One of the long-standing problems of number theory, appealing to professional and recreational mathematicians alike, is the existence of Mersenne primes. These puzzling primes, for example 7, 31, 127 and 8191, are of the form 2P - 1, where p is itself a prime. The problem of their existence originated some 2400 years ago with the early Greek mathematicians' quest for the so-called perfect numbers, those like 6 and 28 which are the sum of their proper divisors. The connection was given in Euclid's Elements in 300 BC: if 2P - 1 is prime, then 2P-1(2P - 1) is a perfect number. Much later, Euler proved that all the even perfect numbers correspond to Mersenne primes. So the interest for many years has been in finding Mersenne primes. Only 39 are known, including several monsters discovered in recent years using thousands of PCs coordinated via the internet (see [1] for information on the project). Many of us would like to know if there is any way of predicting which exponents yield Mersenne primes and whether there is an infinite number of them.

It has been argued from time to time [1] that the special theory of relativity is suitable for first year undergraduates reading mathematics or physics at British Universities. The pedagogical value of this subject is considerable, since it introduces conceptual problems, whose solutions require clear thought, but only elementary mathematics. The teaching of this subject can therefore bring about a useful broadening in a student’s intellectual equipment.

The following paper is an attempt to explain the action of a class of three-dimensional mechanisms, first from a theoretical standpoint and afterwards to suggest some practical applications. One or two rather more general mechanisms are investigated than have previously been studied and well-known mechanisms are deduced as special cases of these. There is no modern textbook or recent publication dealing with these mechanisms and their properties are not generally known.

In the May 1954 issue of the Gazette Daniel F. Ferguson challenged readers to devise their own proof for what he described as a curious and somewhat pleasing sum (see [1])

In the Gazette, No. 248 (February, 1938), pp. 50-57, Mr. W J Dobbs gave a very interesting and complete descriptive account of the 18 equilateral “Morley” triangles of a given triangle. The article is especially memorable for its diagrams (1-8) and the present reader is asked to have them at hand for reference.

Cyclotomic polynomials, (xp - l)/(x-1), have (p-1) zeros exp(2πiq/p) (q = 1, 2, ... ,p - 1). We consider related polynomials whose zeros are given by cos(qπ/p) (q = 1, 2, ... ,p - 1) and indicate their relation to polynomials Pn (ζ) that satisfy the ‘double angle’ condition Pn(2ζ2 - l) = Pn(ζ)Pn(-ζ). The double angle condition is generalised to a chain of double angle conditions Qi(2ζ2 - 1) = Qi+1(ζ)Qi+1(-ζ), i = 1, 2, ...,n(Q1 ≡ Qn+1) For purposes of illustration, we will focus on P4(ζ), demonstrating how construction of the fourth order polynomial subject to the double angle condition yields an equation for cos . We also discuss more general features of the polynomials Pn (ζ). Furthermore, we illustrate how a simple geometrical construction can be used to derive a polynomial for cos .

As recently as ten years ago, professional mathematicians did most of their work at a desk, using paper and pencil. Today’s mathematician still sits at a desk, but facing a cathode-ray screen. The paper and pencil are still there, but a lot of the mathematician’s activities now involve use of the computer. In particular, powerful computer packages like Mathematica and Maple advertise themselves as systems for ‘doing mathematics on a computer’. This rapid transformation has changed the nature of doing mathematics in a fundamental way. The computer does not simply ‘assist’ the mathematician in doing business as usual; rather, it changes the nature of what is done. In particular, the logical structure of mathematical reasoning carried out with the aid of an interactive computer system is different from the structure of the more traditional form of mathematical reasoning. This article is a logician’s view of the ‘new maths’ of the computer age.

One of the oldest and best known of chessboard problems is to place the largest possible number of similar pieces on the board such that no two of these pieces are attacking each other. In his book Amusements in mathematics [1], Dudeney considered this problem on a generalised square chessboard containing n 2 cells, and proved that for rooks, queens and bishops this maximum number is equal to n, n and 2n – 2 respectively.

It is perhaps somewhat remarkable that in such frequently recurring and comparatively simple processes as those of multiplication and division of decimals there should still be no final settling down by teachers of Arithmetic into one standardized method and arrangement of work, shown by experience to be the simplest, safest, least laborious, and most easily understood and assimilated by the pupil. Indeed there seems a tendency in some quarters, after trying in succession all the various modifications of method and set-out which have been suggested, each as an improvement on its predecessor, to come round full circle to the older methods in use, say, fifty years ago.

There are two well known constructions of the real numbers from the rationals—namely the Dedekind cuts method in which a real number is defined as a class of rationals, and the Cantor–Cauchy completion method in which a real number is defined as an equivalence class of Cauchy sequences of rational numbers. In this article we give a new construction of the reals from the rationals in which a real number is defined as an equivalence class of sets of natural numbers.

A danger in teaching familiar material is oversimplification. In this article, we shall explore an example of this.

One of the authors observed a colleague (at another institution) introduce his students to the ordering of the real numbers by stating that < is a relation on ℝ which satisfies the following axiom system:

“For all x, y, z in ℝ:

1. 1. Exactly one of x = y, x < y, y < x holds.

2. 2. If x < y and 0 < z, then xz < yz.

3. 3. If x < y, then x + z < y + z.”

A few years ago, Shail [1] described earlier computer-assisted work on the Lester circle. Shail then showed how a Cartesian proof was ‘just feasible by hand’ but that it was more convenient to use computer algebra systems.

However, not relying on computer assistance does have the advantage that it forces us to search for improvements in our methods. In this instance, using vectors referenced to a special set of three coplanar axes enables us to avoid involved computations.