May 5, 2010

Mystery Solved!

I suspect that few regular readers of my blog have read about digital invariants in the Recreational Mathematics section of Lionel Deimel’s Farrago. The discussion there is about numbers such as 153, a pluperfect digital invariant (PPDI), whose digits bear a peculiar relationship to the number itself:

13 + 53 + 33 = 1 + 125 + 27 = 153

Such numbers are also called Armstrong numbers, presumably named after someone whose name was Armstrong. None of the literature or mathematicians I have consulted have been able to identify the mysterious Armstrong. However, I received an e-mail message the other day from a Michael Armstrong that appears to clear up the mystery:
You asked:
“PPDIs are sometimes called Armstrong numbers, though I have been unable to ascertain the source of this term. The term is likely older than pluperfect digital invariant, though it seems less useful. If you know the origin of the name Armstrong number, I would like to hear from you.”

In the mid 1960s -- probably around 1966 -- I was teaching an elementary course in Fortran and computing in general at The University of Rochester, and “invented” Armstrong Numbers as an exercise for my students. I still have the original coffee-stained paper that was the master copy for the homework assignment and would be happy to send you a copy if the silverfish in the garage haven’t totally devoured it. (Full Disclosure: My memory being what it is(n’t), I can’t say that I sat down and invented them out of whole cloth, but I certainly don’t remember reading about them anywhere. The paper and assignment were meant as a spoof on serious mathematical papers that often didn’t seem to have much purpose. In any event, I am reasonably certain that this was the first association of the name with the numbers.)

I remember there were Armstrong Numbers of several Kinds and Orders, but don’t remember much detail, which is pretty much true of most of my life in the 60s. The students tried to compute almost all of them, and the sharper ones quickly realized that Fortran wasn’t the best way to do the job. They rewrote their algorithms first in assembly language (for an IBM 7000 series machine), and later in hard machine language to get the last bit of speed possible. As a reward, we ran the winning algorithm as the system’s idle process for a few nights, resulting in a very long list of Armstrong numbers (of the first Kind, anyway).

As serendipity would have it, I was in Australia at a meeting in February, 1988, when a short piece on “Armstrong’s Numbers” by Tim Hartnell, one of their regular columnists, was printed in The Australian (Tuesday, February 23rd). I immediately dashed off a note to him asking if he was talking about MY Armstrong Numbers, or some other Armstrong’s Armstrong numbers. We carried on a brief and cordial correspondence, and he published a followup article about finally finding out who “The Great Man” was in the April 19 issue. I guess that was my 15 minutes of fame.

I discovered your website as a consequence of researching some Tom Lehrer songs. Go figure; where would we be without the Internet?

Best regards,

Michael Armstrong
Polk City, Florida
There is little reason to doubt that Michael Armstrong is indeed the Armstrong of Armstrong number, though it is unclear how his name became attached to the concept, as he seems not to have discussed such numbers in any public forum. The numbers were first identified at least a quarter century earlier than Armstrong’s assignment, though he seems to have been the first person to explore the concept in at least a little depth. (I hope the garage may yield some interesting material.)

I plan to credit Michael Armstrong in my discussion of digital invariants, but I am first writing this post to see if anyone knows how, after giving his assignment, Armstrong’s name came to be attached publicly to the numbers in the assignment. Perhaps one of his students was responsible, or perhaps he discussed the numbers with a colleague who than used “Armstrong number” in print somewhere. Can anyone clear up this point?


  1. Lionel, I really liked your blog and Armstrong on the numbers, I think you certainly did not know the English mathematician Godfrey H. Hardy knew the concept of them. you just go to the link below and check in MathWorld everything I said

    I also liked this letter from Michael Armstrong, which gives more truth to them ...

    Goodbye and see you soon ... !!!!!!!!!

  2. I wonder who first noticed that Armstrong's identity is the first of a sequence, with a typical entry being 166^3 + 500^3 + 333^3 = 166,500,333.


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