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:

Does this theorem have a name? I discovered a variation of it years ago when solving a problem given to me by James Aaronson, and applied it again this weekend to reduce an infinite problem to a finite one. One nice corollary of this theorem is that any arithmetic progression of numbers can be partitioned into two subsets, *A* and *B*, such that:

*A* and *B* contain equally many elements;
- The sums of the elements of
*A* and *B* are equal;
- The sums of squares of elements of
*A* and *B* are equal;
- (ellipsis)
- The sums of the
*n*th powers of elements of *A* and *B* are equal.

By iterative application of this theorem, we can partition an arithmetic progression of numbers into subsets with this property. This is similar in principle to the idea of a multimagic square, but strictly less impressive.

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If it doesn’t have a name, might I suggest ‘Flanelle’s Theorem’? ;P

I submitted this on math.stackexchange.com . No answer yet, but three upvotes.

http://math.stackexchange.com/questions/467605/does-this-theorem-have-a-name

It now has an answer, but not a name.

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