Digital Invariants

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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 Numbers and operators 153, which is the sum of the cubes of 1, 5, and 3.) Since Michael did not have access to the requisite computing power, 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 be a few months 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. 2/18/2016


The pages below should be read sequentially for a systematic treatment of digital invariants. Visitors with some knowledge of the topic may have reason to jump to individual pages.


Distributions—remarks on the distribution of digital invariants

PPDIs—tables of all PPDIs in various bases

Observations & Theorems—mostly theorems about digital invariants

Recent Results—more theorems and conjectures

Armstrong Numbers—new information and theorems about digital invariants

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