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

   mod p


I am in debt to Andrew Granville for providing proofs.On Feb 9th 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. 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. To get a very small idea, consult Gauss. I have begun a collaboration with Karl Dilcher, and I expect that Karl and I will produce many papers over the coming years.

Note added February 2011. Since Karl Dilcher and I began our collaboration in May 2007 we have completed four papers; for details see Recent Publications. Working with Karl has been the highlight of my entire mathematical life.

(Incidentally, as a 'fun thing', my Erdős number is 3 (see MathSciNet Collaboration Distance site). Using that site I find that my Andrew Granville number is also 3, my Doron Zeilberger number is 4, my Yves Gallot number is 5, my Bruce Berndt number is 4, my Kenneth Williams number is 5, my Roger Baker number is 4, my Paul Goodey number is 5, my Michel Waldschmidt number is 4, my Sinéad Breen number is 7, my Tom Laffey number is 5, ... , and my Andrew Wiles number is 5.)