Vol. 121, No. 5, pp.379-468.

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## ARTICLES

### Unknotting Unknots

Allison Henrich and Louis H. Kauffman

A knot is an embedding of a circle into three-dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a standard circle. In essence, an unknot is a knot that may be deformed to a standard circle without passing through itself. By representing knots via planar diagrams, we discuss the problem of unknotting a knot diagram when we know that it is unknotted. This problem is surprisingly difficult, since it has been shown that knot diagrams may need to be made more complicated before they may be simplified. We do not yet know, however, how much more complicated they must get. We give an introduction to the work of Dynnikov, who discovered the key use of arc-presentations to solve the problem of finding a way to detect the unknot directly from a diagram of the knot. Using Dynnikov’s work, we show how to obtain a quadratic upper bound for the number of crossings that must be introduced into a sequence of unknotting moves. We also apply Dynnikov’s results to find an upper bound for the number of moves required in an unknotting sequence.

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### A Geometric Representation of Continued Fractions

Alan F. Beardon and Ian Short

Inspired by work of Ford, we describe a geometric representation of real and complex continued fractions by chains of horocycles and horospheres in hyperbolic space. We explore this representation using the isometric action of the group of Möbius transformations on hyperbolic space, and prove a classical theorem on continued fractions.

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### Rethinking Set Theory

Tom Leinster

Mathematicians manipulate sets with confidence almost every day, rarely making mistakes. Few of us, however, could accurately quote what are often referred to as ‘the’ axioms of set theory. This suggests that we all carry around with us, perhaps subconsciously, a reliable body of operating principles for manipulating sets. What if we were to take some of those principles and adopt them as our axioms instead? The message of this article is that this can be done, in a simple, practical way (due to Lawvere). The resulting axioms are ten thoroughly mundane statements about sets.

Due to a printer error page 403 of the May Monthly is in the wrong order. It appears after page 415. All electronic versions have the pages in the correct order.

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### MATHBIT: An Elementary Application of Brouwer’s Fixed Point Theorem to Transition Matrices

Allan J. Kroopnick

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### Repeatedly Appending Any Digit to Generate Composite Numbers

Jon Grantham, Witold Jarnicki, John Rickert, and Stan Wagon

We investigate the problem of finding integers $$k$$ such that appending any number of copies of the base-ten digit $$d$$ to $$k$$ yields a composite number. In particular, we prove that there exist infinitely many integers coprime to all digits such that repeatedly appending any digit yields a composite number.

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### On Wallis-type Products and Pólya’s Urn Schemes

Iddo Ben-Ari, Diana Hay, and Alexander Roitershtein

A famous “curious identity” of Wallis gives a representation of the constant $$\pi$$ in terms of a simply structured infinite product of fractions. Sondow and Yi [*Amer. Math. Monthly* **117** (2010) 912–917] identified a general scheme for evaluating Wallis-type infinite products. The main purpose of this paper is to discuss an interpretation of the scheme by means of Pólya urn models.

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## NOTES

### The Primes that Euclid Forgot

Paul Pollack and Enrique Treviño

Let $$q_{1}=2$$. Supposing that we have defined $$q_{j}$$ for all $$1\leq j\leq k$$, let $$q_{k+1}$$ be a prime factor of $$1+\prod_{j=1}^{k}q_{j}$$. As was shown by Euclid over two thousand years ago, $$q_{1}, q_{2}, q_{3},\dots$$ is then an infinite sequence of distinct primes. The sequence $$\{q_{i}\}$$ is not unique, since there is flexibility in the choice of the prime $$q_{k+1}$$ dividing $$1+\prod_{j=1}^{k}q_{j}$$. Mullin suggested studying the two sequences formed by (1) always taking $$q_{k+1}$$ as small as possible, and (2) always taking $$q_{k+1}$$ as large as possible. For each of these sequences, he asked whether every prime eventually appears. Recently, Booker showed that the second sequence omits infinitely many primes. We give a completely elementary proof of Booker’s result, suitable for presentation in a first course in number theory.

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### Solution of Sondow's Problem: A Synthetic Proof of the Tangency Property of the Parbelos

Emmanuel Tsukerman

In a recent paper titled "The parbelos, a parabolic analog of the arbelos," Sondow asks for a synthetic proof to the tangency property of the parbelos. In this paper, we resolve this question by introducing a converse to Lambert’s Theorem on the parabola. In the process, we prove some new properties of the parbelos.

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### Mathbit: A Difference Equation Leading to the Irrationality of $$\sqrt{2}$$

José Ángel Cid Araújo

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### A Connection between Furstenberg’s and Euclid’s Proofs of the Infinitude of Primes

Nathan A. Carlson

In 1955, Furstenberg gave a curious topological proof of the infinitude of primes. Cass and Wildenberg, as well as Mercer, dispensed with the topological language in this proof to uncover the essential number theory. In this note, we observe that Furstenberg’s proof has an important and intriguing connection to Euclid’s well-known original proof.

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### An Elementary Proof of a Generalization of Banach’s Mapping Theorem

Ming-Chia Li

We give an elementary proof of a generalization of Banach’s mapping theorem, which says that for any two mappings $$f:A\rightarrow B$$ and $$g:B\rightarrow A$$, there exists a subset $$A_{0}$$ of $$A$$ such that $$g(B\setminus f (A_{0})) = A\setminus A_{0}$$.

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### A Technique in Contour Integration

M. L. Glasser

A rotation of an integration contour is shown to lead, in some cases, to interesting integral identities. Elementary and non-elementary examples are provided.

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### Large-Deviation Bounds for Sampling without Replacement

Kyle Luh and Nicholas Pippenger

We give a simple argument, based on drawing balls from urns, showing that the exponential bound on the probability of a large deviation for sampling with replacement applies also to sampling without replacement. This result includes as a special case the relationship between the binomial and hypergeometric distributions.

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## PROBLEMS AND SOLUTIONS

Problems 11775-11781

Solutions 11644, 11649, 11650, 11652

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## REVIEWS

*Math on Trial: How Mathematics Is Used and Abused in the Courtroom*. By Leila Schneps and Coralie Colmez.

Reviewed by Daniel Ullman

To purchase from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.05.463