With one exception, namely {2, 3, 5, 7}, it is impossible to have four consecutive primes p1, p2, p3, p4 with p4 - p1 < 8. An interval of seven or less cannot contain more than three odd numbers unless one of them is a multiple of three. On the other hand, groups of four primes p, p + 2, p + 6, p + 8, usually called prime quadruplets, are fairly common. The first is {5, 7, 11, 13}, followed by {11, 13, 17, 19}, {101, 103, 107, 109}, {191, 193, 197, 199}, {821, 823, 827, 829} and so on. Just as with prime twins, pairs of primes p, p + 2, it is conjectured that the sequence of prime quadruplets goes on for ever. Indeed, the apparently simpler prime twin conjecture is currently an unsolved problem of mathematics although in 1973, Jing-Run Chen proved a weaker form: There are infinitely many primes p such that p + 2 is either prime or the product of two primes (See Halberstam & Richert [1]). A similar result holds for quadruplets [1, Theorem 10.6]: There exist infinitely many primes p such that (p + 2) (p + 6) (p + 8) has at most 14 prime factors. The prime quadruplet conjecture would then follow if we could reduce ‘14’ to ‘3’ but this seems be a problem of extreme difficulty.

In this paper a magic cube of order n is a 3-dimensional matrix Qn = {qn(i,j,k); 1 ≤ i,j,k ≤ n} containing natural numbers 1,2, ... n3 such that the sums of the numbers along each row (n-tuple of elements having the same coordinates on two places) and also along each of its four great diagonals are the same. This contrasts with a previous article [1] which considered magic cubes with no requirement for the diagonal sums.

A new and general proof of theorem [1] is presented, which accommodates directed angles and covers all possible configurations of line and conic. A sign convention for the use of directed angles in this setting is developed, and examples are given showing how this fundamental theorem can be used to establish, much more directly and succinctly, the many tangent/focus properties of conies.

How big is n!? For example, to pluck a number out of thin air, what is the order of magnitude of 272! ?

This is the third note of a series on the factorial. The two previous notes [1] and [2] dealt with the factorial function x! while here we only consider n! when n is an integer, but its main results also apply to non-integer arguments. This note is longer and more elaborate than the previous ones, but the principal result is in the opening section. Only aficionados need go any further.

Figures 1–4 set the scene, with configurations of tangent circles that may perhaps be new. The labels are curvatures: the curvature or bend being the reciprocal of the radius. By convention, a circle that surrounds another has negative bend and radius. In the first part of this paper I give some easy formulae for configurations like these (see Note a). I show how we can develop fractal structures—I call them froths—having an infinity of tangent circles in each ‘triangular’ region; and give methods for finding integral froths. The second part extends these ideas to systems of tangent spheres, which may also be integral. Throughout, an integral object is one whose bend is an integer, circles and spheres are generally referred to by their bends, and examples are marked with bullets (•).

In his thought-provoking Note 82.45 (November 1998) about products of vectors, Bob Ardler asks (p. 454) “Why does dot thrive in n dimensions, but cross only in 3?” and (p. 456) “Which, if any, generalisations of the cross product have been pursued?”

Both of these matters have been treated fairly recently in several journals, including the Gazette, but the subject was first discussed and completely settled in 1942 by B. Eckmann in [1] (see especially pp. 338-339); cf. also [2].

From the date of my own A level studies (many moons ago!) I have been dissatisfied with the presentation of the law of elastic restitution. The title Newton’s ‘Experimental’ Law implies that it is a purely empirical phenomenon, yet if there is a wide range of conditions in which it is approximately valid (and it isn’t worth using at all if this isn’t true), then there must be some reason for it. Equally unsatisfactorily, the law implies loss of energy in most circumstances. Even in my school days I could see that the usual argument about conversion into heat, sound etc didn’t really hold water, as this process couldn’t conceivably be instantaneous. A recent article by O’Connor [1] stimulated me to crystallise my thoughts on the subject [2], and I was later able to develop a simple and plausible mechanical model exhibiting restitution, which also conserved energy [3]. My presentation in [3] was mainly focussed on the physical consequences of the model, and solution of the differential equations involved was numerical. Yet, in the simplest case, the model is amenable to fairly elegant exact analytical techniques, which it is the aim of this paper to present. Apart from the interest of the model, the methods used are a nice illustration of the role of normal modes in the solution of simultaneous linear differential equations—a topic approachable by A level students of Further Maths, or by first-year undergraduates.

Let us first consider the 2 x 2 matrices and Their sum and product are given by

Here the entries a, b, c, d, e, f, g, h can come from a field, such as the real numbers, or more generally from a ring, commutative or not.

For a natural number n ≥ 3 and a positive real number ξ let Pn (ξ) denote the area of the regular n-gon with side of length ξ and let P(ABC … Z) be the area of a convex polygon ABC … Z.

In 1996 Josip Kovačević observed [1] that in every triangle ABC with sides a, b and c and with the angle A equal to π/3 radians the following relation holds:

Similarly, if the angle A is equal to 2π/3 radians then

On the other hand, when the angle A is π/2 radians, then Pythagoras’ theorem implies which could be rewritten as so

because P(ABC) = bc/2. The last observation is the starting point for Veljan’s results in [2] and [3] that

when the angle A is π - 2π/n radians and that

when the angle A is 2π/n radians.

Recursively defined sequences (in other words difference equations) appear in modelling systems changing in discrete time steps. The asymptotic behaviour of such sequences tells us information about the state of the modelled system after a long time. The simplest difference equations are linear, but the standard solving method of even these simple equations involves eigenvalue and eigenvector computation, complex numbers and linear algebra.

On 15 March 1997, when boarding an aeroplane bound for Spain, I was handed a copy of the Independent newspaper. One page of the paper was devoted to the Hale-Bopp comet; from two pieces of information about the comet’s orbit I set out to find out as much as I could about the orbit using A level mathematics, and in some cases further mathematics. I was interested in finding the fastest and slowest speeds of the comet, and the size of its orbit in relation to the planetary orbits. I also found speeds and times at other points on the orbit, where distances from the sun could be calculated. These findings, along with an approximate calendar for the next orbit of the comet, are presented in a diagram at the end of this article. During the course of my working, I made two discoveries of a more general nature. The first of these is that the radius of curvature of a conic at the point nearest to a focus is equal to the semi-latus rectum. The second was one of Kepler’s laws, that for an elliptical orbit T2 ∝ a3, where T is the time of a complete orbit, and a is the semi-major axis.

One of the more interesting uses of mathematics and statistics is their application to everyday situations in assisting people to make informed choices about their courses of action. Nowhere is this more evident than in the application of basic mathematical principles to the range of gambling opportunities. Indeed, a new university statistics course that I give in Australia on Gambling and Sport has attracted an enrolment of over 400 students in its first year alone. In this subject we discuss the intricacies of games such as Craps, Keno, Roulette, poker machines, horse racing, and a wide variety of lotteries. The sports content of the course relates largely to using mathematics to formulate optimal playing strategies.

The Belgian Jesuit, Gregory of St Vincent (1584 - 1667), along with Cavaleri, Fermat and Descartes was amongst those who prepared the way for the calculus of Newton and Leibniz. He was the author of one long book Opus Geometricum (1647) in which he gave a more precise account of progressions and limits than had been previously available and which Leibniz particularly valued. The term ‘exhaustion’, for the limiting process of the Greeks, is due to Gregory of St Vincent. This book contains a section on the hyperbola in which Gregory studied the area between the curve and an asymptote and this section provided the basis for Anthony de Sarasa's claim in 1649 that the area here was like a logarithm.