Smith Numbers are Infinite in Number

Theorem
There are infinitely many Smith numbers.

Algorithm
They can be generated from a $9$-repdigit as follows:


 * $(1): \quad$ Choose any $n \ge 2$ and factor the $9$-repdigit $m = 10^n - 1$.
 * $(2): \quad$ Compute $\map {S_p} m$, the sum of the digits of the prime decomposition of $m$, and set $h = 9 n - \map {S_p} m$.
 * $(3): \quad$ Solve $x = h - 7 b$ for $b, x \in \Z$, $2 \le x \le 8$.
 * $(4): \quad$ Find $t \in \set {2, 3, 4, 5, 8, 7, 15}$ such that $\map {S_p} t = x$.
 * $(5): \quad$ Then $M = t m \times 10^b$ is a Smith number.