Construction of Smith Number from Prime Repunit

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Theorem

Let $R_n$ be a repunit which is prime where $n \ge 3$.

Then $3304 \times R_n$ is a Smith number.


$3304$ is not the only number this works for, neither is it the smallest, but it serves as an example of the technique.


Proof

Let $\map S n$ denote the sum of the digits of a positive integer $n$.

Let $\map {S_p} n$ denote the sum of the digits of the prime decomposition of $n$.

Then $\map S n = \map {S_p} n$ if and only if $n$ is a Smith number.


Let $n \ge 3$.

We have that:

$3304 = 2 \times 2 \times 2 \times 7 \times 59$

and so for a prime repunit $R_n$:

$3304 \times R_n = 2 \times 2 \times 2 \times 7 \times 59 \times \underbrace {111 \ldots 11}_{n \text { ones} }$

and so:

$\map {S_p} {3304 \times R_n} = 2 + 2 + 2 + 7 + 5 + 9 + n \times 1 = n + 27$


Note that:

\(\ds \) \(\) \(\ds 3304 \times R_n\)
\(\ds \) \(=\) \(\ds 3304 \sum_{k = 0}^{n - 1} 10^k\) Basis Representation Theorem
\(\ds \) \(=\) \(\ds 3 \sum_{k = 0}^{n - 1} 10^{k + 3} + 3 \sum_{k = 0}^{n - 1} 10^{k + 2} + 4 \sum_{k = 0}^{n - 1} 10^k\) Basis Representation Theorem
\(\ds \) \(=\) \(\ds 3 \sum_{k = 3}^{n + 2} 10^k + 3 \sum_{k = 2}^{n + 1} 10^k + 4 \sum_{k = 0}^{n - 1} 10^k\) Translation of Index Variable of Summation
\(\ds \) \(=\) \(\ds 3 \times 10^{n + 2} + 6 \times 10^{n + 1} + 6 \times 10^n + 10 \sum_{k = 3}^{n - 1} 10^k + 7 \times 10^2 + 44\)
\(\ds \) \(=\) \(\ds 3 \times 10^{n + 2} + 6 \times 10^{n + 1} + 6 \times 10^n + \sum_{k = 4}^n 10^k + 7 \times 10^2 + 44\) Translation of Index Variable of Summation
\(\ds \) \(=\) \(\ds 3 \times 10^{n + 2} + 6 \times 10^{n + 1} + 7 \times 10^n + \sum_{k = 4}^{n - 1} 10^k + 7 \times 10^2 + 44\)
\(\ds \) \(=\) \(\ds [367 \underbrace {11 \dots 1}_{\paren {n - 4} \text { ones} } 0744]\) Basis Representation Theorem

This gives:

\(\ds \map S {3304 \times R_n}\) \(=\) \(\ds 3 + 6 + 7 + \paren {n - 4} + 0 + 7 + 4 + 4\)
\(\ds \) \(=\) \(\ds n + 27\)
\(\ds \) \(=\) \(\ds \map {S_p} {3304 \times R_n}\)

Hence the result.

$\blacksquare$


Sources

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