An arithmetic note: powers of integers and decimal digits
Abstract
n number theory, several arithmetical questions have been formulated and answered concerning the representation of integers in base 10. For example, Kempner (1914) shows that by suppressing in the harmonic series all terms that do not have a given digit, it converges. Recently, Maynard (2019) has shown that the primes that do not have a given digit in their decimal representation are infinite. In connection with the representation of digits in base ten, a wide range of questions can be asked as the following ones.
For a fixed integer n, can one anticipate the type of digits that appear when n is raised to various powers? Given a block of digits, is there u, a positive integer, such that in the decimal representation of n^u that block of digits appears? For which primes p, is there an integer u such that in the decimal representation of p^u, given blocks of π appear? The aim of this note is to provide answers to the above questions.
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References
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