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Putnam
1989 Putnam
A6
A6
Part of
1989 Putnam
Problems
(1)
power series in F2
Source: Putnam 1989 A6
8/23/2021
Let
α
=
1
+
a
1
x
+
a
2
x
2
+
…
\alpha=1+a_1x+a_2x^2+\ldots
α
=
1
+
a
1
x
+
a
2
x
2
+
…
be a formal power series with coefficients in the field of two elements. Let
a
n
=
{
1
if every block of zeroes in the binary expansion of
n
has an even number of zeroes
0
otherwise
a_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}
a
n
=
{
1
0
if every block of zeroes in the binary expansion of
n
has an even number of zeroes
otherwise
(For example,
a
36
=
1
a_{36}=1
a
36
=
1
since
36
=
10010
0
2
36=100100_2
36
=
10010
0
2
) Prove that
α
3
+
x
α
+
1
=
0
\alpha^3+x\alpha+1=0
α
3
+
xα
+
1
=
0
.
calculus
power series