# Colin Brian Haselgrove

### Quick Info

Chingford, Essex, England

Manchester, England

**Brian Haselgrove**was an English mathematician who worked in number theory and is best known for his disproof of the Pólya conjecture.

### Biography

**Brian Haselgrove**'s family lived in Chingford, Essex, but also had a large house on the east coast at Frinton, where they used to spend the summer. This was taken over by the government at the beginning of the war in 1939 and the family were evacuated to Somerset, in the southwest. Because of the move Brian went to Blundell's School, Tiverton for the last part of his schooldays.

He won a scholarship to King's College, Cambridge and went up in 1944. There were three Mathematics scholars at King's that year. One other was John Leech and the third was Berthold Hamburger, who caught tuberculosis and died in February 1946, in his fifth term. Brian also caught tuberculosis and was away from Cambridge for two years. He returned in 1947 and took Part II of the Tripos in 1948. As a research student, advised by Albert Ingham, he won the Smith's Prize in 1950 and became a Fellow of King's College and a Senior Assistant in Research (equivalent to a Senior Research Fellow elsewhere!) in the Mathematical Laboratory, which later became the Computing Laboratory. His dissertation was

*Some Theorems in the Analytic Theory of Numbers*. While at Cambridge, he published two articles in the Cambridge University Mathematics Society magazine

*Eureka*. The first was

*A Note on Fermat's Last Theorem and the Mersenne Numbers*in the January/February issue of 1949 and the second was

*Telepathy Experiment*in the October issue of 1950.

Haselgrove's first published research paper was on number theory. This was

*A connection between the zeros and the mean values of*$z(s)$ (1949) followed by

*Some theorems in the analytic theory of numbers*(1951),

*On Ingham's Tauberian theorem for partitions*(1952), and (with H N V Temperley)

*Asymptotic formulae in the theory of partitions*(1954). Among the results proved by Haselgrove in the 1951 paper is an extension of Linnik's method for proving the Goldbach-Vinogradov three prime theorem, namely that any sufficiently large odd integer is the sum of at most three primes.

However Haselgrove was involved in much more than number theory. He was working in the Mathematical Laboratory at Cambridge on the EDSAC 1 computer. Haselgrove's wife Jenifer and his fellow mathematics student John Leech were also working in the Laboratory at the time. In 1953 Haselgrove implemented the first computer program to carry out coset enumeration of subgroups of finite index in a finitely presented group. Leech writes in 1963 that Haselgrove's was:-

... the first attempt to carry out coset enumeration on a digital computer. It was prompted by discussion with Todd after experiments with enumeration of elements of groups by expressing them as words in the computer.An implementation of this procedure, designed by Todd and Coxeter for hand calculation, remains today as one of the main tools in computational group theory. The state of the art implementation of coset enumeration today is the package ACE (available both as a stand-alone program or within all major group theory systems). ACE includes an implementation of the HLT method of coset enumeration named after the work by Haselgrove, Leech and Trotter.

Another topic which Haselgrove worked on at this time was the problem of stellar evolution, collaborating with Fred Hoyle. They wrote a joint paper

*A mathematical discussion of the problem of stellar evolution, with reference to the use of an automatic digital computer*(1956) as a result of this joint work. W F Freiberger writes:-

The problem of stellar evolution is expressed, mathematically, by a set of non-linear partial differential equations describing the variation of density and temperature as a function of time and of distance from the star centre. This problem is tackled in the paper under review on the following assumptions: i) spherical symmetry of the star; ii) hydrostatic equilibrium; iii) negligible stirring of material except where convectively unstable; there, "well mixed". The computer employed was one of only moderate storage and speed, viz. the Cambridge EDSAC I; thus, considerable ingenuity was required in defining subsidiary variables.Jenifer Haselgrove relates an amusing anecdote from this collaboration:-

There was a rheostat up on the wall just outside the machine room door, which was used to adjust the voltage if the mains wasn't right. ... One night even the rheostat adjustment wasn't enough and Fred Hoyle, with whom my husband Brian Haselgrove was working on stellar evolution, rang up the electricity board and said "Can you hike your volts up a bit?". I can't remember if they did!Haselgrove spent four months at the California Institute of Technology in Pasadena in 1956-57, continuing work on computers with Fred Hoyle to model stellar evolution, and returned to take up a post as a Senior Lecturer in computing in Manchester University [2]:-

In 1957 [Max Newman] appointed Haselgrove ... to promote research in mathematical computing in Manchester. Some of the applications of computers were immediately obvious; many well-formulated problems in science and engineering which required numerical solutions could benefit directly from faster computation. This was particularly true in astronomy, a classical area for numerical work, where the research at Jodrell Bank called for major computing support. But there was also the prospect of using automatic machines to solve analytical and logical problems in new ways, which were not simply accelerated versions of existing methods.In 1958 Haselgrove published his most famous number theory result in

