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 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.
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 (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 boys and girls in both institutions please note that I still have all the evidence. One day ...
I was on friendly terms with (Professor) Don McQuillan (entry #7 here) - his publications are
here - who at that time was nearing the end of his tenure as Head of Mathematics in University College Dublin. Don and I were both number theorists, and our friendly relationship went back a long way, to early 1976 in fact (Don was on the interview panel when I successfully applied for a post in Ireland in December 1975). I knew that Don had once served as external examiner in Mathematics for QUB for the very same St. Mary's College, and I spoke with him (according to my diary) on Saturday 23nd June 2005 about the wholesale cheating that QUB and St. Mary's were engaging in against the students. Don agreed with me that their behaviour was entirely scandalous, that I had right on my side, but the gave me a word of warning: they will circle the wagons around themselves and freeze you out.... As indeed they did.
Between them they approached Dr. Ron McCartney (entry #67 here) the previous external examiner and lied to him: they told him that I had resigned as external examiner and asked if he would step into my shoes... I found that out on Wednesday 6th July 2005 (from my diary of that day: "Phoned Ron McCartney. I feared he would put down the phone. He didn't.")
Having spoken with Ron McCartney I wrote that very day to Mr. Peter Finn (Acting Principal at St. Mary's), Dr. Margaret Reynolds (Dean of Education and 'line manager' for Mathematics at St. Mary's), and Dr. John Sweeney (Senior tutor at St. Mary's), and I copied my letter to three others: Dr. Ron McCartney himself, the above named Mr. Roy Ferguson, and a Mr. Norman Russell (QUB's Academic Registrar). None of those six acknowledged receipt of my letter.
Eleven days later I did resign in this letter addressed to a Mr. Roy Ferguson (Head of QUB's Academic Council, the Council which had invited me in the first instance to act as external examiner), and copied to eight persons named in my letter (the most senior of those being a Professor Peter Gregson, the university's Principal and Vice-Chancellor). Once again, none of those nine acknowledged receipt of my letter, not even Mr. Ferguson to whom my letter was addressed.
At page 4 of my resignation letter 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."
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...
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.
