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

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

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":

This led to an interest in exact integer programming. (See H. Flanders,

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

Interested viewers can also experiment with some 3x+1 type programs.

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. (See [paper].)

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 turned to a study of the diophantine equation x

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.

Both remain static.

I also enjoy writing bc number theory programs and BCMath programs.

- Adam Segal, Sandy Segal and Rima Segal (31/7/84)

- Conference dinner, Château Larose Trintaudon: Gerhard Niklasch, Keith Matthews, Chris Smyth, Eira Scourfield, Francis Rayner, Radhakrishnan Nair, Andrew Pollington

* Last modified 12th May 2023*