5 5 5 5 4150 = 4 + 1 + 5 + 0 = 1024 + 1 + 3125 + 0.

We call 4150 an order-5 (for the exponent), length-4 (for the number of digits) PDI. Being a PDI is a property of the number and its base (radix). It is easy to verify, for example, that 4150, interpreted as an octal (base-8) number, is not an order-5 PDI. On the other hand, the octal number 4423 is an order-5 PDI because

5 5 5 5 4423 = 4 + 4 + 2 + 3 = 2000 + 2000 + 40 + 363,

where all the arithmetic shown is in octal. Because the radix of the number is significant, we should speak of 4150 as an order-5, length-4, base-10 PDI and of 4423 as an order-5, length-4, base-8 PDI. (Abbreviations are possible; we might say 4150 is an order-5 PDI, understanding that the base is 10 and assuming the length is apparent.)

A PDI whose order is the same as its length is known as a pluperfect digital invariant, or PPDI. The decimal number 8208, for example, is an order-4 PPDI, since

4 4 4 4 8208 = 8 + 2 + 0 + 8 = 4096 + 16 + 0 + 4096.

PDIs and PPDIs have been tabulated, but questions remain about their properties and distribution. (It is not known if there are PPDIs of orders other than 1 in all bases, although there are non-trivial PPDIs in nearly all bases.) These are not burning questions of mathematics, but they have received a degree of attention, particularly from the recreational mathematics community. This program performs exhaustive searches for PDIs and PPDIs.

References:

- Martin Gardner, THE INCREDIBLE DR. MATRIX, pp. 205-209. New York: Charles Scribner's Sons, 1976.
- Lionel Deimel & Michael Jones, "Finding Pluperfect Digital Invariants: Techniques, Results and Observations." JOURNAL OF RECREATIONAL MATHEMATICS 14:2, pp. 87-107, 1981-1982.

This information was extracted from file pdi.a.

David A. Wheeler (wheeler@ida.org)