Mathematical Interests, Past and Present

Early Years (1958-63)

The first number theory book that I studied was L.E. Dickson's "Introduction to number theory". It was available as a cheap Dover book at the time. It must surely rank as the dullest book on number theory ever written. In 1960, I persuaded Prof. Clive Davis to replace a course on projective geometry which used to be given by H.K. Powell, by a course on number theory. This was the first time number theory had been taught here. Clive's course was very stimulating. I went on to study E. Landau's Primzahlverteilung and struggled through Atle Selberg's proof of the Prime Number Theorem and H.N. Shapiro's proof of Dirichlet's theorem, as well as other papers on elementary analytic number theory.
My first discovery, in 1962, after perusing the relevant section in Hardy and Wright's book, was the prime-producing polynomials of Euler and their connection with imaginary quadratic fields with class number one. I soon learned that D.H. Lehmer had already obtained this result in Sphinx, Bruxelles, 1933. Harvey Cohn's A Second Course in Number Theory 1963, also had an account.
In September 1962, I left Sydney by boat for England and Cambridge. I subsequently wasted my first year at Cambridge studying this area, not knowing that fellow student A. Baker and also H. Stark were also attacking this problem. K. Heegner had in fact solved the problem in Diophantische Analysis und Modulfunktionen, Math. Zeit. 56 (1952).
I eventually bowed to my supervisor Harold Davenport's wishes and looked at polynomials over which are near to kth powers and Waring's problem for polynomials over a finite field. The latter was simply a matter of studying Davenport's "Blue Book": Analytic Methods for Diophantine Equations and Diophantine Inequalities (Ann Arbor, 1962) and translating to the discrete situation; but there was always the hope of improving on the classical estimates. I also studied Hardy and Littlewood's Partitio Numerorum III, where the Generalized Riemann Hypothesis is used to study Goldbach's problem. However I did not succeed in obtaining an appropriate error term for the analogue of Vinogradov's three primes theorem. This was done by David Hayes (see Additive Number Theory of Polynomials over a Finite Field, D.R. Hayes and G.W. Effinger, OUP 1991.) I returned to Brisbane in mid 1964 and wrote up my Waring's problem work as a University of Queensland MSc. I didn't publish the thesis, as I considered it a rather technical piece of work and also Davenport told me I had been anticipated by R.M. Kubota, (a student of D.J. Lewis) who had obtained a slightly better result.

(1965-1980: See my publication list.)

More Recently (1981+)

I spent a considerable amount of time helped by collaborators, studying certain aspects of the 3x+1 problem (Collatz problem) and related problems. My starting point was a paper of H. Möller which related the 3x+1 problem to 2-adic analysis.
This lead to an interest in exact integer programming. (See H. Flanders, Scientific Pascal, pp. 175-185, 342-357, Reston 1984, (Second Edition Birkhäuser, 1995).)
I have written a survey (pdf file 206K) of work done with various collaborators on the generalized 3x+1 problem since 1981, presenting a point of view which makes the 3x+1 problem appear as part of a more general class of problems which are equally tantalizing, but about which one can make accurate predictions.
I offer $100 (Australian) for the proof of a conjecture mentioned in Example 2.6 of my survey. (Also see the BCMATH program challenge.)
Interested viewers can also experiment with some BCMATH programs below.
(See 3x+1, 3x+371, generalized 3x+1 functions and their Markov matrices.)

1992+

Some of my more recent research has been with George Havas, applying the modified lattice basis reduction algorithm (MLLL).
At George's suggestion in 1992, I wrote a program to use the MLLL to prevent coefficient explosion in the obtaining the Smith canonical form of an integer matrix. The idea is simple but effective - bring in MLLL when the coefficients exceed a prescribed limit. This usually results in very small row vectors, often unit vectors and is implemented in my CALC program. In practice one gets unimodular P and Q with smallish entries such that PAQ=SNF(A).
The exact integer arithmetic involved in the underlying Gram-Schmidt process slows things down for large matrices of size say bigger than 150x300.
We then turned to the problem of finding small multipliers for the extended gcd problem. We believe we have the best algorithm in existence - a variant of the LLL algorithm. We have also extended the method to finding the Hermite normal form of an integer matrix. A paper on these topics has appeared in Experimental Mathematics.
As an offshoot of this work, I was able to determine analytically the shortest multipliers for m consecutive Fibonacci numbers. (See [paper].)
Recently, prompted by a question from J.H. Silverman, I realised that the above Hermite normal form algorithm provides a simple method for getting short solutions to AX=B. (See manuscript.)
Subsequently I found a refinement of the LLL gcd algorithm, which usually gives shorter multipliers. (See manuscript.)

2000+

In 2000/2001, I studied the diophantine equation x2-Dy2=N, D > 1 and more generally ax2+bxy+cy2=N, D=b2-4ac > 1, D not a perfect square. (See 1, 2 and 3.)
Also BCMath programs patz.html, binary.html and thue.html.
Recently, with John Robertson, we were able to settle an outstanding problem in the above papers 1 and 3, regarding primitivity of the solutions (x,y). (See manuscript.)
I studied a variant of Shanks' algorithm for the continued fraction expansion of logba. (See On Shanks' algorithm for computing the continued fraction of logba, Journal of Integer Sequences 5 (2002) article 02.2.7. Also see the online BCMath program.)

Computer programs

As an outcome of research and teaching, I have written two exact arithmetic calculator programs: an infinite precision number theory package called CALC and a matrix program called CMAT.
I also enjoy writing bc number theory programs and BCMath programs.

Study Leaves

  • Nottingham: (1972/73)
  • Cambridge: (1979/80)
  • Rochester (1984)
  • Brown/Eichstätt/Journées Arithmétiques 18 (Bordeaux)/Cambridge (1993)
  • Technisches Universität, München (August 1997); 13th Czech-Slovak Number Theory Conference, Ostravice (September 1997)
  • University of British Columbia mathematics department (July/August 2004)
  • Victoria University of Manchester mathematics department (September 2004)

    Last modified 11th January 2006