MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

1

# When is the Wendt binomial circulant divisible by 3?

The Wendt binomial circulant determinant $W_n$ can be defined quite simply as a resultant: $$W_n = \operatorname{res}(x^n-1, (x+1)^n-1).$$ Truer to its name, one may also define it as the determinant $\det(A)$ of the circulant matrix with entries $A_{i,j} = \binom{n}{\lvert i-j\rvert}$.

The Wendt determinant was of interest historically to number theory because of its connection to Fermat's last theorem. The sequence is available on the OEIS as A048954, beginning as follows: $$1, -3, 28, -375, 3751, 0, 6835648, -1343091375, \dotsc$$

I have recently become interested in some of the prime factors of the Wendt determinant, a list of which is available online. Specifically, I am wondering: for which $n$, relatively prime to 6*, is $W_n$ divisible by $3$? I am interested in any result that gives a sufficient condition for $W_n$ to not be divisible by 3.

The small $n$, relatively prime to $6$, for which $W_n$ is divisible by $3$ are multiples of 13, 121, 671, and 757 (note that $W_m$ divides $W_n$ if $m$ divides $n$). I was not successful in finding this sequence or any other related sequence in the OEIS.

* I ask for relatively prime to $6$ for some technical reasons. Every sixth entry is zero, and also every even entry is known to be divisible by three. I am also interested in which of the even entries is twice divisible by $3$, ie divisible by $9$.