MathDB

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.

Problem Statement