Bases containing non-trivial PPDIs

Lionel E. Deimel, Jr. and Michael T. Jones ([6,7]) have also proven a series of interesting results about PPDIs. In particular, they have shown that almost *any* base contains at least one non-trivial PPDI by giving an example of a number that is a PPDI in a base meeting the criteria. A listing of base/example pairs is shown in table 1. In each example a set of curly braces '{}' represents the base-$B$ digit whose value is equal to the expression inside the braces.

Table 1: Examples of PPDIs in various bases from Lionel T. Deimel, Jr. and Michael T. Jones

Applicable Bases	Base $B$ Example
Every odd base ($2k + 1$, $k \geq 1$)	$\{k+1\}\{k+1\}_{2k+1}$
$B = 3k+1$ for $k \geq 1$	$\{k\}\{2k+1\}0_{B}$
$B = 18k + 6$, $k \geq 0$	$\{14k + 5\}\{4k+1\}\{12k+4\}_{B}$
$B = 18k + 12$, $k \geq 0$	$\{10k+6\}\{8k+6\}\{12k + 8\}_{B}$
$B = 18k^{2}$, $k \geq 1$	$\{9k^{2} - 3k\}\{9k^{2}\}_{B}$
$B = 36k^{2}$, $k \geq 1$	$\{12k^{2} - 4k\}\{24k^{2} - 2k\}1_{36k^{2}}$
$B = 90k+18$, $k \geq 0$	$\{18k+4\}\{36k+8\}_{90k+18}$
$B = 90k + 72$, $k \geq 0$	$\{18k+14\}\{36k + 29\}_{90k + 72}$

As a result of these findings, they have been able to provide the following corollary:

Corollary: There is at least one nontrivial PPDI in every base $B > 2$, except possibly where $B = 18k$ and all of the following are true:

1. $k$ is neither a perfect square nor twice a perfect square, and
2. neither $k - 1$ nor $k + 1$ is divisible by 5.

They also point out that the cases which are excluded by this theorem would be extremely difficult to provide examples for; for instance if $k = 5$ and $b = 90$, then the smallest possible nontrivial PPDI would be 8 digits long! One other theorem that they have proven about PPDIs is the following:

Theorem: There are infinitely many numbers that are nontrivial PPDIs in at least two bases.

Proof: $\{k\}\{2k+1\}\{0\}_{3k+1} = \{k\}\{0\}\{2k+1\}_{3k+2}$, $k \geq 1$. They have also provided one other significant theorem which will be presented in the section on PDIs.