Number Theory

Diophantine equations other topics
general Thue and Thue-Mahler p-adic Diophantine approximation
exponential elliptic etc. powerful number sets with small diameter
recurrences binomial abc conjecture 3n+1 conjecture

p-adic Diophantine approximation


Diophantine equations - general


Diophantine equations - exponential


Diophantine equations - recurrences


Diophantine equations - Thue and Thue-Mahler


Diophantine equations - elliptic, etc.

I've always objected against the terminology of integral points on elliptic curves. This is not a well defined concept. One can speak of rational points, as they are kept intact under birational transformations. Integral points are not. One should therefore speak of integral solutions to elliptic equations or something like that.

Diophantine equations - binomial

I've always wondered why, in the overwhelming amount of papers on binomial coefficients, there is so little about the problem of binomial coefficients being equal (I guess, in view of my more recent work on MD5, I should now call this binomial collisions). Here's what I did about it.

powerful number sets with small diameter

This work grew out of a question Frits Göbel once asked me about how exceptional the case of 48, 49, 50 is, being consecutive numbers all divisible by high powers. I started working on it, got stuck, and gave the problem to Christiaan for a Master Thesis. He did a wonderful job, resulting in these two papers.

abc conjecture

In september 1985 the abc conjecture was formulated at a conference in the UK, in which my supervisor professor Tijdeman participated. Immediately after his return in Leiden he told me about the conjecture, as at the time I was working (for my PhD thesis) on solving x + y = z in S-integers. So to compute a small list of good examples, I just had to add a few lines of code to the program I already had, and to run it again. This immediately produced the record example at the time: 112 + 32 56 73 = 221 23, with quality 1.62599.
One day later, on September 20, 1985, I went to professor Tijdeman to show him the fresh record. His reaction was to grab his pocket calculator and to type in the numbers, saying: "I want to see the miracle happen."
This small list was published in my 1987 Journal of Number Theory paper (Section 5E) and in my PhD Thesis (Chapter 6, p. 130).
In 1987 Eric Reyssat from Caen found a better example. Mine is still in second place. For up to date lists of abc-examples see Bart de Smit's table and Abderrahmane Nitaj's tables.

In 1999 Niklas Broberg came with good examples for the uniform algebraic number field case. From his work it was immediately apparent to me that many of his examples have to do with exceptional solutions to Diophantine equations and/or exceptional values in recurrence sequences. And that also immediately brought to mind the possibility that the well known exceptional zero in the Berstel ternary recurrence was a good candidate for being a very good abc-example. This turned out to be the case, giving me again a record example, as can be read in my note A new extreme abc-example, published only on the web in 1999.
Then the history repeated itself, when a few years later Tim Dokchitser found a better example, leaving my example again at the second place, where it still is. Dutchmen call this being "eternal second" the Zoetemelk-syndroom. See my web note A newer extreme abc-example due to Tim Dokchitser, 2003.

At a conference in Edinburgh in 1996 I attended a talk by Dorian Goldfeld about elliptic curves, in particular Frey-Hellegouarch curves, with large Tate-Shafarevich group Ш (the proper spelling for "Shafarevich" is "Шафаревич", hence the symbol for this group). As Frey-Hellegouarch curves are linked to abc-examples, it turns out that good abc-examples lead to exceptionally large Ш's. In the coffee break after the lecture I started up my brand new fancy laptop computer (running Apecs on Windows 95), and in a few minutes I had a record size Ш. The following paper is a result of the subsequent research. My wife still remembers me sitting at the kitchen table with my laptop and getting excited each time a new large Ш popped up.
Needless to say that shortly after this paper appeared, others, notably Abderrahmane Nitaj, found better examples and took over the record.

The newest development in this area is the emergence of the xyz conjecture, in a 2009 paper by Lagarias and Soundararajan.
For abc-examples (coprime positive integer triples a, b, c such that a + b = c) they introduced the quality function Q*(a,b,c) = 3/2 (loglog c) / (log P), where P is the largest prime factor of abc. The xyz conjecture then states that lim sup Q* = 1.
I tested the "ABC@home" database for this new quality function. This yielded the new record
1 + 4374 = 4375 with Q* = 1.63904.
Impressive, isn't it? I challenge you all to find a better example, and throw me back to second place again.
Here's a short report on my further findings.

3n+1 conjecture

A cycle with m local minima is called an m-cycle. Steiner's famous result from 1977 is that there is no nontrivial 1-cycle. Our Acta Arithmetica paper proves that there are no nontrivial m-cycles with m ≤ 68. The update improves this to m ≤ 75. As the paper explains, with our methods further improvements are not to be expected soon.
See John Simons' research webpage for his followup papers.

This unpublised note criticizes a proof attempt by a serious mathematician.

This paper grew out of Thijs' Bachelor Thesis.

Slides of an overview talk (in Dutch): "Het 3n+1-vermoeden ", PWN Vakantiecursus, August 23, 2013, Eindhoven and August 30, 2013, Amsterdam.
The chapter from the course syllabus (in Dutch). A somewhat modified version of this chapter has appeared as paper in the Nieuw Archief voor Wiskunde, 2014 (self-plagiarism).

Slides of an overview talk (in Dutch): "Het 3n+1-vermoeden ", 25e InterTU-Studiedag, 3TU.AMI, June 19, 2013, Eindhoven.