In the 1930s several mathematicians, principally Alonzo Church (1903-1995), Stephen Kleene (1909-1994), Emil Post (1897-1954) and Alan Turing (1912-1954), began investigating the notion of effective calculability. (A function from natural numbers to natural numbers is effectively calculable if there is some finite rule or mechanism which will calculate the value of the function for any natural number.) Central to this activity was the notion of recursiveness. Loosely, recursion is a process of defining a function by specifying each of its values in terms of previously defined values.

In this article we show how to obtain by recurrence all primitive Pythagorean triples from the single basic (4, 3, 5) triple. It is all done by repeated application of two integer unimodular transformations that leave the indefinite metric x2 + y2 - z2 invariant, together with the additional transformations (i) that change the sign of x and (ii) that change the sign of y. These alone would restrict the triples to those in which x is even, y is odd and z is positive, so we then include two further transformations (iii) that exchange x and y and (iv) that change the sign of z, thereby accounting for all primitive triples. There is a bonus in extending the method beyond the first objective in that the theory, which we outline, of the Lorentz transformations involved not only provides necessary background explanation of the first part, but it also enables us to show how to solve by recurrence Diophantine equations of the form x2 + y2 = z2 + n, where n is any integer.

In this article we usually omit limits since, in all the cases which we consider, the integration variables are in the interval (-∞, ∞).

Few mathematics students in their first year at university would be likely to list linear algebra as part of their tool box for solving problems in the calculus class. Asked to show that

the majority, at this stage of their careers, would probably not realise that techniques which they had recently learned in linear algebra had any bearing on the problem. The computer experts in their midst might even argue that there was little merit in learning how to perform such calculations since high speed computers had eliminated the need to develop the technical skills which only a few years ago would have been regarded as essential for anyone aspiring to become a mathematician.

One of the questions that I set my students in a recent A level revision lesson was to express (√2 - 1)6 in the form a + b√2, where a and b are integers. While they were calculating the answer, it occurred to me that, as a consequence of the fact that (√2 - 1)n → 0 as n → ∞ (since 0 < √2 - 1 < 1 ), a and b could be used to give us a ‘good’ rational approximation to √2. Within a minute or so everyone agreed that (√2 - 1)6 = 99 - 70√2, allowing us to write √2 ≈ 99/70.

This article explains the variety of patterns seen in sequences defined by for bilinear functions of the form . The initial confusion of very different patterns can be understood by generalising from the real numbers to the whole complex plane, and simplified further by projecting each number from the plane onto the surface of a sphere. The sphere is finally projected onto a Mercator map on which the iterative sequences take on the simplest possible pattern, creating a link with the problem of rhumb-line navigation. The examples given below are typical, and should first be tried as a reminder of what can happen.

Almost everybody has a light in their house that can be switched on and off from two or more different places. If your electrician set up the wiring properly, you are able to toggle the light with every single switch, independent of the others' current configuration.

We may think of the switches as implementing a parity function. If we label the two positions of a switch by 0 and 1, the light is on if the sum of switch positions is even and off if this sum is odd (or vice versa). So in the common case of only two switches, ‘light on’ is just the binary relation xor. (Not all real-life installations are of this kind. We also find the binary and implemented in a few houses—which can be very annoying, and even dangerous, at nights.)

In this paper we analyse a game that was published in the 1996 Burger King Kid’s Club calendar. This calendar included an activity for each month, and for the month of September the activity was called the Mountain Bike Rally. To begin the game, each player places a marker on day 1. A turn consists of each player flipping a fair coin. Each player whose coin comes up heads moves his marker forward one day. If a player’s coin shows tails, then he moves his marker back one day, unless his marker is on day 1, in which case it stays there. The object is to be the first player to reach square 31. We never played the game because it was clear that it would take a long time to complete.

This number arises naturally, of course, as the hypotenuse of a right triangle with legs of length 1. Notoriously, √2 was found to be irrational by the Pythagoreans in around 500 BC. Their mathematics and philosophy was based upon natural numbers and small number ratios, and thus this discovery would have been very troubling: the Pythagoreans wouldn’t be the only ones to react badly to the imposition of the irrational.