A107751
Edwin Clark
eclark at math.usf.edu
Sat Oct 13 22:26:03 CEST 2007
On Fri, 12 Oct 2007, Robert Israel wrote:
> You may have noticed that (at least for n up to 200) whenever your polynomial
> x^n+x+1 factors, one of the factors is x^2+x+1. I don't
> know if this is true for all n, but if so there's doubtless a good
> reason for it. In any case it's easy to see that for n >= 2, x^n+x+1
> is divisible by x^2+x+1 if and only if n == 2 mod 3.
>
According to
http://www.math.niu.edu/~rusin/known-math/95/trinom:
The polynomial x^k+x+1 has the factor x^2+x+1 if k is 2 (mod 3) and is
otherwise irreducible.
The reference given there by David Eppstein is:
Ernst Selmer, "On the irreducibility of certain trinomials", Math.
Scand. 4 (1956) 287-302
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