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According to Wolfram's MathWorld article "Narcissistic Number", such numbers are also called "Armstrong numbers".

Such a number is an $n$-digit number $N$ such that: $$N = {d_1}^n + {d_2}^n + \cdots + {d_n}^n$$ where $d_1, d_2, \ldots, d_n$ are the digits in $N$.

They crop up over and over again in collections of recreational mathematics problems.

The question that I can't find the answer to is: who is (or was) "Armstrong"?

The best guess I can find is a line on ProofWiki stating that "there are rumours that he may have been a Michael Armstrong of Polk City in Florida", but I can't find anything corroborating this.

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As you have pointed out, Armstrong numbers seem to have been named after Michael Armstrong of Polk City in Florida. This is clear from an email he wrote to a Web Blogger in which he talks about giving the problem of “Armstrong numbers” to his computing class at University of Rochester. However, it is not known how his name became attached to the class of numbers. Below is his email from Lionel Deimel's blog:

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.

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    $\begingroup$ Year of graduation found in Rochester Review, September–October 2014, Vol. 77, No.1 : Michael Armstrong ’63 Polk City, Florida $\endgroup$
    – njuffa
    Nov 27, 2021 at 10:47
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    $\begingroup$ Found in "The University of Rochester One Hundred and Thirteenth Annual Commencement, Sunday, June 9, 1963": Bachelor of Science [...] Electrical Engineering as Major [...] Michael Frederick Armstrong $\endgroup$
    – njuffa
    Nov 27, 2021 at 10:55
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    $\begingroup$ Obituary: Michael Frederick Armstrong passed away on July 20, 2020, in Hansville, Washington [...] Michael was born on March 6, 1941 to Frank and Virginia (Hawkins) Armstrong. He received his Electrical Engineering degree from the University of Rochester in 1963 and worked for the University of Rochester Computing Center for fifteen years. [...] $\endgroup$
    – njuffa
    Nov 27, 2021 at 10:59
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    $\begingroup$ First mention of Armstrong number in print that I could find: F. D. Federighi and Edwin D. Reilly, "Computer Science Laboratory Exercises", Schenectady, New York: Reidinger and Reidinger 1971, p. 9: "An n-digit number is an Armstrong number if the sum of the nth power of the digits is equal to the original number. Therefore, 371 is an Armstrong number [...]". There is a footnote (1) attached to the term Armstrong number, but Google won't let me see the footnote. My hypothesis is that the footnote explains after whom the term is named. $\endgroup$
    – njuffa
    Nov 27, 2021 at 11:20
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    $\begingroup$ Crux Mathematicorum, Volume 4-5, p. 276: In base 10, the smallest Armstrong number is 153 , and the largest we have been able to find in the literature is 4679307774 : 4679307774 = 4¹º +6¹º + 7¹º +9¹º +3¹º +0¹º + 7¹º + 7¹º + 7¹º + 4¹º . This extraordinary number was discovered in 1963 by Harry L. Nelson, according to [...] $\endgroup$
    – njuffa
    Nov 27, 2021 at 11:35

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