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Abstract A fundamental theorem of linear algebra asserts that every basis for the vector space has n elements. In this expository note we present a theorem of W. G. Leavitt describing one way in which this invariant basis number property can fail when one does linear algebra over rings, rather...
Abstract Let and be the set of natural numbers and prime numbers, respectively. For a positive integer n, define the following two representation functions and An old result of Erdős showed that is unbounded. As an elementary application of the Maynard–Tao theorem, we prove that is also...
Abstract Legendre published the first attempted proof of the law of Quadratic Reciprocity. In its final form (1797), however, it had a gap in the form of an unproven hypothesis. Some 125 years later, Herman Teege published the first rigorous proof of that hypothesis. Then, 48 years later,...
Abstract For two matrices A and B, and large n, we show that most products of n factors of and n factors of are close to . This extends the Lie-Trotter formula. The elementary proof is based on the relation between words and lattice paths, asymptotics of binomial coefficients, and matrix...
Abstract Constants of motion in Mechanics are usually inferred from groups of symmetry transformations of the given system, as, for example, a Lagrangian function that is time-invariant implies the conservation of energy. Here we wish to show that useful properties of a mechanical system can...
Proposed problems, solutions, and classics should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed problems must not be under consideration concurrently at any other journal, nor should they be posted to the internet before the deadline date for solutions....
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