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It is not difficult to see that there can be but a finite number of PPDIs for any given base. In particular, for sufficiently large order m, the smallest positional value possible is larger than the largest possible summation value. This fact allows us to enumerate all PPDIs in a given base, so long as there is sufficient computing power available. One of the major objectives of the search that Michael Jones and I undertook years ago was to find all PPDIs in bases 2 through 10. The resulting list was published in our Journal of Recreational Mathematics paper, which also includes a number of HPDIs. The next page includes complete listings for PPDIs in bases 2 through 12.

It is not immediately obvious that the number of PDIs in a given base is finite, since one can always increase the summation value by increasing k. In fact, we were able to prove that many bases contain an infinite number of PDIs.

 

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