TITLE: Polynomials identity, factoring $x^{2^n}-1$
QUESTION [7 upvotes]: There is a proof that I can't solve.
Show that for any integer $k$, the following identity holds:
$$(1+x)(1+x^2)(1+x^4)\cdots(1+x^{2^{k-1}})=1+x+x^2+x^3+\cdots+x^{2^k-1}$$
Thanks for your help.

REPLY [11 votes]: This equation is a fancy way of stating existence and uniqueness of binary representation.