Metamathematics is the mathematical study of mathematics itself. Two of its most famous theorems were proved by Kurt Gödel in 1931. In a simplified form, Gödel's first incompleteness theorem states that no reasonable mathematical system can prove all the truths of mathematics. Gödel's second incompleteness theorem (also simplified) in turn states that no reasonable mathematical system can prove its own consistency. Another famous undecidability theorem is that the Continuum Hypothesis is neither provable nor refutable in standard set theory. Many of us logicians were first attracted to the field as students because we had heard something of these results. All research mathematicians know something of them too, and have at least a rough sense of why ‘we can't prove everything we want to prove’.

Bayesian statistics is included in few elementary statistics courses, and many mathematicians have heard of it, perhaps through collateral readings from popular literature or [1], selected as an Editor's Choice in the New York Times Book Review. ‘Bayesian statistics’ provides for a way to incorporate prior beliefs, experience, or information into the analysis of data. Bayesian thinking is natural, and that is an advantage. For example, on a summer morning, if we see dark rain clouds up in the sky, we leave home for work with an umbrella because prior experience tells us that doing so is beneficial. In general, the idea is simple; schematically, it looks like this:

(prior belief) + (data: new information) ⇒ (posterior belief).

Thus, we begin with a prior belief that we allow to be modified or informed by new data to produce a posterior belief, which then becomes our new prior, and this process is never-ending. We are always willing to update our beliefs according to new information.

What is the likelihood P(n, T) that at least two people in a gathering of n people are born within a given time T of each other? In particular, for T = 24 hours what is the smallest n for which P(n, T) is at least 50%? The well-known classic version of the birthday problem asks for the smallest number of people needed to give a better than 50% chance of at least one birthday match with the assumptions that the birthdays are independently selected from a discrete uniform distribution over 365 days. Using the calculus tool of the limit, we refine two characterisations for P(n, T) and show that they give consistent results with each other and with the classic birthday problem as well. At first thought P(n, 24 hours) should answer the classic birthday question. Yet consider this experiment: suppose Andrea was born at noon on June 1, then potential matching birth times (for the other n - 1 people) with her birth time extend from noon on May 31 to noon on June 2, a period of 48 hours! What this period suggests is that P(n, 24hours) exceeds the classic answer. But, by how much? Read on.

There are many families of polynomials in mathematics, and they often occur naturally in pairs. The Fibonacci polynomials and the Lucas polynomials, for example, are generated by the same recurrence relation but with different starting values, and there are many identities that link the two families [1]. The same is true for the Chebyshev polynomials of the first and second kinds, Tn (x) and Un (x) [2], respectively. There are two further polynomial families that are less well-known, the Chebyshev polynomials of the third and fourth kinds, Vn (x) and Wn (x) [3], respectively. Each of the four kinds is an example of an orthogonal polynomial family Pn (x), where for some appropriate weight function W (x), whenever n ≠ m. The families Tn (x) and Un (x) in particular are ubiquitous in their mathematical uses, in approximation theory, in differential equations, and in solving the Pell equation, to name but three. There are also many connections between Tn (x), Un (x), Vn (x) and Wn (x), some of which are explored here, and some of which we hope are new.

Consider the integral

where b > a > 0. First, let us clarify why it even exists. Of course, convergence at infinity is ensured by the exponential terms, but the integrals of and e–ax/x and e–bx/x, taken separately, are divergent at 0, since these integrands equate asymptotically to 1/x as x → 0. However,

so (e–ax – e–bx)/x tends to the finite limit b – a as x → 0 and there is no problem integrating it on intervals of the form [0, r].

A neat way to evaluate I1 starts by expressing the integrand itself as an integral:

(1)

Inserting this into I1 converts it into a double integral.

We shall use the word ‘polyhedron’ to mean a connected, simply-connected 3-polytope of positive (non-zero) volume. The idea of the net of a polyhedron P is well known. For example, the regular octahedron has eleven distinct nets, three of which are shown in Figure 1. A net consists of three parts:

• (a) A plane connected and simply-connected polygon Q (denoted by heavy lines in the diagrams), known as the carrier of the net;

  (b) A set of lines known as, fold-lines in the interior of Q;

  (c) A labelling of the edges of Q.

  If one cuts Q out of paper or similar material, folds it along the fold-lines, and then pastes together edges with matching labels, one obtains a model of the polyhedron P. We say that a net is convex if, and only if, its carrier Q is convex.

In a recent paper [1], the authors gave a combinatorial interpretation to sums of equally spaced binomial coefficients. Others have been interested in finding such sums, known as multisection of series. For example, Gould [2] derived interesting formulas but much of his work involved complicated manipulations of series. When the combinatorial approach can be implemented, it is neat and efficient. In this paper, we present another approach for finding equally spaced sums. We consider both infinite sums and partial finite sums based on generating functions and extracting coefficients.

While generating functions were first introduced by Abraham de Moivre at the end of seventeen century, its systematic use in combinatorial analysis was inspired by Leonhard Euler. Generating functions got a new birth in the twentieth century as a part of symbolic methods. As a central mathematical tool in discrete mathematics, generating functions are an essential part of the curriculum in the analysis of algorithms [3, 4]. They provide a bridge between discrete and continuous mathematics, as illustrated by the fact that the generating functions presented here appear as solutions to corresponding differential equations.

