# Publications

The following is a list of my joint papers with Karl Dilcher, together with slides of talks given at various meetings or universities.

- Publications
- Extensions of the Gauss-Wilson Theorem
- Mod \(p^{3}\) Analogues of Theorems of Gauss and Jacobi on Binomial Coefficients
- The Multiplicative Orders of Certain Gauss Factorials
- An Introduction to Gauss Factorials
- Pairs of Reciprocal Quadratic Congruences Involving Primes
- On a Congruence of Emma Lehmer Related to Euler Numbers
- Sums of Reciprocals Modulo Composite Integers
- The Gauss–Wilson Theorem for Quarter-Intervals
- The Multiplicative Orders of Certain Gauss Factorials II
- A Role for Generalized Fermat Numbers

In February 2005 I gave a talk to the Trinity College Dublin Mathematical Society; it is available in active Maple format (`.mws`

) and also in html text format from the Jacobi section of my archived (and edited) original web site. My talk concerned primes that later featured in some of the papers listed below: first in our Monthly paper (#4), later in #9 (where they were shown to be 'non-exceptional', a critical point), and finally in #10, where they materialised as being "level 0 (aka 'standard') Jacobi primes" of paper (a critical point in the paper). Having used the term 'standard' I will only remark here that in our paper #10 - where we introduce the novel notion of a **'Jacobi' prime** - it transpires that these primes occur at determined 'levels' (0, 1, 2, 3, 4, ... ), and it further transpires that the primes of my TCD talk are those at level 0 (there are \(215105\) of them up to \(10^{14}\)), there are **none** at level 1, just **one** (13) at level 2, and then a further **forty-four** of level 3 or higher up to \(10^{14}\). It is because of this distribution that we deemed the former to be 'standard', and all the others 'non-standard'.

My TCD Math. Soc. talk was aimed at students with perhaps little or no knowledge of Number Theory, and, to convey the spirit of my talk, I quote the following from it:

"A theorem is never arrived at in the way that logical thought would lead you to believe or that posterity thinks. It is usually much more accidental, some chance discovery in answer to some kind of question. Eventually you can rationalise and say that this is how it fits. Discoveries never happen as neatly as that. You can rewrite history and make it look more logical, but actually it happens quite differently."

Michael Atiyah (from M. Raussen and C. Skau, Interview with Michael Atiyah and Isadore Singer, Notices of the American Mathematical Society, Feb 2005, Vol. 52, Number 2, 225-233. The full interview is available here from the AMS web site.)

On 14th May 2007 I gave at talk - "Gauss-4 primes " - at the Dalhousie Mathematics Colloquium, and then began my collaboration with Karl. Those 'Gauss-4 primes' later became the 'Gauss primes' of paper 8 below ('The Gauss-Wilson theorem for quarter intervals').

Mathematics Colloquia, 2007 (PDF, 151.07 KB)

## #Extensions of the Gauss-Wilson Theorem

**#1**. EXTENSIONS OF THE GAUSS-WILSON THEOREM, INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A39. The paper is available in pdf format, directly from the journal here.

On 26th March 2008 I gave a talk to the University College Dublin Number Theory Seminar; the Abstract of my talk is here.

The Maple worksheet for a talk - Extensions of the Gauss-Wilson Theorem - that I gave at the Rutgers Experimental Mathematics Seminar on Thurs. 10th April 2008 is available from that Rutgers page.

Two number theorists - Xiumei Li (China) and Min Sha (Australia) - have published (Sept. 2017) a paper "Gauss factorials of polynomials over finite fields" - in the International Journal of Number Theory. Their paper was motivated by this, our INTEGERS paper (in which we introduced the term 'Gauss factorial', the first of our seven papers on Gauss factorials), and they referenced six papers of ours. The arxiv version of their paper is here.

## #Mod \(p^{3}\) Analogues of Theorems of Gauss and Jacobi on Binomial Coefficients

**#2**. Mod \(p^{3}\) analogues of theorems of Gauss and Jacobi on binomial coefficients, Acta Arithmetica, Vol. 142, No. 2, 103-118, 2010.

A copy of our paper is available here.

I gave two talks on this - and more besides - at Brigham Young University, Provo, Utah in February 2009. I prepared a massive Maple worksheet in advance of my talk, but it was in an earlier version (5, I think) than the one I now have (16 and 17), and it is proving problematic to convert it to pdf format. I shall post it here should I ever mend it.

The slides of a talk by Karl Dilcher on this topic at the Fields Institute of the University of Totonto on 22nd. Sept. 2009 are available here.

On the 7th October 2009 I gave a talk with title "The (new) world of Gauss factorials" at the (most beautiful university campus I ever expect to see) Boğaziçi Üniversitesi, İstanbul, Turkey.

Then, early in the following year, I gave essentially the same talks in two of my favourite places to visit (Cork and Galway, here in Ireland):

"The (new) world of Gauss factorials", University College Cork, 19th Feb, 2010, and

"The (new) world of Gauss factorials", National University of Ireland Galway, 25th Feb, 2010.

