Half-factorials (just three examples to show all actual (proven) behaviours) 

In the following quarter-factorials section the are: 

,  

and recall that, when , is simply the familiar (the order of with respect to the modulus ) 

Quarter-factorials (just three examples to show all apparent behaviours ...) 

A  brief summary of what computations suggested to be true for these : 

etc 

However, this is what a theoretical analysis uncovered, for these same : 

•   OR  
  

•   OR  
  

•   OR  

etc 

In a pre-computer age one would have been tempted to conjecture that the second alternative (the 'OR') never held ... In fact, in the pre-computer age one might never even have arrived at conjecturing the first alternatives ...  

We derived a criterion (too involved to relate here) for the second atlernative to hold, and it transpired that at the prime , one had: 

(which is )

(which is ) 

so that  

and we deemed this prime to be (in this context) 'exceptional'. 

Using the criterion we had obtained we tested (using Maple) to , and then, using a very fast program written specially for us by Yves Gallot, we extended the search to , without finding another exceptional prime (for quarter Gauss factorials, that is). 

In later years we discovered (with proof) a more beautiful (and much faster) criterion which enabled us to extend the search to , and still no new exceptional prime (so the term 'exceptional' is well-chosen).  

I shall give that test later; it came directly - and quite unexpectedly - from our extension of the Gauss binomial coefficient congruence. It's all written up in our Acta Mathematica Hungarica paper: 

THE GAUSS-WILSON THEOREM FOR QUARTER INTERVALS,

Acta Mathemetica Hungarica, Vol. 142 (2014), no. 1, 199-230 

You should understand that I am keeping my remarks exceptionally brief here, as there is so much more to be said.