One of the many definite integrals that Euler was the first to evaluate was

(1)

He did this, almost as an afterthought, at the end of his short, seven-page paper catalogued as E675 in [1] and with the matter-of-fact title, On the values of integrals from x = 0 to x = ∞. It is a beautiful Euler miniature which neatly illustrates the unexpected twists and turns in the history of mathematics. For Euler's derivation of (1) emerges as the by-product of a solution to a problem in differential geometry concerning the clothoid curve which he had first encountered nearly forty years earlier in his paper E65, [1]. As highlighted in the recent Gazette article [2], E675 is notable for Euler's use of a complex number substitution to evaluate a real-variable integral. He used this technique in about a dozen of the papers written in the last decade of his life. The rationale for this manoeuvre caused much debate among later mathematicians such as Laplace and Poisson and the technique was only put on a secure footing by the work of Cauchy from 1814 onwards on the foundations of complex function theory, [3, Chapter 1]. Euler's justification was essentially pragmatic (in agreement with numerical evidence) and by what Dunham in [4, p. 68] characterises as his informal credo, ‘Follow the formulas, and they will lead to the truth.’ Smithies, [3, p. 187], contextualises Euler's approach by noting that, at that time, ‘a function was usually thought of as being defined by an analytic expression; by the principle of the generality of analysis, which was widely and often tacitly accepted, such an expression was expected to be valid for all values, real or complex, of the independent variable’. In this article, we examine E675 closely. We have tweaked notation and condensed the working in places to reflect modern usage. At the end, we outline what is, with hindsight, needed to make Euler's arguments watertight: it is worth noting that all of his conclusions survive intact and that the intermediate functions of one and two variables that he introduces in E675 remain the key ingredients for much subsequent work on these integrals.

The so-called Archimedean iterative algorithm for calculating π uses a method involving the two equations

(1)

and

(2)

(Note that (1) is the harmonic mean.) Imagine two regular polygons each with the same number of sides, circumscribed and inscribed to a circle of diameter one. The larger one has perimeter a0, the smaller has perimeter b0. (Archimedes used hexagons with and b0 = 3, but regular polygons of any number of sides can begin the iterations.) Since the perimeter of the circle is π we have b0 < π < a0. Now consider regular polygons with twice the number of sides that circumscribe and inscribe the circle and call their perimeters a1 and b1 respectively. These can be calculated, from (2) and (3). (See [1] for a derivation or many explanations on the web.) Continuing in this way we generate the perimeters of inscribed and circumscribed regular polygons, and in each case the number of sides is twice the sides of the previous polygon.

In this paper we will discuss the lemniscate curve and show that its arc length can be bisected and trisected using classical ruler and compasses construction. The method dates back to 1718 when Count Giulio Fagnano (1682-1766) first published these constructions [1]. Fagnano was self-educated in mathematics and treated the subject as a hobby. Euler was impressed by his work on this topic and recommended his admission to the Berlin Academy of Science. Euler then generalised Fagnano's work on integrals. In addition, Fagnano was employed to assist in reinforcing the dome of Saint Peter's which was in danger of collapse. Pope Benedict IV rewarded him for his work by publishing his mathematical papers. Fagnano achieved considerable international fame as a mathematician, and rightly so given the outstanding contributions which he made on a number of different topics.

It is well known that there are precisely five integer-sided triangles which have equal area, Δ, and perimeter, P. These triangles are called equable Heronian triangles.

A proof of this result was given by Whitworth [1]. Since Whitworth's time, much attention has been given to triangles whose areas are integer multiples of their perimeters, for example [2, 3]. However, as this paper will show, Heronian triangles with areas less than their perimeters have some mathematical interest.

A continued fraction is an expression of the form

and we will denote it by the notation [f0, (g0, f1), (g1, f2), (g2, f3), … ]. If the numerators gi are all equal to 1 then we will use a shorter notation [f0, f1, f2, f3, … ]. A simple continued fraction is a continued fraction with all the gi coefficients equal to 1 and with all the fi coefficients positive integers except perhaps f0.

The finite continued fraction [f0, (g0, f1), (g1, f2),…, (gk–1, fk)] is called the k th convergent of the infinite continued fraction [f0, (g0, f1), (g1, f2),…]. We define

if this limit exists and in this case we say that the infinite continued fraction converges.

Cyclotomy is concerned with the division of a circle into a given number of equal segments, amounting to the construction of a regular polygon, say a q-gon, so that we need to deliver the angle α = 2π / q, or the length cos α. The construction is by Euclidean means, which make use of only ruler and compasses. Now, from given lengths, sums and differences of lengths are easy to obtain and, with the compasses, products and quotients of lengths can be obtained from similar triangles using parallel lines; indeed even the length can be obtained by applying the intersecting chord theorem to a circle with diameter 1 + a. However, there is not much else one can do with the compasses, so that the length cos α has to come from the real roots of a sequence of quadratic equations with ‘suitable’ coefficients — the meaning of being suitable will be made clear later.

Gauss made the first significant contribution to the classical theory of cyclotomy in Article 365 of his famous Disquisitiones Arithmeticae [1] in 1801. He showed that the construction is possible if q = p is a Fermat prime, that is a prime of the form 22n + 1; see §7 for a necessary and sufficient condition for q. The only known Fermat primes are p = 3, 5, 17, 257, 65537; the cases p = 3, 5 and 17 correspond to the construction of the equilateral triangle, the regular pentagon, and the regular heptadecagon, the details for which Gauss gave.

Recall the integral definition of the gamma function: for a > 0. By splitting this integral at a point x ⩾ 0, we obtain the two incomplete gamma functions:

(1)

(2)

Γ(a, x)is sometimes called the complementary incomplete gamma function. These functions were first investigated by Prym in 1877, and Γ(a, x) has also been called Prym's function. Not many books give these functions much space. Massive compilations of results about them can be seen stated without proof in [1, chapter 9] and [2, chapter 8]. Here we offer a small selection of these results, with proofs and some discussion of context. We hope to convince some readers that the functions are interesting enough to merit attention in their own right.