[Turkmath:6595] Bogazici Universitesi Matematik Seminerleri - ilan ekte

[ferit.ozturk at boun.edu.tr]

Seminerler Ingilizce duzenlenmektedir.

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Bogazici Universitesi Matematik Seminerleri

The (new) world of Gauss factorials

John Cosgrave
St. Patrick's College, Ireland


Date: Wednesday, October 7, 2009
Time:	14:00
Place:	TB 250, Boğaziçi Üniversitesi

Abstract: I will report on joint work with Karl Dilcher. Gauss's
generalisation 
of Wilson's theorem was given in his renowned classic, the Disquisitiones

Arithmeticae (1801). In [1] Karl Dilcher and I gave the first extension of the 
Gauss-Wilson theorem, the key new object of interest being what we call a
"Gauss 
factorial". This entirely elementary object opens up a whole new world of

interest, revealing many new results in Number Theory. By a pleasing 
co-incidence this new object has important consequences for another classic 
theorem of Gauss: his (1828) "beautiful (mod p) binomial coefficient
congruence" 
(quote of Bruce Berndt and others), and an equally beautiful, closely related 
congruence of Jacobi (1837). The former concerns primes that are 1 mod 4, while

the latter concerns primes that are 1 mod 3. In a 1983 Paris seminar Frits 
Beukers conjectured a mod p squared extension of Gauss's binomial
coefficient 
congruence, and that conjecture was settled in 1986 by S. Chowla, B. Dwork and 
R. Evans. In the late 1980's, R. Evans and K. M. Yeung independently
proved a 
mod p squared extension of Jacobi's congruence. No mod p cubed extension
of 
either Gauss's or Jacobi's congruences had been conjectured, but
last year, as 
a side outcome of another investigation of ours, we formulated and proved
mod p 
cubed extensions of both the Gauss and Jacobi congruences [2]. In this entirely

elementary talk - requiring no number theory background - I will introduce you 
to as much as possible of the above, and inform - if time allows - of some 
work-in-progress.

[1] John B. Cosgrave and Karl Dilcher, Extensions of the Gauss-Wilson theorem, 
Integers: Electronic Journal of Combinatorial Number Theory, 8, (2008)
[2] John B. Cosgrave and Karl Dilcher, Mod p^3 analogues of theorems of Gauss 
and Jacobi on binomial coefficients, Acta Arithmetica (to appear)

Tea and coffee will be served at 15:00

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