Since Karl Dilcher and I began our collaboration in May 2007 we have completed four papers:

A. John B. Cosgrave and Karl Dilcher, Extensions of the Gauss-Wilson 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, 103-118, 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.

1. Remark on Euclid’s Proof of the Infinitude of Primes, American Mathematical Monthly, p. 239-241, Vol 96, No 4, April 1989.
   
   

2. Teaching Mathematics by Questioning – The Socratic Method, Newsletter of Irish Mathematics Teachers Association, Nos 81-82, 1993, 32-47.
   
   

3. A Halmos Problem and a Related Problem, American Mathematical Monthly, p. 993-996, Vol 101, No 10, December 1994.
   
   

4. From divisibility by 6 to the Euclidean Algorithm and the RSA cryptographic method, The American Mathematical Association of Two-Year Colleges Review, Vol 19, No 1, Fall 1997, 38-45.
   
   

5. 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).
   
   

6. 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), Springer-Verlag, 2000, pp 124-143. That paper is now available from the US Naval Academy web site here. The core content of my 3rd year Number Theory and Cryptography course is available here.
   
   

7. 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 (56 KB), or fermat.zip (15 KB). 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 R-approximations to quadratic irrationalities. The Maple files of my lecture: ictm4.mws (34 KB) , or ictm4.zip (9 KB).
   
   

8. A Prime For The Millennium, Folding Landscapes, 2000.
   
   

9. 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.
   
   

10. 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 (135 KB), or fer_1640.zip (35 KB). The html version is available here.
    

11. At 4.00 P.M. on Thursday 23rd March, 2000, in Room AL8 of 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 10¹⁰⁰⁰⁰⁰ 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 open-minded 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 (1870-1952))

    Maple files of that talk are available here.
    

12. 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.
    

13. An unpublished paper on e squared etc.
    
    

14. In Chicago, October-November 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.
    
    

15. 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.