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 rebuild this new publications page at this, my new site, my earlier publications page was last archived in April 2026 by the (remarkable) WaybackMachine here.

(Aside. We are all in debt to the WaybackMachine, and I believe that everyone should consider making a modest financial contribution to its continued existence (and to Wikipedia also). Occasionally some anonymous, highly principled person matches dollar-for-dollar small individual donors like myself up to a maximum of five million dollars. Life being what it is fewer than two percent of users contribute... . Do please consider becoming a donor.)

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 (I give evidence) and a gutless trio of Editors ("we're not 'experts' in your field")) intervened (see comments at paper #9 below).

Before I list our papers (all may be downloaded) and provide some background, I would like to make 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:

In late 2008/early 2009 we submitted our second paper (the MOD p³ ANALOGUES OF GAUSS AND JACOBI ON BINOMIAL COEFFICIENTS paper below) to the prestigious Polish journal Acta Arithmetica. 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" Zhi-Hong Sun's paper Congruences involving Bernoulli and Euler numbers, available here.

Where did our paper overlap with Zhi-Hong Sun's? Our paper contained eight theorems: #1 (Gauss), #2 (Chowla-Dwork-Evans, extending #1), #3 (Ours, extending #2), #4 (Jacobi), #5 (Evans, Yeung, extending #4), #6 (Ours, extending #5), #7 (Ours, extending #1 to general prime power modulus), and #8 (Ours, extending #4 to general prime power modulus).

Question. Did any of our theorems #3, #6, #7 or #8 overlap in any way with any of the content of Zhi-Hong Sun's J.N.T. paper? Answer: NO!

Question. How did our paper's content - in the words of the anonymous referee - "overlap too much" with the contents of Zhi-Hong Sun's paper?

Answer. We derived our Theorem 3 from our Theorem 7, and a proof of the former required establishing four Lemmas. Of those four:

the second had three elements ((4.5), (4.6) and (4.7)), the last two of which were parts of theorems in Sun's paper (incidentally we had given better proofs),

the third had three elements ((4.8), (4.9) and (4.10)), all of which followed from special cases of parts of theorems in Sun's paper (incidentally we had given better proofs),

and the fourth had six elements elements ((5.6) through to (5.11)), of which (5.6) and (5.8) were once again parts of theorems in Sun's paper (once again, incidentally, we had given better proofs).

It was a very sharp and knowledgeable referee indeed who noticed all of that!!

Question. How did this well-informed referee form the judgement that our paper overlapped too much with the contents of Zhi-Hong Sun's J.N.T. paper? (Incidentally, this Zhi-Hong Sun should not be confused with his better known twin brother Zhi-Wei Sun, who has widely published on binomial coefficient congruences, indeed the latter has referenced us on several occasions. Xie Xie Zhi-Wei Sun!)

How indeed. It was as if someone had pointed out that there was was too much overlap between Shakespeare's "The man that hath no music in himself, Nor is not moved with concord of sweet sounds, Is fit for treasons, stratagems, and spoils; The motions of his spirit are dull as night, And his affections dark as Erebus. Let no such man be trusted." and, let's say, something like "The man got up on his bicycle and cycled down to the nearby sea. While there he read a marvelous proof by the Chinese mathematician Chen, a proof establishing that every sufficiently large even number is the sum of a prime and an integer that is either prime or is the product of two primes."

(An aside. I am well aware that an Editor very much depends on the academic integrity of the chosen referee, and that in the case of A.A. the Editor is likely to have sent our paper to someone with a well-established reputation. What else could the Editor at the time do but reject our paper? However, I would like to suggest that alarm bells ought to have rung for the Editor: the referee didn't give a single instance of the asserted 'overlap', and perhaps didn't have the time to check for himself.

An experienced reader will appreciate how shocked we were, and how quickly we sought Zhi-Hong Sun's paper to observe the 'too much overlap'. At this remove (some sixteen years later) I cannot recall if we laughed or not... Karl, as the 'corresponding author', had the simple task of writing to the Editor to inform that the referee had got it badly wrong, and shortly afterwards our paper was accepted.

I would like to think that the Editor struck the referee (an absolute scoundrel) off the list of Acta Arithmetica referees. Halmos would have given him a permanent red card.)

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 referee (only one) did read it (you might think that an obvious remark, but in view of a referee of our ninth paper - one who "skimmed" our 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, Evans, 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.
   
   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.


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.
   
   This fifth paper (and then our sixth and seventh) marked a temporary diversion from Gauss factorials, the central topic of our interests. There was a certain prime modulus congruence of which I wanted to know what the composite modulus analogue might be (that was at a time when I was planning what our fourteenth/fifteenth paper might be about; I was forever taking a temporary break from our current work, seeing ahead... , then returning to the present, returning to an upcoming visit to Karl in Halifax) - and I wrote to ask Karl is he knew of my possible congruence. Karl - who always had a million-and-one preoccupations - sent me two papers, one by Cai, Fu and Zhou (see our paper #6), the other by Cao and Pan (see our paper #7), . Karl thought he had refereed those papers, and that I might find what I was looking for in one or the other of them...
   
   I didn't find what I was looking for, but very quickly we produced the contents of papers #6 and 7, greatly extending Cai, Fu and Zhou had done, and also what Cao and Pan had done. In the case of Cai, Fu and Zhou, extending to general composite modulus what they had done for prime power moduli, and in the case of Cao and Pan, finishing what they couldn't complete.
   
   Tucked away in the midst of the detail of our extension of Cai, Fu and Zhou's work we had a pleasing minor lemma (Lemma 6), and Karl felt it would get 'lost' in the midst of all the detail of paper #6. In fact we greatly extended that minor observation into a much more substantial work, and it became this, our paper #5.
   
   


6. Our sixth paper - On a Congruence of Emma Lehmer Related to Euler Numbers - was accepted by Acta Arithmetica, and is available here.
   
   The referee's report to the editor of Acta Arithmetica is available here, the most significant part of which is "The paper is doubtless worth publishing. Its publication in Acta Arithmetica is supported by the fact that Cai et al. in their paper mentioned above and published in AA say that they were not able to do the generalization the present authors now did."


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 (but rejected by) the I.J.N.T. I will return here to write about what happened...
   
   Shortly afterwards it was published by Functiones et Approximatio Commentarii Mathematici, and is available here.
   
   


10. I should preface my remarks on this paper by explaining what had been my personal attitude to 'generalized Fermat numbers'...
    
    First, a question: what exactly is a generalized Fermat number? Given that the (classic) Fermat number F_n is given by F_n = 2⁽²ⁿ⁾ + 1.
    
    I HAVE TO RETURN HERE TO FURTHER DEVELOP.
    
    Our tenth paper - A Role for Generalized Fermat Numbers was accepted by Mathematics of Computation, and is available here.
    
    Originally I had suggested giving this paper the name 'Gauss Factorials, Jacobi Primes, and Generalized Fermat Numbers, but I argued against myself by saying that potential readers might be put off by (probably) not knowing what a 'Gauss factorial' was, had never heard of a 'Jacobi' prime (we introduced it in this paper), and...
    
    


11. We didn't submit our eleventh paper for publication (it was published in the Punjab University Journal of Mathematics), rather what happened was this: Karl was invited to visit the Mathematics Department of the University of Lahore, and to deliver some special lectures there. Karl provided them with a menu of topics from which they could choose, and they opted to hear about work that Karl and I had done. (He had a very pleasant and agreeable time there.)
    
    When Karl finished his lecture course he was asked if he would write a summary for inclusion in the just named journal, and he did so. It is available here. It is essentially a distillation of our eight, ninth and tenth papers.