Problem 5

List

A

contains the decimal numbers

10^k

for integers

k\ge1

. Lists

B

and

C

contain these same numbers written in bases

2

and

5

, respectively. Prove that for every integer

n>1

, exactly one number in exactly one of

B

C

has exactly

n

digits.

Step 4 of 4: Apply Beatty's theorem

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\{\lfloor k\alpha\rfloor:k\ge1\}\cup\{\lfloor k\beta\rfloor:k\ge1\}=\mathbb Z_{>0}

Detailed analysis

Beatty's theorem says the sequences $\lfloor k\alpha\rfloor$ and $\lfloor k\beta\rfloor$ partition the positive integers. These are exactly $b_k-1$ and $c_k-1$ , so every $n>1$ occurs exactly once as a digit count in exactly one list.