From: rcs@xmission.com

This looks like an interesting puzzle. His sum of k = z^-3 + ... + z^3 can also be read as the sum of a few cosines, or as the sum of a geometric series.  It should be easy to gather more data.

I've already posted something very similar to NMBRTHRY and the OP. Here's a slightly faster Pari/GP script to aid the gathering of more data: ? least=.001;forstep(i=1,2000,2,z1=exp(2*Pi*I/i);for(k=1,i\2,if(gcd(k,i)>1,,z=z1^k;s=1;for(j=1,i\2,s+=z^j;r=2*real(s);d=abs(r-round(r));if(d>1e-20&&d<=least,least=d;print(i" ^"k" "j"\t"d)))))) Phil