Gilbreath Conjecture

Within the card magic community the Gilbreath Principle is a well-known but much misunderstood mathematical principle. Few magicians know much about its creator, Norman Gilbreath, and in particular they are unaware of his other mathematical work. Following a recent email conversation with him about the principle (always go to the source!) he kindly sent me an offprint of a recent paper on the Gilbreath Conjecture. The Gilbreath Conjecture is a conjecture about primes and is fairly easy to state. Consider the sequence of primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … Now work out the (absolute) difference between neighbouring terms 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, … Work out the absolute difference between terms for this sequence and keep doing this: 1, 0, 2, 2, 2, 2, 2, 2, 4, 4, … 1, 2, 0, 0, 0, 0, 0, 2, 0, … 1, 2, 0, 0, 0, 0, 2, 2, … 1, 2, 0, 0, 0, 2, 0, … 1, 2, 0, 0, 2, 2, … 1, 2, 0, 2, 0, … 1, 2, 2, 2, … 1, 0, 0, … 1, 0, … 1, … The conjecture is that the first term on a line, after the first line, is always a 1. Gilbreath’s paper, Processing process: The Gilbreath conjecture, has recently been published in the Journal of Number Theory. (You can find it at http://dx.doi.org/10.1016/j.jnt.2011.06.008 but unless you have a subscription to the journal you will have to pay for it.) The introduction states the following There is one very important aspect of history that is often left out – the process. This is even true of the history of mathematics. I will give an example. A number of years ago...

Oct24

New book – Variational Problems in Differential Geometry...

Along with my colleagues Roger Bielawski and Martin Speight, I have edited the proceedings of the Workshop on Variational Problems in Differential Geometry which we held here at Leeds in 2009. The book has just been published by Cambridge University Press in the prestigious London Mathematical Society Lecture Notes Series. It’s intended for researchers working in differential geometry and can be bought from the publishers or via Amazon.co.uk or Amazon.com. Needless to say, it’s a must-have if you work in the area. And if you work in a university, then don’t forget to recommend it to your library!...

Oct03

The scandal of academic journal profits...

[I thought I had posted this last Monday! Obviously I forgot to hit the right button.] The scandal of research journal profits seems to be gaining visibility in the world outside academia with articles in the press and a government working group. The proposed solutions seem to miss an important point which I’ll come to later. First, what is the problem? Well, governments throughout the world pay researchers in universities to do fundamental research. This can take numerous forms, either the money is paid to the university in general money or to specific researchers to do specific research via bodies such as EPSRC. Once the research is done and new results have been found, the researchers put these in a paper and send it to a journal. The editor of the journal sends the paper to an anonymous referee who assesses the paper (are the results new? Are they worthy of publication? and so on). If accepted, the paper is published in the journal. Researchers who want to read the paper can do so by consulting a copy from their university library (online or in hard copy). The problem becomes apparent when you follow the money. The tax payers give money to the researcher. The researcher gives the results to a company to publish. The editor and the referees are not paid to edit and referee – generally academics work for free on those. The libraries get money from the taxpayers to buy the journal (often at a very high price). So what is happening is that tax payer ends up paying twice for the same research. We researchers do the research, give it away and then buy it back. Obviously something is wrong here. This is compounded by the fact that the publishing...

May25

Back from Poland

Really I should be marking essays and exams but I sat through a day of student project presentations so I guess I deserve a rest. I had a good time at the conference in Poland. There were a few delays on the way but as I was travelling with my students, two gentlemen of middle eastern origin, I guess this is to be expected in the current times. The modern conference centre is in the grounds of a picturesque palace which is pictured below. The interior is quite nice too. But what about the talks? Well, one was so bad that I wished that I had brought a video camera. In the coming years I will be teaching our students how to give presentations and this talk was a perfect example of how to alienate an audience. The speaker gave a 55 minute talk in which used more than one slide per minute – all slides densely packed with maths naturally – and read them in a monotone, his eyes fixed to the screen except for three occasions where he looked towards us. Three times in an hour! And those times were all in the first half hour. How can anyone, especially a non-novice, believe that they can give a good talk without engaging in the audience? The conference finished on Friday and as our flight home was on Saturday evening, we went to the centre of the nearby city, Poznan. While we were there we saw a bubble festival. Not sure why they were having a bubble...

May15

Conference in Poland

I’m currently in a small village in Poland about to take part in a conference: http://bcc.impan.pl/sga/ The general public may find it hard to believe but new mathematics is being produced all the time. Not only that the volume of new mathematics is huge. So huge that even the best mathematicians know only a very small proportion of it. This week’s conference focuses on the geometric side of my area of interest, Singularity Theory (it’s a descendent of Newton’s calculus). I’ll be giving a short talk at the end of the week. Haven’t written it...

Mar23

Milnor wins Abel Prize...

As you may know, I’m a big fan of John Milnor, I posted about his video on D’Arcy Thompson. Well, today he won the Abel Prize. My own research has been greatly influenced by Milnor. His seminal work on the topology of complex hypersurface singularities is the foundation for the work in my PhD thesis. What is amazing is that he seems to have come into the subject in the late 60s, knocked out a book on the subject and then left the field to do something else groundbreaking. He was great at exposition. Really great. His early 60s book on Morse Theory has not been surpassed in the field for its clarity and choice of material. For me it’s one of the great books of the 20th century. It’s up there with Orwell’s 1984. As a postgraduate I read his book on the h-cobordism theorem when I was on holiday. How nerdy am I? (It’s also a great book but a bit more challenging shall we say.) More information about his achievements, certainly more than I can fit here, can be found at the official winner’s page:...