Critical Observation

While Schwartz's proof makes a search for all PPDIs in a given base possible in theory, the amount of numbers that need to be tested is too large to make such a search even remotely feasible. However, there is one very simple observation that drastically reduces the size of the set of numbers to be checked, and makes a computer search possible. The key is to separate the numerical value of the number from it's representation in base $B$. That is, instead of thinking of a number as a value, think of it instead as a collection of digits. For it to be a PPDI (or a PDI), the sum of each of those digits raised to a power $t$ must be some permutation of the original digits. However, the sum does not depend on the order of the values being added; that is, *any* permutation of those digits will equal the same sum. Therefore, if a collection of digits raised to a power produces a PPDI, it will only produce a PPDI for a single permutation of those digits and no other.

As an example, take the base 5 number 2124. $2^{4} + 1^{4} + 2^{4} + 4^{4} = 2124$ in base 5. Therefore, it is a PPDI. However, we now know that any other permutation of the digits 2, 1, 2 and 4 in base 5 is not a PPDI, because no matter what order the digits are in, the sum of each of them raised to the $4^{th}$ power will still equal 2124 in base 5. This reduces the problem of searching for all PPDIs in base $B$ from checking $B^{L} - 1$ numbers to checking

$$\sum_{i = 1}^{L-1}\left( \begin{array}{c} B i \end{array}\right)$$

numbers, where $L$ is the constant calculated previously.