitness in (0,1). All the sites are updated simultaneously at every discrete time. At...
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A Derivation of the Infinitude of Primes Am. Math. Monthly (IF 0.5) Pub Date : 2023-10-11 Hector Pasten
The well-known analogy between polynomials and integers breaks down when it comes to considering the polynomial derivative. This is rather unfortunate since derivatives are a powerful tool for doin...
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A Short Simple Proof of Closedness of Convex Cones and Farkas’ Lemma Am. Math. Monthly (IF 0.5) Pub Date : 2023-10-12 Wouter Kager
Proving that a finitely generated convex cone is closed is often considered the most difficult part of geometric proofs of Farkas’ lemma. We provide a short simple proof of this fact and (for compl...
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Tiling with Monotone Polyominos Am. Math. Monthly (IF 0.5) Pub Date : 2023-10-11 István Tomon
A monotone polyomino is a set of grid cells pierced by a continuous monotone function f:[a,b]→R. We prove that the minimum number of monotone polyominos in a tiling of the n×n lattice square is n. ...
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100 Years Ago This Month in The American Mathematical Monthly Am. Math. Monthly (IF 0.5) Pub Date : 2023-10-09 Edited by Vadim Ponomarenko
Published in The American Mathematical Monthly (Ahead of Print, 2023)
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The Eighty-Third William Lowell Putnam Mathematical Competition Am. Math. Monthly (IF 0.5) Pub Date : 2023-09-25 Daniel H. Ullman, Paul Zeitz
Published in The American Mathematical Monthly (Vol. 130, No. 8, 2023)
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Lagrange Inversion Formula by Induction Am. Math. Monthly (IF 0.5) Pub Date : 2023-09-15 Erlang Surya, Lutz Warnke
We present a simple inductive proof of the Lagrange Inversion Formula.
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Reviews Am. Math. Monthly (IF 0.5) Pub Date : 2023-09-14 Reviewed by Stephen Kennedy
Published in The American Mathematical Monthly (Vol. 130, No. 9, 2023)
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Problems and Solutions Am. Math. Monthly (IF 0.5) Pub Date : 2023-09-05 Daniel H. Ullman, Daniel J. Velleman, Stan Wagon, Douglas B. West
Proposed problems, solutions, and classics should be submitted online at americanmathematicalmonthly.submittable.com/submit.Proposed problems must not be under consideration concurrently at any oth...
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In Search of Comrade Agronomof: Some Tribonacci History Am. Math. Monthly (IF 0.5) Pub Date : 2023-08-23 Hans J. H. Tuenter
A note from May 1914 by a certain Agronomof in the Belgian journal Mathesis is frequently cited in the context of the mathematical history of the Tribonacci number sequence. We round out the histor...
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A New Perspective on Impartial and Unbiased Apportionment Am. Math. Monthly (IF 0.5) Pub Date : 2023-08-17 Ross Hyman, Nicolaus Tideman
How to fairly apportion congressional seats to states has been debated for centuries. We present an alternative perspective on apportionment, centered not on states but “families” of states, sets o...
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A Quixotic Proof of Fermat’s Two Squares Theorem for Prime Numbers Am. Math. Monthly (IF 0.5) Pub Date : 2023-08-15 Roland Bacher
Every odd prime number p has exactly (p+1)/2 different expressions as a sum ab + cd of two ordered products ab and cd such that min(a,b)>max(c,d). An easy corollary is a proof of Fermat’s Theorem e...
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Problems and Solutions Am. Math. Monthly (IF 0.5) Pub Date : 2023-08-14 Edited by Daniel H. Ullman, Daniel J. Velleman, Stan Wagon, Douglas B. West
Published in The American Mathematical Monthly (Vol. 130, No. 8, 2023)
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A Tale of Two by Two Matrices Am. Math. Monthly (IF 0.5) Pub Date : 2023-08-14 Dan Kalman, Robert Mena
We formulate the Euclidean algorithm using directed graphs on integer points in the plane and operations on a particular semigroup of two by two matrices. The properties of the graphs and the semig...