TCD2017.mws1.html

> ### extension of Gauss' binomial coefficient congruence.mws

I dedicate this talk to the memory of Jon Borwein (1951-2016)

https://en.wikipedia.org/wiki/Jonathan_Borwein

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Chorus, in the Prologue of Shakespeare's Henry V:

" ... jumping o'er times

Turning the accomplishment of many years

Into an hour glass : "

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Title, speaker, aim/structure of talk

Procedures (anyone using this 'live' needs to exectute this section to perform independent computations)

Photos of Gauss, Karl, Beukers, Chowla-Dwork-Evans, and some web links

Before I tell you Gauss' theorem (and its extension by Chowla-Dwork-Evans, and then ourselves, Cosgrave-Dilcher) I need to remind you of certain earlier work - Fermat (two instances), Wilson, Lagrange

Fermat's two-square theorem (with the 'signed' element incorporated into its statement)

In the following section - which has several sub-sections - I have added more details than in my actual talk, and I include them now for your independent reading:

Some classic congruences (Fermat, Wilson, Lagrange, and some others ... ), and the Gauss- Wilson theorem

And now we are ready for the remarkable:

Gauss' binomial coefficient congruence, and its extension ("suggested by Beukers"), proved by Chowla-Dwork-Evans

And now the notion of a Gauss factorial (which we introduced in our first paper):

A very, very fast introduction to Gauss factorials

Cutting corners: introducing the gamma-values

View examples of the gamma-structure of prime powers for half- and quarter-factorials, and 'exceptional' prime

How we went from one of our gamma-structure results - helped by Chowla-Dwork-Evans - to the start our new ...

And the continuation, and finalisation

And a brief return to 'exceptional' primes