Suggestion for a sequence: weights on a circle

[Edwin Clark]

On Wed, 14 Sep 2005, Brendan McKay wrote:

> 
> Edwin has produced strong evidence that there are no solutions when
> n is a prime power.  

I think a proof is not hard. For p a prime, the pth cyclotomic polynomial 
is F_p(x) = x^(p-1) + x^(p-2) + ... + x + 1. and if q = p^k then the qth 
cyclotomic polynomial is F_q(x) = F_p(x^(p^(k-1))) = x^(p^k-p^(k-1)+ 
x^(p^k-2p^(k-1) + ... + x^(p^(k-1)) +1. If you multiply F_q by a generic 
polynomial g of degree p^(k-1) - 1. Then the product will just have the 
p^(k-1) coefficients of g repeated p times. 


> I can't prove that, but it is easy to show that
> there are solutions when n is not a prime power.  Every such number
> can be written as m*n where m,n are coprime.  Now continue like in 
> this example (n=5,m=3):
> 
>     Write a permutation of {1,2,...,n} m times:
>        5  1  3  4  2  5  1  3  4  2  5  1  3  4  2
>     Add 0,n,2n,...,(m-1)n,0,... cyclically:
>        5  6 13  4  7 15  1  8 14  2 10 11  3  9 12
>     That's a solution.
> 
>     Hugo's example for n=15 suggests another way to do the last step:
>     Replace the value v in position i by m*v+(i mod m).
> 
> Brendan.
> 

---------------------------------------------------------
  W. Edwin Clark, Math Dept, University of South Florida
           http://www.math.usf.edu/~eclark/
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