The following theorems are famous landmarks in the history of number theory.

Theorem 1 (Fermat-Euler): A number is representable as a sum of two squares if, and only if, it has the form PQ2, where P is free of prime divisors q ≡ 3 (mod 4).

Theorem 2 (Lagrange): Every number is representable as a sum of four squares.

Theorem 3 (Gauss-Legendre): A number is representable as a sum of three squares if, and only if, it is not of the form 4a (8n + 7).

This Article is on the discrete Fourier transform (DFT) and the fast Fourier transform (FFT). As we shall see, FFT is a slight misnomer, causing confusion to beginners. The idiosyncratic title will be clarified in §4.

Computing machines are highly efficient nowadays, and much of the efficiency is based on the use of the FFT to speed up calculations in ultrahigh precision arithmetic. The algorithm is now an indispensable tool for solving problems that involve a large amount of computation, resulting in many useful and important applications: for example, in signal processing, data compression and photo-images in general, and WiFi, mobile phones, CT scanners and MR imaging in particular.

Biomedical moral enhancement, or BME for short, aims to improve people's moral behaviour through augmenting, via biomedical means, their virtuous dispositions such as sympathy, honesty, courage, or generosity. Recently, however, it has been challenged, on particularist grounds, that the manifestations of virtuous dispositions can be morally wrong. For instance, being generous in terrorist financing is one such case. If so, biomedical moral enhancement, by enhancing people's virtues, might turn out to be counterproductive in terms of people's moral behaviour. In this chapter, we argue, via a comparison with moral education, that the case for the practice of biomedical moral enhancement is not weakened by the particularists’ stress on the variable moral statuses of the manifestations of our virtues. The real challenge from the particularists, we argue, lies elsewhere. It is that practical wisdom, being essentially context-sensitive, cannot be enhanced via biomedical means. On the basis of this, we further argue that BME ought to be used with great caution, for it may wrongly enhance, for instance, a terrorist financier's generosity, a robber's courage, or an undercover detective's honesty. Finally, we sketch how boundaries can be set on the use of BME, and address some potential objections to our position.

Some forty years ago my wife Christine and I considered the problems in this article, which involves a fair amount of computation. Computing facilities were not good then, so we considered instead the problems in [1] in which we showed, without using computers, that there were 64703 ways to make up £1 using coins; this was before the introduction of the 20p and £1 coins, and the ½p coin was in circulation. If Christine were still with us, this would have been another piece of joint work. I therefore dedicate this article to her memory.

The design of a coinage system depends on considerations we give to various criteria: for example, the number of denominations for the coins, the maximum number of coins required to deliver any given amount in a range, or the required number of coins averaged over the range; see also §3.