Introduction to PDIs and PPDIs

The number 153 has the follwing special property - if you sum the cube of each of its digits, you end up with 153, which was the original number. i. e. $1^{3} + 5^{3} + 3^{3} = 153$. In a similar way, if you sum each of the digits in 1470 raised to the 5th power, you get 1470: $1^{5} + 4^{5} + 7^{5} + 0^{5} = 1470$. The general term used to describe numbers that can be expressed as a function of their digits is Narcissistic Numbers. More specifically, the term Perfect Digital Invariant(PDI) is used to denote a number that is equal to the sum of it's base $B$ digits raised to a power. If the power is equal to the number of digits in the base $B$ representation of the number, then it is further qualified to a PluPerfect Digital Invariant (PPDI). When talking about PDIs and PPDIs, the term order refers to the power that each digit is raised to. Thus, 153 is an order-3 PPDI in base 10, and 1470 is an order-5 PDI in base 10.

There is more that is known about PPDIs than about PDIs. The reasons for this are two-fold-

• It is easier to construct proofs about PPDIs, since the exponent is constrained to a very specific value. In contrast, the lack of a restriction on the exponent for PDIs removes the possibility of many of the assumptions that are necessary for the PPDI proofs. Instead of definitive proofs, all that can be said about PDIs is that if a PDI were to exist, it would have to fall within a set of criteria.
• It will be demonstrated later in this paper that for any base, there is an upper bound on the order of any PPDIs in that base. Although empirical evidence suggests that this bound overestimates the true limit, its existence means that an exhaustive search can be done (with the aid of a computer) to determine all PPDIs for any base. On the other hand, for PDIs, not only is no such bound known to exist, but many people conjecture that the opposite is true - that there is an infinite number of PDIs for all bases other than 2.