 
[A note of late February 2017.
This is my first opportunity to update this page of my old web site  'old' meaning the one I had when I was in fulltime employment (until 2007)  and the most recent papers (ten in all) that Karl Dilcher and I published may be viewed in the
Publications corner of my new web site.]
Since
Karl Dilcher and I began our collaboration in May 2007 we have completed four papers
(now ten, see here):
A. John B. Cosgrave and Karl Dilcher, Extensions of the GaussWilson theorem, Integers:
Electronic Journal of Combinatorial Number Theory, Vol. 8, #A39, 2008.
B. John B. Cosgrave and Karl Dilcher, Mod p^3 analogues of theorems of Gauss and Jacobi
on binomial coefficients, Acta Arithmetica, Vol. 142, No. 2, 103118, 2010.
C. John B. Cosgrave and Karl Dilcher, The multiplicative orders of certain Gauss factorials, accepted March 2010 by the International Journal of Number
Theory.
D. John B. Cosgrave and Karl Dilcher, An Introduction to Gauss factorials. Submitted November 2010, after my fourth research visit to Karl Dilcher.
Note of February 2011. Seminar talks (based on work from above papers) given since 2007
:
Dalhousie university colloquium, Mon. May 14th 2007.
Gauss4 primes (a beautiful new sequence of primes)
University College Dublin, Wed 26th March 2008.
Extensions of the GaussWilson theorem
Rutgers university colloquium, Thurs. 10th April 2008.
Extensions of the GaussWilson theorem (link to prepared Maple worksheet of talk)
City University of New York (CUNY), Fri. 11th April 2008. Repeat of previous day's Rutgers talk.
Brigham Young University, Mon. and Wed. 2nd and 4th Feb. 2009. Two classic theorems of
Gauss
(the GaussWilson theorem, and Gauss's binomial coefficient congruence)  Then and Now".
(While visiting BYU I also gave my
Bill Clinton, Bertie Ahern, and digital signatures talk , on Tues. 3rd Feb. 2009).
Bogazici university (Istanbul, Turkey) colloquium, Wed. 7th Oct. 2009.
The (new) world of Gauss factorials.
University College Cork colloquium, Fri. 19th Feb. 2010. GaussJacobi advances.
NUI Galway. Thurs. 25th Feb. 2010.
GaussJacobi advances.
Manchester university colloquium, Wed. 13th Oct. 2010.
What is a Gauss factorial?
Dalhousie university colloquium, Mon. 8th Nov. 2010.
What is a Gauss factorial?
Some earlier papers of mine :

Remark on Euclid's Proof of the Infinitude of Primes, American Mathematical Monthly, p. 239241, Vol 96, No 4, April 1989.

Teaching Mathematics by Questioning: The Socratic Method,
Newsletter of Irish Mathematics Teachers Association, Nos 8182, 1993, 3247.

A Halmos Problem and a Related Problem, American Mathematical Monthly, p. 993996, Vol 101, No 10, December 1994.

From divisibility by 6 to the Euclidean Algorithm and the RSA cryptographic method, The American Mathematical Association of TwoYear
Colleges Review, Vol 19, No 1, Fall 1997, 3845.

An Introduction to Number Theory with Talented Youth, USA School Science and Mathematics, Vol 99, No 6, October 1999
(Special issue devoted to gifted and talented Mathematics and Science students).

Number Theory and Cryptography (using Maple), in David Joyner USNA (Ed.), Coding Theory and Cryptography: From Enigma to
Geheimschreiber to Quantum Theory (Unites States Naval Academy Conference), SpringerVerlag, 2000, pp 124143.
That paper is in the
Springer book here.
The core content of my 3rd year Number Theory and Cryptography course is available
here.

Two papers read at at the fourth International Conference on the use of Technology in Mathematics Teaching (ICTMT4), in Plymouth
University (UK) August 1999:
(a) Tuesday 10^{th} August 1999 I gave a talk  The mathematical context of the recent (25^{th} July 1999)
discovery of the largest known composite Fermat number. The Maple files of my lecture:
fermat.mws.
A html version of that lecture is available
here.
(b) on Thursday 12^{th} August 1999 I gave a talk 
Using Maple programming to investigate L and Rapproximations to quadratic irrationalities.
The Maple files of my lecture:
ictm4.mws.

A Prime For The Millennium ,
Folding Landscapes, 2000.

Fermat's 'little' theorem, a paper read (rather introduced) at the Fifth International Conference on Technology in Teaching Mathematics (ICTM5, Klagenfurt University, Austria, August 2001) is available
here.

On Thursday 28^{th} October, 1999, I gave a public lecture (using Maple) entitled The history of Fermat numbers from August 1640.
Here are the Maple files of my lecture:
fer_1640.mws.
The html version is available
here.

On Thursday 23rd March, 2000, in University College Cork (UCC), I gave a talk entitled:
Could there exist a sixth Fermat prime? I believe it is not impossible.
An abstract:
Fermat believed the Fermat numbers {F[n]} (F[n] = 2^(2^n) + 1) to be prime for all n = 0,1,2,3,...), but, as is well known, they are prime
for n = 0, 1, 2, 3 and 4, and for no other known value of n. An orthodoxy has now developed that F[n] is composite for all n>4, but with not
the remotest sign of any proof. Only in September 1999 did Richard Crandall's team settle that F[24] is composite (now the smallest
undecided case is F[31]), and last July  as a participant in Yves Gallot's Proth Prime Search  I fortuitously discovered the largest known
composite Fermat number, F[382447] (having more than 10^{100000} digits). Last March I rediscovered a generally unknown unification of
Fermat and Mersenne numbers, an outcome of which was to make an observation which  I believe (though my point of view has opponents)  should
cause openminded people to question the above orthodoxy. This talk will be of an entirely elementary nature, and will be suitable for undergraduate
students.
Also on that day I gave a talk to the UCC student mathematics society:
A Prime For The Millennium (primality testing, with special reference to a wonderful idea of Henry Cabourn Pocklington
(18701952))
Maple files of that talk are available
here.

To mark the 400^{th}anniversary (on 17^{th} August 2001) of the birth of Pierre de Fermat I presented a survey paper –
using Maple – on his renowned 'little' theorem.
 Fermat's little theorem  at the 5^{th} International Conference  6^{th} to 9^{th} August 2001 on the
use of Technology in Mathematics Teaching, Klagenfurt University, Austria. I chose this topic partly because of my personal admiration for this
fundamental theorem, but mainly because of the time coincidence that August 17^{th} 2001 will be the 400^{th} anniversary of
Fermat's birth.
That Maple work  in mws and html versions  may be accessed
here .

An unpublished paper on e squared etc.

In Chicago, OctoberNovember 2003, at the 16^{th} International Conference on the use of Technology in Collegiate Mathematics
(ICTCM16)
I gave a
Bill Clinton, Bertie Ahern, and digital signatures (96 KB in size) talk. A html version is available
here.

My best work begins in the
Jacobi section of my site, and continues at
Gauss.
I have begun a collaboration with
Karl Dilcher,
and I expect that Karl and I will produce many papers over the coming years. Note added February 2011: see above for some details.


