DE-BUNDESWETTBEWERB 2023 Problem 4

Given a real number $\alpha$ in whose decimal representation $\alpha=0,a_1a_2a_3\dots$ each decimal digit $a_i$ $(i=1,2,3,\dots)$ is a prime number. The decimal digits are arranged along the path indicated by arrows in the accompanying figure, which can be thought of as continuing infinitely to the right and downward. For each $m\geq 1$ , the decimal representation of a real number $z_m$ is formed by writing before the decimal point the digit 0 and after the decimal point the sequence of digits of the $m$ -th row from the top read from left to right from the adjacent arrangement. In an analogous way, for all $n\geq 1$ , the real numbers $s_n$ are formed with the digits of the $n$ -th column from the left to be read from top to bottom. For example, $z_3=0,a_5a_6a_7a_{12}a_{23}a_{28}\dots$ and $s_2=0,a_2a_3a_6a_{15}a_{18}a_{35}\dots$ .

Show:

(a) If $\alpha$ is rational, then all $z_m$ and all $s_n$ are rational. (b) The converse of the statement formulated in (a) is false.

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