Colin Mallows's problems
Edwin Clark
eclark at math.usf.edu
Sun Jun 12 01:25:35 CEST 2005
On Sat, 11 Jun 2005, Max wrote:
>
> Could you please compute also the number of polynomials with coefficients in {-1,0,1} that divide x^n-1 ?
>
If I haven't made a mistake:
Here are the number of monic divisors of x^n - 1 with coefficients in
{0,1,-1}. Multiply by 2 to get all that have coefficients in {0,1,-1}.
2, 4, 4, 8, 4, 14, 4, 16, 8, 14, 4, 48, 4, 14, 14, 32, 4, 50, 4, 48, 14,
14, 4, 162, 8, 14, 16, 48, 4, 136, 4, 64, 14, 14, 14, 286, 4, 14, 14, 160,
4, 136, 4, 48, 48, 14, 4, 550, 8, 50, 14, 48, 4, 186, 14, 164, 14, 14, 4,
1124, 4, 14, 48, 128, 14, 136, 4, 48, 14, 136, 4, 1546, 4, 14, 49, 48, 14,
136, 4, 532, 32, 14, 4, 1138, 14, 14, 14, 160, 4, 1192, 14, 48, 14, 14,
14, 1882, 4, 50, 48, 296
Note that many of these are equal to 2^tau(n) where tau(n) is the number
of positive divisors of n = number of irreducible factors of x^n - 1 for
many values of n. This is connected to the fact that for small values of n
the coefficients of the nth cyclotomic polynomial has coefficients in
{0,1,-1}. But it is unlikely that any simple formula gives the sequence
for all n (IMHO)...
Please submit the sequence Max if you wish. It might be a good idea to
have someone else check my computation.
--Edwin
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