Let P(x) be a polynomial of degree n. Let H(i) represent the number of 1's in the binary expansion of the integer i. Although reasonably easy to prove, it may seem surprising that the following identity holds:
$\sum_{i=0}^{2^{n+1}-1} (-1)^{H(i)}P(i) = 0$
This was asked here on the "Complex Projective 4-Space" blog by apgoucher.
I don't know if the theorem has a name. I found the result (with variations) from the very nice book Elementary Number Theory - A Problem Oriented Approach written in beautiful calligraphy by Joe Roberts, p. 88 to be precise. I warmly recommend that book to aspiring students from high school age and up.
Anyway, he indicates that the result was first published by M.E. Prouhet in an article titled Mémoire sur quelques relations entre les puissances des nombres that appeared in Comptes Rendus in 1851.