I made this question https://math.stackexchange.com/questions/113760/binary-representation-of-powers-of-3 at Mathstackexchange but since i didnt got a sutisfacting answer i decided to ask it also here.

We write a power of 3 in bits in binary representation as follows. For example $3=(11)$, $3^2=(1001)$ which means that we let the $k$-th bit from the right be $1$ if the binary representation of this power of 3 contains $2^{k-1}$, and $0$ otherwise.

1. Prove that the highest power of 3 that has a palindromic binary representation is $3^3 = (11011)$.

2. Prove that $3 = (11)$ is the only power of 3 with a periodic binary representation (in the sense that it consists of a finite sequence of $1$s and $0$s repeated two or more times, like "$11$" consists of two repetitions of the bitstring "$1$").

I asked this question at Mathematics Stack Exchange but since I didn't got a satisfactory answer I decided to ask it here as well.

We write a power of 3 in bits in binary representation as follows. For example $3=(11)$, $3^2=(1001)$ which means that we let the $k$-th bit from the right be $1$ if the binary representation of this power of 3 contains $2^{k-1}$, and $0$ otherwise.

1. Prove that the highest power of 3 that has a palindromic binary representation is $3^3 = (11011)$.

2. Prove that $3 = (11)$ is the only power of 3 with a periodic binary representation (in the sense that it consists of a finite sequence of $1$s and $0$s repeated two or more times, like "$11$" consists of two repetitions of the bitstring "$1$").

I made this question http://math.stackexchange.com/questions/113760/binary-representation-of-powers-of-3 at Mathstackexchange but since i didnt got a sutisfacting answer i decided to ask it also here.

We write a power of 3 in bits in binary representation as follows. For example $3=(11)$, $3^2=(1001)$ which means that we let the $k$-th bit from the right be $1$ if the binary representation of this power of 3 contains $2^{k-1}$, and $0$ otherwise.

1. Prove that the highest power of 3 that has a palindromic binary representation is $3^3 = (11011)$.

2. Prove that $3 = (11)$ is the only power of 3 with a periodic binary representation (in the sense that it consists of a finite sequence of $1$s and $0$s repeated two or more times, like "$11$" consists of two repetitions of the bitstring "$1$").

I made this question https://math.stackexchange.com/questions/113760/binary-representation-of-powers-of-3 at Mathstackexchange but since i didnt got a sutisfacting answer i decided to ask it also here.

We write a power of 3 in bits in binary representation as follows. For example $3=(11)$, $3^2=(1001)$ which means that we let the $k$-th bit from the right be $1$ if the binary representation of this power of 3 contains $2^{k-1}$, and $0$ otherwise.

1. Prove that the highest power of 3 that has a palindromic binary representation is $3^3 = (11011)$.

2. Prove that $3 = (11)$ is the only power of 3 with a periodic binary representation (in the sense that it consists of a finite sequence of $1$s and $0$s repeated two or more times, like "$11$" consists of two repetitions of the bitstring "$1$").