For all three of these talks I had prepared a very extensive Maple worksheet (with far too much material, and it would never have been realistic to cover it all), but in reality I ended up (as is so often the case) not using what I had prepared... instead reverting to traditional "chalk-and-talk". Sometime, when I have time on hand, I might convert the Maple worksheet to pdf format, and place it here for interested readers.

On 16th August 2012 Karl gave a talk "A mod \(p^{3}\) analogue of a theorem of Gauss on binomial coefficients" at the ELAZ (Elementare und Analytische Zahlentheorie) meeting in Schloss Schney, Germany. The slides of Karl's talk are here.

And here are two photos of Karl taken at his talk: photo#1, photo#2.

On 8th March 2017 I gave a talk - The (remarkable and beautiful) binomial coefficient theorem of Gauss, and more besides - to the Trinity College Dublin Mathematical Society, the aim of my talk was to introduce this theorem of Gauss, and then to outline (cutting corners of course) how Karl Dilcher and I extended this to the Theorem 7 of our "Mod \(p^{3}\) analogues ... " paper. I prepared that talk in a Maple worksheet, and a htlm conversion of it may be accessed in the Public and other lectures corner on my archive web site here. For anyone who has Maple, here is the full active original file (here follow Maplesoft's advice: download this file to your computer, and then open it).

## #The Multiplicative Orders of Certain Gauss Factorials

**#3**. The multiplicative orders of certain Gauss factorials, International Journal of Number Theory, 7 (2011), 145–171.

A copy of our paper is available here.

At the time we wrote this paper it appeared (to us) it would be our final word on the topic, but we later extended it in quite an unexpected manner: see our paper #9 below.

## #An Introduction to Gauss Factorials

**#4**. An introduction to Gauss factorials, American Mathematical Monthly, 118 (2011), 810–828. On 13th October 2010.

Our aim in writing this paper was precisely what the title suggested: we wished to introduce non-specialists to the notion of a Gauss factorial (which we had introduced in our first three papers). At the same time we included some new results, especially our Lemma 2 (with its perhaps difficult proof).

The cover of that issue of the Monthly is here, a copy of our paper is available here.

I gave a talk with this title at the annual University of Manchester (England) Colloquium on Wed. 13th. October 2010.

The slides of a talk - entitled Gauss Factorials (Properties and Applications) - given by Karl Dilcher at the Jon Borwein 60th birthday fest in May 2011 are available here.

A video of that Jon Borwein fest talk of Karl's is available at this vimeo link.

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.)

## #Pairs of Reciprocal Quadratic Congruences Involving Primes

**#5**. PAIRS OF RECIPROCAL QUADRATIC CONGRUENCES INVOLVING PRIMES, Fibonacci Quarterly, Volume 51, Number 2, May 2013, 98-111

## #On a Congruence of Emma Lehmer Related to Euler Numbers

**#6**. On a congruence of Emma Lehmer related to Euler numbers, Acta Arithmetica, Vol. 161 (2013), no. 1, 47-67

A copy of our paper is available here.

The slides of a talk - entitled A congruence of Emma Lehmer related to Euler numbers - given by Karl Dilcher at the CMS Winter Meeting, Montréal, December, 2012 - are available here.

## #Sums of Reciprocals Modulo Composite Integers

**#7**. SUMS OF RECIPROCALS MODULO COMPOSITE INTEGERS, Journal of Number Theory, Vol. 133 (2013), no. 11, 3565-3577.

A copy of our paper is available here.

The slides of a talk - entitled Sums of reciprocals modulo composite integers - given by Karl Dilcher at the Canadian Number Theory Association Meeting, Lethbridge, June, 2012 are available here.

## #The Gauss–Wilson Theorem for Quarter-Intervals

**#8**. THE GAUSS–WILSON THEOREM FOR QUARTER-INTERVALS, Acta Mathematica Hungarica, Vol. 142 (2014), no. 1, 199–230.

I gave a talk on that work - Quarter Gauss factorials assuming simplest value - at the Dalhousie Number theory Seminar on 26th. Oct. 2011.

On 11th June 2012 I gave a talk "Gauss primes, a new class of primes intimately related to Gauss factorials" at the **National University of Ireland Galway Mathematics Seminar (2011-2012)**. It was a great pleasue for me that one of those attending was Geoff(rey) Mason, a friend from my days at Royal Holloway College (London), whom I hadn't met since 1969, and who was visiting Galway at that time. Here's a photo of the two of us (Geoff on the left) taken by Michael Tuite in his office (Geoff and Michael collaborate).

On 5th November 2013 I gave the same talk at University College Cork. From my records I see that I prepared an immense amount of Maple worksheets, which - as usual - I didn't really use in my talk. One day - time permitting - I may tidy those, and upload here.

