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[John Conway]

On Wed, 3 Sep 2003, benoit wrote:

> 
> Many thanks for this very instructive answer. Furthermore, I built an 
> array defined as follows :  T(p,k) is the least integer s such that p 
> divides k^s-1,  p is an odd prime and k is ranging from 1 to p-1 :
> 
> [1, 2]
> [1, 4, 4, 2]
> [1, 3, 6, 3, 6, 2]
> [1, 10, 5, 5, 5, 10, 10, 10, 5, 2]
> [1, 12, 3, 6, 4, 12, 12, 4, 3, 6, 12, 2]
> [1, 8, 16, 4, 16, 16, 16, 8, 8, 16, 16, 16, 4, 16, 8, 2]
> [1, 18, 18, 9, 9, 9, 3, 6, 9, 18, 3, 6, 18, 18, 18, 9, 9, 2]
> [1, 11, 11, 11, 22, 11, 22, 11, 11, 22, 22, 11, 11, 22, 22, 11, 22, 11, 
> 22, 22, 22, 2]
> [1, 28, 28, 14, 14, 14, 7, 28, 14, 28, 28, 4, 14, 28, 28, 7, 4, 28, 28, 
> 7, 28, 14, 7, 7, 7, 28, 28, 2]
> [1, 5, 30, 5, 3, 6, 15, 5, 15, 15, 30, 30, 30, 15, 10, 5, 30, 15, 15, 
> 15, 30, 30, 10, 30, 3, 6, 10, 15, 10, 2]
> 
> I just noticed 2 simple rules :
> 
> (i) T(p,1)=1 and  T(p,p-1)=2
> 
> (ii) T(p,k) 1<k<p-1  take values among divisors >2  of (p-1) (does each 
> divisor appear at least once?)

   Yes, in fact the list is a permutation of the denominators (in least
terms) of the numbers  1/(p-1), 2/(p-1), ... , (p-2)/(p-1).

   Regards, John Conway