Recent Publications


[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 full-time 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 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.

Note of February 2011. Seminar talks (based on work from above papers) given since 2007 :

Dalhousie university colloquium, Mon. May 14th 2007. Gauss-4 primes (a beautiful new sequence of primes)

University College Dublin, Wed 26th March 2008. Extensions of the Gauss-Wilson theorem

Rutgers university colloquium, Thurs. 10th April 2008. Extensions of the Gauss-Wilson 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 Gauss-Wilson 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. Gauss-Jacobi advances.

NUI Galway. Thurs. 25th Feb. 2010. Gauss-Jacobi 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

  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 in the Springer book 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 10th August 1999 I gave a talk - The mathematical context of the recent (25th 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 12th 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.

  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 28th 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.

  11. 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 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 400thanniversary (on 17th 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 5th International Conference - 6th to 9th 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 17th 2001 will be the 400th 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 16th 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.

Contact details. jbcosgrave at