TCD2017.mws.mw

• 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
• Fermat's two-square theorem (with the 'signed' element incorporated into its statement)
• Some classic congruences (Fermat, Wilson, Lagrange, and some others ... ), and the Gauss- Wilson theorem
  • Fermat's 'little' Theorem, and introducing a standard term: the Fermat quotient, q[p](a)
  • An exercise concerning Sum(1/i, i = 1 .. p-1), and the half and quarter sums Sum(1/i, i = 1 .. (1/2)*{p-1}) and Sum(1/i, i = 1 .. (1/4)*{p-1})
  • Wilson's Theorem, and introducing another standard term: the Wilson quotient, w[p]
  • The Lagrange path
  • The Gauss-Wilson theorem
• Gauss' binomial coefficient congruence, and its extension ("suggested by Beukers"), proved by Chowla-Dwork-Evans
• A very, very fast introduction to Gauss factorials
• Cutting corners: introducing the gamma-values gamma[1], gamma[2], gamma[3], gamma[4], `...`
• View examples of the gamma-structure of prime powers for half- and quarter-factorials, and 'exceptional' prime
  • Half-factorials (just three examples to show all actual (proven) behaviours)
    • p = 3 Here the gamma-values gamma[1], gamma[2], gamma[3], gamma[4], `...` alternate 1, 2, 1, 2, `...`
    • p = 5 Here the gamma-values gamma[1], gamma[2], gamma[3], gamma[4], `...` are 4, 4, 4, 4, () .. ()
    • p = 7 Here the gamma-values gamma[1], gamma[2], gamma[3], gamma[4], `...` alternate 2, 1, 2, 1, () .. ()
  • Quarter-factorials (just three examples to show all apparent behaviours ...)
    • p = 5(the order values appear to go: gamma, 2*gamma*p, p^2, 2*gamma*p^3, p^4, 2*gamma*p^5, `...`
    • p = 13(the order values appear to go: gamma, gamma*p, p^2, gamma*p^3, p^4, gamma*p^5, () .. ()
    • p = 37(the order values appear to go: gamma, (1/2)*gamma*p, p^2, (1/2)*gamma*p^3, p^4, (1/2)*gamma*p^5, `...`
• 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