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x_{n+1} = x_n + P(x_n), P(x) is product of digitss, x_n bounded?

Source: Austrian Polish 1982 APMC

April 30, 2020
number theorySequencerecurrence relationboundedunboundedDigitsproduct of digits

Problem Statement

Let P(x)P(x) denote the product of all (decimal) digits of a natural number xx. For any positive integer x1x_1, define the sequence (xn)(x_n) recursively by xn+1=xn+P(xn)x_{n+1} = x_n + P(x_n). Prove or disprove that the sequence (xn)(x_n) is necessarily bounded.
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