Our Acta. Math. Hung. paper may be downloaded here. It had a 'typo' which was pointed out to us by (a great hero of mine) N.J.A. Sloane (see "The Connoiseur of Number Sequences" in this Quanta profile/interview). Neil observed that '145' was omitted from the line of text immediately following (1.4) of our paper (it was a copy and paste that had gone wrong), and we are very happy that Neil himself (on Dec 08 2013) created sequence "A232986" in his (remarkable) ON-LINE ENCYCLOPEDIA OF INTEGER SEQUENCES), from which I quote:

A232986 Numbers n == 1 (mod 4) such that the Gauss factorial ((n-1)/4, n)! == 1 (mod n)
5, 145, **205, 725, 1025, 1105**, 1145, 1205, 1305, 1313, 1365, 1405, 1469, 1745, 1785, 1845, 1885, 1989, 2145, 2249, 2405, 2465, 2545, 2665, 2745, 2805, 3005, 3045, 3145, 3161, 3205, 3393, 3445, 3485, 3545, 3601, 3625, 3705, 3885, 3893, 3965 [... ]

LINKS .

J. B. Cosgrave and K. Dilcher, An introduction to Gauss factorials, Amer. Math. Monthly, 118 (2011), 810-828.

J. B. Cosgrave and K. Dilcher, The Gauss-Wilson theorem for quarter-intervals, Acta Mathematica Hungarica, Sept. 2013.

## #The Multiplicative Orders of Certain Gauss Factorials II

**#9**. The multiplicative orders of certain Gauss factorials II, Functiones et Approximatio Commentarii Mathematici, Volume 54, Number 1, March 2016, Pages 73-93.

A copy of our paper is available here. (The first referee of this paper - on behalf of the International Journal of Number Theory (in which our **multiplicative I** - paper #3 above - had been published) - wrote this: " I have skimmed the present paper. While the results appear correct, I do not believe this paper meets the standards required for publication in the International Journal of Number Theory. " This 'report' was sent to us by the 'Managing Editors' (Berndt, Sujatha and Zannier) just **six days** after our paper's submission, and it made me decide to discontinue with attempted publication of work after our tenth paper - below - was published. That is **why** there are no other papers of ours after our number 10, even though we already had in place the to-be-written-up papers 11, 12, 13 and 14, and a veritable mountain of other work-in-progress for papers 15 through to 21.) Here, in complete detail, is the IJNT referee's report.

I gave a talk - The multiplicative orders of certain Gauss factorials (again) - on that work at the Dalhousie Number Theory Seminar on 25th Oct. 2012.

The slides of a talk - The multiplicative orders of certain Gauss factorials - given by Karl Dilcher at the Urbana Number Theory Seminar on 19th. Feb. 2015 are available here.

## #A Role for Generalized Fermat Numbers

**#10** . A role for generalized Fermat numbers, accepted (Sept. 2015) for publication in Mathematics of Computation. Here Karl and I went out on a high as far as a referee's report (contrast with the initial referee at #9 above) was concerned. Our lengthy and very detailed paper was (essentially) accepted within two weeks of submission - the referee's initial two page report was very detailed and evidenced a really close reading, and made some valuable suggestions with regard to making the contents clearer for a reader. We were both quite busy at the time and it took some two months before we submitted the improved version, which led to a reply from the referee which began: **"This is an extremely well-written paper, with the prose flowing very smoothly. Moreover interesting examples are interjected at just the right points to help the reader grasp what is being proven. The mathematics is quite intricate, especially the surprising relationships between (what the authors call) Jacobi primes, divisors of generalized Fermat numbers, and congruence conditions for Gauss factorials. ... "**. It is not often that one can say "thank you" to an anonymous referee.

I gave a talk on this work (with the then too-lengthy title) - **Gauss factorials, Jacobi primes, and divisors of generalized Fermat numbers** - at the Dalhousie Number Theory Seminar on 19th March. 2014, and again in St. Patrick's College (Drumcondra, Dublin), and also (in a marathon 90-minute talk) in the Alfréd Rényi Institute of Mathematics **(Budapest) on Tues. 9th December 2014**.

**The slides of a (beautifully organised) short talk - Generalized Fermat numbers and congruences for Gauss factorials - by Karl Dilcher on this subject, at the Australian Mathematical Society Meeting in Adelaide on October 1st 2015 (Karl's 61st birthday), and then later also given at the Canadian Mathematical Society Winter Meeting December 4-7 2015 are available here.**

**We wanted our paper to include "Dedicated to our wives Mary and Anna", but it is Math. Comp. policy not to allow such dedications. The final version of our paper #10 (with the dedication included) is available here.**

**On 26th Sept 2016 Karl gave another beautifully organised talk "Generalised Fermat Numbers: Some Results and Applications" at the (Australian) Number Theory Down Under Meeting (there are many linked photographs) marking the 70th birthday of Richard Brent. The slides of Karl's talk are here, and the Abstract(s) here.**