# Difficult text problem

### Difficult text problem

Given is a real number α,
in its decimal representation
α = 0,a1a2a3 ... each decimal digit ai(i=1,2,3,...) is a prime number
is. The decimal digits are
along the line indicated by arrows in the adjacent figure,
to the right and downward infinitely continued to think way
to the right and downward. For each m ≥ 1, the decimal representation of a real number
zm is formed by placing the digit 0 in front of the decimal point and after the decimal point the
the sequence of digits of the m-th line from above, read from left to right, is written after the decimal point.
is written in the adjacent arrangement. In an analogous way
for all n ≥ 1 the real numbers sn with the digits of the n-th column to be read from top to bottom.
digits of the nth column from the left.
For example, z3 = 0,a5a6a7a12a23a28 ... and s2 = 0,a2a3a6a15a18a35 ... .
Show:
(a) If α is rational, then all zm
and all sn are rational.
(b) The converse of the statement in (a)
is false.
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Kokez

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