In migrating my former .com site to this, my new .ie site, I have given up on my previous Publications page. In the meantime, while I reorganise my new site, my earlier Publications page was last archived in April 2026 by the WaybackMachine here. From time-to-time the normally reliable WaybackMachine displays this message: "Temporarily Offline, Internet Archive services are temporarily offline, ... , We apologize for the inconvenience."

I will have more to add at this page when I have time.

The following eleven papers were written with my German/Canadian friend Karl Dilcher. We would have published more but Fate (in the guise of two ignorant referees and a gutless trio of Editors ("we're not 'experts' in your field")) intervened (see comments at paper #9 below).

An important (I believe) illustrated comment on how a (leading) journal editor can be misled (indeed was misled, but subsequently corrected) by an unscrupulous referee:

We submitted our second paper (the MOD p³ ANALOGUES OF GAUSS AND JACOBI ON BINOMIAL COEFFICIENTS paper below) to Acta Arithmetica late 2008/early 2009 and waited... After several months of not receiving a report from the Editor with the referee's comments, Karl said it was a good sign, as a rejection would normally come quite quickly ...

Then, suddenly, out of the blue: a rejection! But why? The editor informed us that a referee had rejected our paper on the grounds that - wait for it - "their paper's contents overlaps too much with [the following paper]" Zhi-Hong Sun's paper Congruences involving Bernoulli and Euler numbers, available here.

Where did our paper overlap with Zhi-Hong Sun's?

1. Our first paper - EXTENSIONS OF THE GAUSS-WILSON THEOREM - was published by the electronic journal INTEGERS, and is available here.
   
   Initially we submitted it to the Journal of Number Theory (JNT) where it was very quickly rejected. The (only one) referee did read it (you might think that an obvious remark, but in view of a later referee of a later paper, it needs pointing out), praised it slightly, observed that its merit was that the contents were correct, but opined that it didn't meet the high standard required for acceptance by the JNT, and suggested rejection.
   
   Karl was about to give a talk on that very work at the 18th Czech and Slovak International Conference on Number Theory (see 'Participants' at conference page) when I uncovered a major error (so much for 'correct') is the central proof, one which I was fortunately able to repair before Karl gave his talk...
   
   Two Chinese mathematicians Xiumei Li and Min Sha wrote a paper Gauss factorials of polynomials over finite fields - available here - and in that work of their they reference our papers #1, #2, #3, #4, #8 and #10. It was pleasing to see that we had at least two appreciative readers!
   
   


2. Our second paper - MOD p³ ANALOGUES OF GAUSS AND JACOBI ON BINOMIAL COEFFICIENTS - was accepted by Acta Arithmetica, and is available here.
   
   Some comments. The MOD p³ of our title came from our improving on a MOD p² paper by S. Chowla, B.Dwork, and R.Evans.
   
   Sadly the legendary Chowla and Dwork both died many years before our paper appeared, leaving only Evans... . I presume that's why he was invited to review our paper for the AMS's Math Reviews.
   
   It would be an understatement to say that Evans' review lacked a certain generosity of spirit, and singularly failed to inform readers of our paper's contents: he didn't make a single mention of the role of 'Gauss factorials' (a notion latched onto by several Chinese number theorists in later works). When Karl sent me a copy of the Evans review (as I don't have access to Math Reviews) I remarked that had I been unfamiliar with the paper's contents, and was a reader relying on it to influence my reading of it, then I wouldn't have bothered to read the paper.
   
   Nor did Evans properly inform readers in his pseudo-review that our Theorems 7 and 8 gave mod p^α extensions of the well-known classical binomial coefficient congruences of Gauss and Jacobi (work that he, along with Bruce Berndt and Kenneth Williams, wrote about in their CMC monograph Gauss and Jacobi Sums). And while he did hint at something with his singularly uninviting "Also proved is a general congruence...", he could have mentioned that the authors had obtained a best possible extension of those classic Gauss and Jacobi congruences in the language of Gauss factorials. (Perhaps he didn't realise we had done that?)
   
   Concerning our just mentioned Theorems 7 and 8 I am reminded of Chandrashekhar B. Khare's Notes on Writing a Mathematical Memoir which was published in the June/July issue of the Notices of the American Mathematical Society (see here). At the top right of page 645 C.B.K wrote "... is one of the many reasons why I believe more mathematicians should write about the path to their results" [my bold]. I couldn't agree more.
   
   Whenever I give talks to students I try to convey how results were arrived at, a relevant example being a talk I gave on 8th March 2017 to the Student Mathematical Society of Trinity College Dublin, a talk with title The remarkable (and beautiful) binomial coefficient theorem of Gauss, and more besides. I used Maple to prepare my talk, exported it to html, and it's available at the top of this page. Viewing the contents one may see that I included some photographs (Gauss, Karl [Dilcher], Beukers, Chowla-Dwork-Evans) and I sent a link to 'Professor Evans'. Evans replied (he Hi John-ed me as if we were bosom pals), didn't make any comment about the content, only remarking that I had inserted an incorrect photo of him (taken fron his website). It was years before I was able to correct my error, I sent Evans the link to the corrected version, but he didn't acknowledge. (A well-known mathematician once remarked to me in an email: "John, the are only two kinds of mathematician: good mannered ones and bad mannered ones.)


3. Our third paper - THE MULTIPLICATIVE ORDERS OF CERTAIN GAUSS FACTORIALS - was accepted by the International Journal of Number Theory, and is available here.
   
   


4. Our fourth paper - An Introduction to Gauss Factorials - was accepted by the Mathematical Association of America, and is available here.
   
   

   The Monthly cover of November 2011:

   

   For many years the Monthly had been my favourite Mathematics publication, and I need hardly tell other mathematicians about its history. Its founder Benjamin Franklin Finkel was a remarkable person, someone with a vision.
   
   Franz Lemmermeyer - in his review for for Zentralblatt MATH - wrote "Starting from Wilson's theorem, the authors lead the readers slowly up to Gauss's surprising theorem that ... in this beautiful article ... ". Thank you Franz Lemmermeyer. (What a comfort and joy it is to have a sympathetic and understanding reader.)


5. Our fifth paper - Pairs of Reciprocal Quadratic Congruences Involving Primes - was accepted by Fibonacci Quarterly and is available here.
   
   


6. Our sixth paper - On a Congruence of Emma Lehmer Related to Euler Numbers - was accepted by Acta Arithmetica, and is available here.
   
   


7. Our seventh paper - SUMS OF RECIPROCALS MODULO COMPOSITE INTEGERS was accepted by the Journal of Number Theory, and is available here.
   
   


8. Our eight paper - THE GAUSS-WILSON THEOREM FOR QUARTER-INTERVALS - was accepted by Acta Mathematica Hungarica, and is available here.
   
   


9. Our ninth paper - The Multiplicative Orders of Certain Gauss Factorials II was initially submitted to the I.J.N.T. here.
   
   I will return here to write about what happened...


10. Our tenth paper - A Role for Generalized Fermat Numbers was accepted by Mathematics of Computation of the here.
    
    


11. here.