This article sets out to explore some of the connections between two seemingly distinct mathematical objects: trigonometric functions and the integer sequences composed of the Fibonacci and Lucas numbers. It establishes that elements of Fibonacci/Lucas sequences obey identities that are closely related to traditional trigonometric identities. It then exploits this relationship by converting existing trigonometric results into corresponding Fibonacci/Lucas results. Along the way it uses mathematical tools that are not usually associated with either of these objects.

The number π (as all—or very nearly all—the world doth know) signifies the ratio of the circumference of a circle to its diameter and is also equal to the ratio of the area to the square of the radius. This suggests two generalisations of π which apply to plane figures of any shape:

What sentence in this Mathematical Gazette contains seven a’s, four c’s, thirty-two e’s, eight f’s, five g’s, ten h’s, twelve i’s, three l’s, three m’s, fifteen n’s, eight o’s, one q, seven r’s, twenty-seven s’s, twenty-five t’s, four u’s, eight v’s, seven w’s, four y’s & two z’s?

Problems concerning the set

of numbers which are representable as sums of two squares have a long history. There are statements concerning W in the Arithmetic of Diophantus, who seemed to be aware of the famous identity

which shows that the set W is ‘multiplicatively closed’. Since a square must be congruent to 0 or 1 (mod 4), it follows that members of W cannot be congruent to 3 (mod 4). Also, it is not difficult to show that a number of the form 4k + 3 must have a prime divisor of the same form dividing it an exact odd number of times. However, the definitive statement (see, for example, Chapter V in [1]) concerning members of W, namely that they have the form PQ2, where P is free of prime divisors p ≡ 3 (mod 4), was first given only in 1625 by the Dutch mathematician Albert Girard. It was also given a little later by Fermat, who probably had a proof of it, but the first published proof was by Euler in 1749.

It must be remembered that one does not get a leap year at the turn of a century unless the year is divisible by 400.

A few years ago I tried making a model of an object from its photograph. Realising that, of course, one could not take direct measurements from the photograph, I had to try to guess the relative dimensions of the object. I ended up, unfortunately, with a model which did not seem to be in the correct proportions. Would it have been possible, I wondered, to find the precise ratios of the different lengths?

Call the ends of a ruler A and Z, and (towards A) place on it a mark X. Draw a line BC the length of AX, and a square BCDE. Erect a perpendicular bisector of BC. With centre C and radius CE draw a circle. Now place the ruler (Fig. 1) so that AZ passes through B, with A on the perpendicular bisector of BC, and with X on the circle. Then ∠BAC = π/7.

If you enjoy magic squares and their unusual and fascinating numerical properties, you are in good company. Indeed, many a professional mathematician, scientist and amateur has shared the same interest. Technical research work on the properties of magic squares is available aplenty in the literature or in books concerned with the recreational aspects of mathematics. Studies of magic squares naturally lead to some elements of group theory, of lattices, of Latin squares, of partitions, of matrices, of determinants, and so on …

By definition, the general n x n magic square is a square array of n2 numbers {aij; i, j = 1,2,... , n) whose n rows, n columns, and two main diagonals have the same sum s. It is therefore natural to think of a magic square as a matrix and we shall do this in the present note, simply referring to it as a magic matrix.

In the performance of digital communication systems over fading channels, the error analysis is typically modelled using a Gaussian probability distribution. One function central to the analysis is what engineers routinely refer to as the (Gaussian) Q-function and is defined by

1

This is the canonical representation used for the function. In this paper a number of derivations of an important alternative representation for the Q-function known as Craig's formula will be given.

This article concerns the evaluation of the ‘logsine’ integral

We shall encounter it in several guises. Indeed, standard integration techniques used below readily show that (1) has the same value as the following integrals:

En passant, it is worth noting that forms (6) and (7) are the best behaved for numerical integration.

I first met the logsine integral as a callow youth in that strange hinterland of results that you may not have met at school but were not guaranteed to meet later on either.

The defining property of a rectangle is that it is a quadrilateral whose angles are all equal (or are right angles, which amounts to the same thing).

The opposite sides of a rectangle are equal. If a and b are the lengths of adjacent sides of a rectangle, such a rectangle can be described as an abab rectangle (or an (ab)2 rectangle). The area of the rectangle is ab.

As far as I understand the latest version of the national curriculum and the new A level syllabuses, which make no serious reference to irrational numbers, students leaving school will not necessarily know that there are two sorts of rectangle: those coverable by a set of identical squares, which can therefore be drawn on squared paper; and those not, such as the golden rectangle. An (ab)2 rectangle is coverable by squares if, and only if, a/b is rational.