The history of mathematics is an abundant field for making surprising discoveries. Some luck is necessary, but whereas the literary tradition has been extensively explored by scholars, finding a nugget in the history of mathematics is not so difficult as turning up a new fragment by Jane Austen, or a lost piece by Mozart. In my case I struck lucky in the theory of matrices.

As is well known, the theory of matrices arose during the 1850s – a useful benchmark is the publication of Arthur Cayley's celebrated memoir of 1858, A memoir on the theory of matrices, a paper which generations of mathematicians have taken as signalling the beginning of matrix theory. Even today the opening paragraphs serve as a useful introduction to the subject.

This paper contains many opportunities to investigate, conjecture and verify results by writing simple computer programs, and the reader is strongly urged to participate in this way throughout.

For each positive integer n, let F(n) be the sum of the squares of the digits of n; for example,

In a recent issue of the Gazette S. Parameśwaran defined a positive integer n to be an S•P number if it is the product of the sum of its digits and the product of its digits; for example, 135 and 144 are S•P numbers because

His conjecture that there are only finitely many such numbers was proved in where it was shown that any S•P number is less than 1060. Here, we prove the following result.

The problem of finding rational integral numbers which could be made the sides of right triangles, is said by Proclus to have been solved by Pythagoras, by a method which is equivalent to the use of the formula,

where m is any odd integer.

Heath (1) quotes Proclus’ statement as follows:

“The method starts from odd numbers. For it makes the odd number the smaller of the sides about the right angle; then it takes the square of it, subtracts unity, and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side the hypotenuse.”

Note 74.11 from Nick Lord in the March 1990 Gazette looks at the fascinating Simpson's paradox starting from a complex number situation; and in the same issue John Haigh, reviewing Cox and Snell's Analysis of Binary Data, gives a (fictitious) example; but the fascination of the paradox stems from the real occurrences.

Formulae for Σl=1n ik when k is a positive integer are well known, and many methods for finding them have been described. This article gives an account of an approximate, but remarkably accurate, formula which I have found for the case k =1/2.