Jacobi
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A note of late March 2017 concerning the opening of the Maple worksheets below, indeed my Maple worksheets in general. You may encounter a problem opening my Maple worksheets (as indeed I do myself) here at my web site - it would appear to depend on the internet browser being used. Thus, if I attempt to open one of my worksheets using Interner Explorer there is never a problem, whereas if I use Firefox then all that one sees - this is just an example - is something like this: {VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 0 128 0 128 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 128 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" and so on (almost at infinitum)...

I asked Maplesoft for advice on this and they recommended doing the following (and I found it worked):

  1. Don't attempt directly to open the worksheet (by clicking on my link), instead right click and save the worksheet to (say) the download folder.

  2. Now that the worksheet is in the download folder (cut it out if you wish and put into whatever folder you wish) you may open the worksheet in the usual way (I should add that while I see this 'works', I have absolutely no idea as to why it does... ) [END OF NOTE].

In December 2004 I discovered (after a three week struggle) a complete classification of all primes p = 1 (mod 3) such that

   mod p


At the time I made this discovery I was entirely ignorant of something really vital - Jacobi's remarkable binomial coefficient congruence - the critical clue to proving the found result (I cannot imagine that there is any other way of doing so).

It was Andrew Granville who introduced me to this wonderful Jacobi theorem, and Andrew provided the proof of my discovered classification (I had known of Gauss's binomial coefficient congruence from my school days; I read it in my copy of Davenport's The Higher Arithmetic, though Davenport didn't call it by that name), and I couldn't have known at the time the role this theorem would play in my later work with Karl Dilcher.

On Feb 9th 2005 I gave a Maple-based talk (which I don't regard as being complete) to the Trinity College Dublin Mathematical Society on some of this work. Here are the Maple files of my talk (I didn't get to cover everything...). I had a problem with the html version, and some elements of it may be incomplete:

  1. mws format (232KB, output not removed)

  2. html format (Maple converts all outputs, in-line equations etc, to gif files, and there are quite a few of those)

I was in the process of writing the work up for submission, but an unfortunate episode (the worst, and most dismaying event of my entire working life) in connection with my being external examiner at the Queen's University of Belfast delayed my efforts. (Unprincipled elements in QUB, and in its affiliated St. Mary's College, conspired to cheat the Mathematics students in St. Mary's; administration boys and girls in both institutions please note that I still have all the evidence. One day ... )

Then, at the end of November 2005 I returned, hoping to complete... I thought I just needed to tidy up one small idea, when quite suddenly my work went off in an entirely new direction ("Gauss factorials").

Note added March 2017. Since Karl Dilcher and I began our collaboration in May 2007 we have published ten papers; for details see Recent Publications. Working with Karl has been the highlight of my entire mathematical life.


 

Contact details. jbcosgrave at gmail.com