Apéry's constant is the value of ζ (3) where ζ is the Riemann zeta function. Thus

This constant arises in certain mathematical and physical contexts (in physics for example ζ (3) arises naturally in the computation of the electron's gyromagnetic ratio using quantum electrodynamics) and has attracted a great deal of interest, not least the fact that it was proved to be irrational by the French mathematician Roger é and named after him. See [1,2].

Numerous series representations have been obtained for ζ (3) many of which are rather complicated [3]. é used one such series in his irrationality proof. It is not known whether ζ (3) is transcendental, a question whose resolution might be helped by a study of an appropriate series representation of ζ (3).

This paper is about the classification of frieze groups. A frieze is a decorative strip of paper (or wood, stone,…) on which a pattern is produced by the periodic repetition of a picture along the strip. A symmetry of a frieze is an isometry of the plane that leaves the pattern unchanged, and a frieze group is the group of symmetries of some frieze. A popular exercise is to start with a given frieze and then try to identify its frieze group. However, in order to do this we need to know that there are only seven possible frieze groups, and what these groups are. Seven friezes, with different frieze groups, are illustrated below, in such a way that each pattern is invariant under the same translation, namely x → x + 1. Our task is to show that (up to a change in the motif, and a simple change of coordinates in the plane) these are the only frieze patterns.

Every student of calculus learns that one typically solves a differential equation by integrating it. However, as Euler showed in his 1758 paper (E236), Exposition de quelques paradoxes dans le calcul intégral (Explanation of certain paradoxes in integral calculus) [1], there are differential equations that can be solved by actually differentiating them again. This initially seems paradoxical or, as Euler describes it in the introduction of his paper:

Here I intend to explain a paradox in integral calculus that will seem rather strange: this is that we sometimes encounter differential equations in which it would seem very difficult to find the integrals by the rules of integral calculus yet are still easily found. not by the method of integration. but rather in differentiating the proposed equation again; so in these cases, a repeated differentiation leads us to the sought integral.

Given two triangles ΔABC and ΔDEF, if each side of ΔDEF contains a vertex of ΔABC, then we call ΔDEF an outscribed triangle of ΔABC. Given ΔABC, let ΦΔABC be the set of all outscribed equilateral triangles of ΔABC. Clearly ΦΔABC is non-empty. In the following we will determine the area of the largest member of ΦΔABC when each angle of ΔABC is smaller than 120° and show that this largest member can be constructed by ruler and compass from ΔABC. The corresponding problem on quadrilaterals has been considered in [1].

How many times must a die be thrown to get four consecutive sixes? More generally, and more precisely,

Q1. What is the expected number of times we must throw a die to get n consecutive sixes?

The answer, rather suggestively for the numerologically inclined, turns out to be

A1. The expected number of throws is 6 + 62 + 63 + … + 6n.

In some well-known dice games players throw several dice at the same time: in craps a pair (brace) of dice is thrown, while in yahtzee, yam and balut, up to five dice are thrown at a time. However, Q1 is not the same as

What is it that makes us judge two proofs of the same theorem to be the same or different? This is not an idle question: one central aspect of judging mathematics is the novelty of the mathematics presented. This is important everywhere, from the peer-review system, to assigning international prestige, to funding agencies’ grant decisions. It even matters to some extent in examinations, to avoid accusations of collusion. Surprisingly, philosophers of mathematics have not paid the question of novelty much attention. In this Article, we will consider the appealing conjecture that the main ideas that make up the proof, the essence of a proof, can indeed be identified and that very different styles of proofs can share common main ideas. Further, that a particular theorem can be proved using quite different, independent main ideas. As a means of exploring whether this is plausible, we will present a number of novel proofs of the following theorem.

Folded paper road maps are found next to sextants in the pile of obsolete navigation tools. GPS navigation apps like Waze and Google Maps, accurate to within a few metres, are available on all smartphones and most new cars. These apps provide drivers with real time traffic conditions and suggest minimum drive time routes, giving drivers the ability to avoid congestion and delays caused by heavy traffic, accidents, road construction and other hindrances.

Applications of convex coordinates are many and varied in mathematics today. Our first interest in convex coordinates grew out of their use in linear programming, e.g. [1]. That interest has remained quite lively as the study of convex coordinates continues to yield pleasing and sometimes unexpected results, e.g. [2,3]. Another name for these coordinates which may be more familiar to the reader is that of barycentric coordinates as originally given by the German mathematician Möbius. In this article we shall confine our attention to the Euclidean plane, but convex coordinates may be defined in Euclidean spaces of any dimension.

A positive integer, n, which is equal to the sum, S, of its digits multiplied by their product, P, is called an S·P number. S. Parameśwaran gave three examples of such numbers in:

and he conjectured that there are only a finite number of them. This was confirmed when several of us proved that each S·P number has at most 60 digits. There the matter rested for a time. Even if it were possible to test all numbers from 1 to 1060 at the rate of 1040 per second (far beyond the capacity of today’s fastest computer) it would take much longer than the estimated age of the universe to complete the task.