Alan Turing’s mathematical interests were deep and wide-ranging. From the beginning of his career in Cambridge he was involved with probability theory, algebra (the theory of groups), mathematical logic, and number theory. Prime numbers and the celebrated Riemann hypothesis continued to preoccupy him until the end of his life. As a mathematician, and as a scientist generally, Turing was enthusiastically omnivorous. His collected mathematical works comprise thirteen papers, not all published during his lifetime, as well as the preface from his Cambridge Fellowship dissertation; these cover group theory, probability theory, number theory (analytic and elementary), and numerical analysis. This broad swathe of work is the focus of this chapter. But Turing did much else that was mathematical in nature, notably in the fields of logic, cryptanalysis, and biology, and that work is described in more detail elsewhere in this book. To be representative of Turing’s mathematical talents is a more realistic aim than to be encyclopaedic. Group theory and number theory were recurring preoccupations for Turing, even during wartime; they are represented in this chapter by his work on the word problem and the Riemann hypothesis, respectively. A third preoccupation was with methods of statistical analysis: Turing’s work in this area was integral to his wartime contribution to signals intelligence. I. J. Good, who worked with Turing at Bletchley Park, has provided an authoritative account of this work, updated in the Collected Works. By contrast, Turing’s proof of the central limit theorem from probability theory, which earned him his Cambridge Fellowship, is less well known: he quickly discovered that the theorem had already been demonstrated, the work was never published, and his interest in it was swiftly superseded by questions in mathematical logic. Nevertheless, this was Turing’s first substantial investigation, the first demonstration of his powers, and was certainly influential in his approach to codebreaking, so it makes a fitting first topic for this chapter. Turing’s single paper on numerical analysis, published in 1948, is not described in detail here. It concerned the potential for errors to propagate and accumulate during large-scale computations; as with everything that Turing wrote in relation to computation it was pioneering, forward-looking, and conceptually sound. There was also, incidentally, an appreciation in this paper of the need for statistical analysis, again harking back to Turing’s earliest work.
R-prime numbers of degree k
