Write (2000 - 1/2)^N + (2000 + 1/2)^N = [(4000-1)^N + (4000+1)^N]/2^N

For even N, each term in the expression in brackets is a multiple of 4000 and therefore 4, except the last which = 2.  So the bracketed value is twice an odd number and will not be divisible by 2^N.

For odd N, terms with even exponent cancel and terms with odd exponent double.  The smallest of these = 2*N*(4000) which is divisible by 2^6 and no greater power of 2.

So N is limited to 1,3,5.  Checking I find each to be a solution.