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 3 of 4: Check the Beatty parameters

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\frac1{\log_2 10}+\frac1{\log_5 10}=\log_{10}2+\log_{10}5=1

Detailed analysis

Set $\alpha=\log_2 10$ and $\beta=\log_5 10$ . They are irrational and satisfy $1/\alpha+1/\beta=1$ because $1/\log_2 10=\log_{10}2$ and $1/\log_5 10=\log_{10}5$ .