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power series in F2

Source: Putnam 1989 A6

August 23, 2021
calculuspower series

Problem Statement

Let α=1+a1x+a2x2+…\alpha=1+a_1x+a_2x^2+\ldots be a formal power series with coefficients in the field of two elements. Let an={1if every block of zeroes in the binary expansion of n has an even number of zeroes0otherwisea_n=\begin{cases}1&\text{if every block of zeroes in the binary expansion of }n\text{ has an even number of zeroes}\\0&\text{otherwise}\end{cases}(For example, a36=1a_{36}=1 since 36=100100236=100100_2) Prove that α3+xα+1=0\alpha^3+x\alpha+1=0.
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