One day, a student named Michael Jones walked into my N.C. State
University office with a proposal. Michael had been reading one of Martin Gardner’s
books of mathematical curiosities (Gardner, Martin. The Incredible
Dr. Matrix. New York: Charles Scribner’s Sons, 1976.), and he had concluded
that some open questions about interesting but apparently meaningless numbers
called perfect digital invariants (or PDIs) and pluperfect
digital invariants (PPDIs) could be resolved by performing computer
searches for them. (Formal definitions of PDI and PPDI appear below. An example
of a PPDI is the three-digit number  153, which is the sum of the cubes of 1, 5,
and 3.) Since Michael did not have access to the requisite computing
facilities, he suggested that we jointly develop the necessary software to carry out
the proposed searches.
To this day, I’m not sure why Michael came to me. He was not my student, so
he was likely referred by another student or by a colleague. In any case, the
project appealed to me, partly because the techniques that would be needed were
similar to ones I had used to find pentomino
solutions—that is, solutions to another puzzle popularized by Martin
Gardner—as an undergraduate at the University of Chicago.
Michael and I wrote a succession of programs, beginning with a Pascal program
for the Digital Equipment Corporation VAX 11/780 and culminating in a FORTRAN 77
program for the Data General MV/8000. Numbers found by the
search programs were verified using a PL/I program running on an IBM 370/165. (Years later, I re-implemented our search
algorithm in an Ada program, which appeared as part of an Educational
Materials package from the Software Engineering Institute. Click
here to view the package.)
After running our programs 24 hours a day for weeks, we made some
discoveries, I proved some theorems, and we published our results in the Journal
of Recreational Mathematics. There are some references to this work on
the World Wide Web (including also some findings of recent vintage), but someone seeking to
understand what is known of PDIs and PPDIs is at a disadvantage without access to the aforementioned journal. What
is presented here summarizes Michael’s and my article and presents more recent
work.
If you are aware of additional results not presented here, I would very much appreciate being
told about them.
My most recent addition here involves Armstrong numbers. This
term is a longstanding alternate name for PPDIs, but I had been unable,
until recently, to determine its origin. Michael F. Armstrong wrote to
me several years ago with evidence that he is the Armstrong of Armstrong
number fame. You can read what he told me in his first e-mail message to
me on my
blog. The final page listed below is the result of my
correspondence with Mike Armstrong and my own investigations based on
his original definitions.
— LED, 1/30/2003, rev. 7/23/2022
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