In December 2004 I discovered (after a three week struggle) a complete classification of all primes p = 1 (mod 3) such that
over3))1.gif)
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 (a truly great, ego-free mathematician) 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. (Within four years Karl and I gave best possible 'Gauss factorial' extensions of those classic congruences, see Theorems 7 and 8 of paper #2 here.)
On Feb 9th 2005 I gave a Maple-based talk to the Trinity College Dublin Mathematical Society on
some of this work. A html version of my talk
is available here. My talk was aimed at students who knew little or no Number Theory, and my principal aim was to show them how a (beautiful to me) discovery came to be made.
In early 2005 I was in the process of writing up the above work (together with quarter-interval analogues) for submission, but a series of unfortunate episodes (the worst, and most dismaying events of my entire working life) in connection with my being external examiner at the Queen's University of Belfast (QUB) 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 folk in both institutions please note that I still have all the evidence. One day ...
Here I will only record that at page 4 of my (12-page) resignation letter of 17th July 2005 I wrote: "Finally, a purely practical matter: given my experiences with St Mary's University College, I do not wish to receive whatever fee might be due to me for my work with that institution, and I would ask you, Mr Ferguson, to kindly arrange payment of the gross amount, together with whatever fee was due to me for my Stranmillis University College work (that does not imply any dissatisfaction with Stranmillis, a college with whose staff I had an extremely positive association), to the Irish Cancer Society at this address: 43-45 Northumberland Road, Dublin 4."
My letter was addressed to Mr. Roy Ferguson (Head of QUB's Academic Council, the body which had invited me to be external examiner for the years 2004-2007) and copied to Professor Peter Gregson (Principal and Vice-Chancellor QUB), Mr. Norman Russell (Academic Registrar QUB), Professor John Gardner (Dean, Faculty of Legal, Social and Educational Sciences), Mr. Peter Finn (Acting Principal, St. Mary's College), Dr. Margaret Reynolds (Dean, Faculty of Education, and 'line manager' for Mathematics, St. Mary's College), and Dr. John Sweeney (Senior Tutor Academic Affairs, St. Mary's). None of these acknowledged receipt of my letter, not even Mr. Ferguson.
On September 1st I phoned one of QUB's Pro Vice Chancellors, one Professor Kenneth Bell (a former head - no less - of QUB's mathematics department); he seemed to be partially briefed about his university's shenanigans, and as a fellow mathematician (though he was a
physicist)
I (foolishly) expected he would be concerned about academic integrity... No, no, ... . His concern (from my diary entry of that day) was to "defend the university's intrests", and he informed me that QUB had set up an internal committee to investigate and write a report on the entire matter, and would talk to all the interested parties. Would I be one of those who would be spoken to? I would be willing to travel up from Dublin to talk to anyone and everyone. No, no, I wouldn't be asked to participate in the inquiry. Would I be afforded the courtesy of seeing the report? No, I wouldn't get to see the reported findings.
St. Mary's/QUB continued to upset my mental equilibrium for weeks, but resolved during a holiday (7th-21st Sept. 2005) that my wife and I spent in Alghero (Sardinia). On the evening of Friday 16th we attended a flamenco concert by a troupe of Sardinian performers (a dancer, a singer, and two guitarists). I hadn't expected to be moved (or even impressed), foolishly thinking that they were Sardinian, not Spanish... . I couldn't have been more wrong. They were truly remarkable, integrity written all over them. They weren't just performers, they were the living embodiment of what one imagines flamenco to be, the real thing, something that cannot be written down, but has to be experienced. Where are they now, what lives did they continue to lead?
As we sat spellbound under a mediteranean sky I couldn't but reflect upon what these artists were contributing to human happiness, compared to the wretched shower still earning their undeserved salaries in Belfast. (One of them was later appointed 'Directory of Quality' at another university...)
Then, at the end of November 2005 I returned to my Jacobi and Gauss work, 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").
Years later I wondered if QUB had acted on my request to pay my external examiner fees (determined by the university, not by me of course) to the Irish Cancer Society (ICS), since QUB had not even informed me about that element of my letter... I phoned the Dublin office of the ICS to enquire, explained why I was contacting them, and this is what I was told (after much searching through their records): yes, they had received a cheque from QUB, one for seven hundred plus pounds sterling, but there was no covering letter, just a cheque, not even something along the lines of we are sending you this on behalf of someone called... . Good old QUB, true to itself, even at the end. And it didn't even occur to them to round their payment up to one thousand.
