Since April 2015 there has been no legal requirement in the UK to purchase an annuity with pension savings [1] while for those who reach state pension age on or after 6th April 2016 the UK Government changed the state pension deferral arrangements [2]. The latter refers to an arrangement whereby a pensioner can receive an enhanced state pension by deferring its uptake for an arbitrary number of years. These two changes raise certain questions for prospective pensioners which are worthy of some mathematical consideration.

An annuity is a guaranteed income for life in exchange for a certain sum of money: the pension pot. An alternative to the annuity since April 2015 is a ‘draw down scheme’ in which the pension pot can be used almost like an ordinary bank account and money periodically withdrawn. These two choices arise from ‘defined contribution’ pension arrangements. By contrast ‘defined benefit’ work-based (company) pensions allow no such choice and are not considered further here apart from the special case of the UK state pension. With an annuity a further option to consider, and one which predates the 2015 changes, is whether to take payments that are fixed or index-linked to inflation. Only the UK state pension offers a late retirement enhanced pension if its uptake is deferred.

The behaviour of the divisor function d (n) is rather tricky. For a prime p, we have d(p) = 2, but if n is the product of the first k primes then, by Chebyshev's estimate for the prime counting function [1, Theorem 414], we have so that

for such n then, d (n) is ‘unusually large’ — it can exceed any fixed power of log n, for example.

In [2] Jameson gives, amongst other things, a derivation of Dirichlet's theorem, which shows that the mean-value of the divisor function in an interval containing n is log n. However, the result is somewhat deceptive because, for most n, the value of d (n) is substantially smaller than log n.

There are many named quadrilaterals. In our hierarchical classification in [1, Figure 10] we included 18, and at least 10 more have been named, but the properties of the latter have only scarcely (or not at all) been studied. However, only a few of all these quadrilaterals are defined in terms of properties of the sides alone. Two well-known classes are the rhombi and the kites, defined to be quadrilaterals with four equal sides or two pairs of adjacent equal sides respectively. The orthodiagonal quadrilaterals are defined to have perpendicular diagonals, but an equivalent defining condition is quadrilaterals where the consecutive sides a, b, c, d satisfy a2 + c2 = b2 + d2. Then it is possible to prove that the diagonals are perpendicular and that no other quadrilaterals have perpendicular diagonals (see [2, pp. 13-14]). In the same way tangential quadrilaterals can be defined to be convex quadrilaterals where a + c = b + d. Starting from this equation, it is possible to prove that these and only these quadrilaterals have an incircle (since this equation is a characterisation of tangential quadrilaterals, see [3, pp. 65-67]).

Finding the roots of cubic equations has been the focus of research by many mathematicians. Omar Khayyam, the 11th century Iranian mathematician, astronomer, philosopher and poet, discovered a geometrical method for solving cubic equations by intersecting conic sections [1]. In more recent times, various methods have been presented to find the roots of cubic equations. Some methods require complex number calculations, a number of techniques use graphical methods to find the roots [e.g. 2, 3] and some other techniques use trigonometric functions [e.g. 4]. The method presented in this paper does not use graphical techniques as in [2] and [3], does not involve complex number calculations, and does not require using trigonometric functions. By using this fairly simple method, the roots of cubic equations can be found in a short time without using complicated formulas.

The sums of powers of the first n positive integers Sp(n) = 1p + 2p + …+np, (p = 0, 1, 2, … )satisfy the fundamental identity

(1)

from which we can successively compute S0 (n), S1 (n), S2 (n), etc. Identity (1) can easily be proved by using the binomial theorem; see e.g. [1, 2]. Several variations of (1) are also well known [3, 4, 5].

In this note, we derive the following lesser-known recursive formula for Sp (n):

(2)

where denote the (unsigned) Stirling numbers of the first kind, also known as the Stirling cycle numbers (see e.g. [6, Chapter 6]). Table 1 shows the first few rows of the Stirling number triangle. Although the recursive formula (2) is by no means new, our purpose in dealing with recurrence (2) in this note is two-fold. On one hand, we aim to provide a new algebraic proof of (2) by making use of two related identities involving the harmonic numbers.

August Möbius introduced the system of barycentric or areal coordinates in 1827 [1, 2]. The idea is that one may attach weights to points, and that a system of weights determines a centre of mass. Given a triangle ABC, one obtains a coordinate system for the plane by placing weights x, y and z at the vertices (with x + y + z = 1) to describe the point which is the centre of mass. The vertices have coordinates A = (1, 0, 0), B = (0, 1, 0) and C = (0, 0, 1). By scaling so that triangle ABC has area [ABC] = 1, we can take the coordinates of P to be ([PBC], [PCA], [PBA]), provided that we take area to be signed. Our convention is that anticlockwise triangles have positive area.

Points inside the triangle have strictly positive coordinates, and points outside the triangle must have at least one negative coordinate (we are allowed negative masses). The equation of a line looks like the equation of a plane in Cartesian coordinates, but note that the equation lx + ly + lz = 0 (with l ≠ 0) is not satisfied by any point (x, y, z) of the Euclidean plane. We use such equations to describe the line at infinity when doing projective geometry.

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.

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.

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.

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.