*A disproof of a conjecture of Pólya*. The conjecture of Pólya claims that for every $x$ > 1 there are at least as many numbers less than or equal to $x$ having an odd number of prime factors as there are numbers with an even number of prime factors. R S Lehman and W G Spohn had verified the conjecture for all numbers $x$ up to 800,000 but Haselgrove found a counterexample using methods based on those developed by Ingham with the help of computations carried out on the EDSAC 1 computer at Cambridge. He also verified the calculations using Manchester University's Mark I computer before publishing the results. In the same paper Haselgrove announced that he had also disproved a number theory conjecture of Turán. Another piece of work, undertaken at Cambridge, but published after he was working in Manchester was his

*Tables of the Riemann zeta function*, work which he undertook jointly with J C P Miller. Derrick Lehmer writes:-

This is the first table of the Royal Society to be produced automatically by electronic computers. Any other method would not have been feasible. A great many terms of the Euler-Maclaurin or Riemann-Siegel series were used to calculate each entry. For some cases 15 figure logarithms were needed as input data to obtain barely 6 decimal accuracy. The large zeros of Table IV required inverse interpolation involving 14th differences.Also in 1958 Haselgrove published

The tables will be very helpful in future exploration of inequalities involving certain numerical functions ...

*Applications of digital computers in mathematics*in

*The Mathematical Gazette*. Together with his wife, he published

*A Computer Program for Pentominoes*in the Cambridge University Mathematics Society magazine

*Eureka*in October 1960.

Haselgrove's work in Manchester is described in [2]:-

Brian Haselgrove's research interests ranged over many areas of mathematics both pure and applied. ... at Manchester he published papers on Dirichlet functions and the Riemann hypothesis, ray paths in the ionosphere, numerical integration using quasi-random numbers, two-point boundary-value problems, and some geometrical puzzles. The paper on boundary-value problems illustrates an early stage of what was to become a major interest of numerical analysts at Manchester and elsewhere, the development of general algorithms. This work involves the elucidation of classes of mathematical problems which are suitable for solution by a standardised approach. The solution methods have to be studied analytically and tested on an extensive range of problems, to determine their applicability and limitations. In the early 1960's the potential for general solvers was becoming apparent, but the programming languages available did not provide enough flexibility for implementation.At Manchester Haselgrove continued his interest in coset enumeration, and implemented a new version of the Todd-Coxeter procedure on the Mercury computer there in 1960. He also studied numerical analysis. In

*The solution of non-linear equations and of differential equations with two-point boundary conditions*(1961) Haselgrove suggests general iterative techniques, based on an $n$-dimensional extension of the Newton-Raphson process. His tables of Dirichlet L-functions were deposited with the Royal Society but not published although he did publish (with D Davies) the paper

*The evaluation of Dirichlet L-functions*(1961) describing the methods used. In

*A method for numerical integration*(1961) he gave the following summary:-

In this paper we shall give an account of some methods developed for the numerical evaluation of multidimensional integrals. These methods are based on the theory of Diophantine approximation. They are suitable for some problems for which the Monte Carlo method is commonly used and, like the Monte Carlo method, are well fitted for use with an electronic digital computer.Also in 1961, together with his wife Jenifer and with R Jennison, he published

*Ray paths from a cosmic radio source to a satellite in orbit*. The paper begins:-

A receiver mounted on a satellite in orbit above the maximum of the F2 layer can receive radiation of frequencies that are totally reflected by the ionosphere. Two effects of reflection in the upper part of the ionosphere are discussed in this paper; both occur particularly when the satellite enters or leaves a region in which it can receive radiation from a point source.Haselgrove's teaching at Manchester is described in [2]:-

On the teaching side, Haselgrove initiated a postgraduate Diploma in Computing in 1959 in collaboration with Tony Brooker and other members of the Computing group. In 1964 Computer Science became a separate Department, and the postgraduate course began to concentrate on the more mathematical aspects of computing, retaining some options from Computer Science. Student numbers were relatively small at first, but the course provided a source of research students in Numerical Analysis. Haselgrove also introduced an undergraduate course in numerical methods and computer programming, but it was not possible to include realistic practical work until autocodes were designed.He died in Manchester in May 1964 from an infiltrating brain tumour, which had been causing him minor epileptic fits for several years. Perhaps because of his illness he did not publish much research himself in these last years but he did supervise several research students.

### References (show)

- J Leech, C Brian Haselgrove, Personal communication (3 November, 2006).
- Numerical Analysis at the University of Manchester, 1957-1979, in
*Manchester Centre for Computational Mathematics, Annual Report: January-December 2003, Numerical Analysis Report No.***449**(May 2004), 20-22.

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Written by Jenifer Leech and Edmund Robertson

Last Update November 2006

Last Update November 2006