Mathematical Interests, Past and Present
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 one of 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,
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
In September 1962, I left Sydney on the Oriana for Southampton and thence to Cambridge. I subsequently wasted my first year at Cambridge studying this area, not knowing that fellow student Alan Baker and also Harold Stark were also attacking this problem. Kurt 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 ℤ 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. (See Waring's problem for Fq[x], R.M. Kubota, Dissertationes Math. 117 (1974), 60 pp.)
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
This led 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 an account 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 BCMATH program challenge.
Interested viewers can also experiment with some 3x+1 type programs.
I worked with George Havas, applying the modified lattice basis reduction algorithm (MLLL) to a number of problems.
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 small 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 and derived 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 was published in Experimental Mathematics.
As an offshoot of this work, I was able to determine the shortest multipliers for m consecutive Fibonacci numbers.
Subsequently, 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 coded a refinement of the LLL gcd algorithm, which usually gives shorter multipliers. (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.)
I turned to a study of 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 papers 1, 2 and 3.)
I wrote some related BCMath programs patz.html, binary.html and thue.html.
From mid 2007-late 2011, I studied the nearest square continued fraction of A.A.K. Ayyangar and other half-regular continued fractions such as the optimal continued fraction of Wieb Bosma. Initially this was prompted by collaboration with Jim White who was attached to the ANU. Subsequently we were joined by John Robertson and several publications resulted. See my continued fractions page. In late 2011, I studied a conjecture of Andrej Dujella, with John Robertson and Jim White. This was followed by a converse of theorems of Nagell and Stolt with John Robertson and Anitha Srinivasan. Research in this area is continuing as of 2017.
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 remains static, but I add to CALC from time to time.
I also enjoy writing bc number theory programs and BCMath programs.
Study Leaves, short visits and overseas conferences
Nottingham: (July 1972/June 1973)
Cambridge: (July 1979-June 1980)
Rochester (July-December 1984)
Eichstätt (July 1993)
Journées Arithmétiques 18 (September Bordeaux),
Cambridge (July - December 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 5th November 2016