Dik Winter NOTE: This page displays the content of the page originally located at http://homepages.cwi.nl/~dik/english/mathematics/armstrong.html. Dik Winter died in Amsterdam on December 28, 2009, and his pages on Armstrong numbers are no longer on the Web. I miss his collaboration.

For the convenience of readers, I have augmented the chart at the bottom of the page with the additional PPDIs found by Winter and originally embedded in the page as HTML comments.

Dik Winter

 

Armstrong numbers

Armstrong numbers (also called narcissistic numbers or pluperfect digital invariants (PPDI)) are well known in recreational mathematics. I started playing with them back in the 80's when a colleague remarked that his housenumber was an Armstrong number (153).

What is an Armstrong number?

An Armstrong number is an n-digit base b number such that the sum of its (base b) digits raised to the power n is the number itself. Hence 153 because 13 + 53 + 33 = 1 + 125 + 27 = 153.

Much more information can be found at the site of Lionel Deimel. The most principal information is that the number of Armstrong numbers for a particular base is finite. So, theoretically, you could list all Armstrong numbers up to a particular base, and that is what I have done, using a program of course. My first program was pretty fast compared to what I have found later in the literature. For instance I found references of weeks of computing all base 10 Armstrong numbers while my program did it at that time in about 34 minutes. Compare that to my current desktop computer (fairly old) which does it in 11 minutes, and my next desktop computer which will perform the same feat in 1.5 minutes! But searching times will be exponential on the base. The last base I did on the old computer (a CDC Cyber) was 12, and it took 36 hours 6 minutes and 30.061 seconds back in 1985. Later (1997) we had faster local computers so I could complete the search until base 16, but it still took quite some time. As far as I know it took slightly less than a year to complete base 16. So I will not continue on this path. The results can be found in the table.

An interesting question that Lionel Deimel raises was if there were non-trivial (i.e. more than one digit) Armstrong numbers for every base. The question is still open and not likely to be solved soon (I think). He gives formulas for basic Armstrong numbers for many bases, but the bases that are a multiple of 18 are difficult, and honestly, I do not see a pattern yet. Remarkable is the smallest base 90 Armstrong number which is 8 digits in length. It took me two days on my current desktop to find that number. Similar is base 270 with a seven digit solution. In the table below I will cover the bases I checked and that are not covered by Lionel Deimel's generic solutions, so now all bases less than 1000 are covered. I give here the smallest solution and the number of solutions of that size. If you want to see all solutions of that size look at the page source; the other solutions are there in html comments. [See note above. óLED]

You may remark that if there is a 2 digit solution pq to base b that there is also a 2 digit solution (b-p)q; this is easily proven with simple algebra. There is no such simple rule with more than 2 digits.

base smallest number of this size additional PPDIs
54 {43}{35}{17} 1  
90 {73}{62}{15}{62}{83}{18}{39}{47} 1  
126 {55}{71}{71} 3 {76}{92}{8}, {120}{24}{55}
180 {118}{62}{125} 4 {167}{13}{91}, {176}{64}{0}, {176}{64}{1}
216 {6}{36} 6 {36}{81}, {66}{100}, {150}{100}, {180}{81}, {210}{36}
234 {14}{56} 2 {220}{56}
270 {19}{143}{63}{63}{12}{30}{118} 1  
306 {34}{56}{144} 3 {60}{144}{135}, {181}{209}{125}
360 {30}{100} 6 {50}{125}, {140}{176}, {220}{176}, {310}{125}, {330}{100}
396 {11}{121}{0} 4 {11}{121}{1}, {248}{16}{287}, {301}{215}{216}
414 {4}{41} 2 {410}{41}
486 {150}{225} 2 {336}{225}
504 {130}{221} 2 {374}{221}
540 {32}{128} 4 {120}{225}, {420}{225}, {508}{128}
594 {258}{420}{27} 4 {360}{432}{0}, {360}{432}{1}, {479}{235}{359}
630 {78}{208} 2 {552}{208}
666 {50}{176} 2 {616}{176}
684 {40}{161} 14 {84}{225}, {94}{236}, {210}{316}, {220}{320}, {234}{325}, {310}{341}, {374}{341}, {450}{325}, {464}{320}, {474}{316}, {590}{236}, {600}{225}, {644}{161}
720 {222}{333} 2 {498}{333}
756 {176}{320} 2 {580}{320}
774 {4}{56} 2 {770}{56}
810 {40}{176} 2 {770}{176}
846 {50}{200} 2 {796}{200}
864 {200}{238}{504} 2 {790}{20}{459}
936 {71}{337}{288} 7 {343}{491}{522}, {368}{322}{621}, {626}{670}{144}, {916}{254}{261}, {918}{222}{271}, {935}{121}{45}
954 {294}{441} 2 {660}{441}
990 {58}{233} 2 {932}{233}