Also known as narcissistic numbers, Armstrong numbers are the sum of their own digits to the power of the number of digits. As that is a slightly brief wording, let me give an example:

153 = 1³ + 5³ + 3³

Each digit is raised to the power three because 153 has three digits. They are totalled and we get the original number again! Notice that Armstrong numbers are base dependent, but we'll mainly be dealing with base 10 examples here.

Armstrong numbers are certainly rare. They cannot have more than 60 digits in base 10, because for n > 60

n · 9^{n} < 10^{n-1}

Since there is an upper limit to their size, it is theoretically possible to find all of them, given sufficient computer time. However, 10^{60} is an unimaginably huge number, so such a "brute force" approach would be unwise. Luckily, D. Winter proved in 1985 that there are exactly 88 base-10 Armstrong numbers, and they must have 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 19, 20, 21, 23, 24, 25, 27, 29, 31, 32, 33, 34, 35, 37, 38 or 39 digits. Of course, the one digit Armstrong numbers are somewhat trivial since clearly 1^{1} = 1, 2^{1} = 2 etc.

The Armstrong numbers up to 10 digits are

1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, 1741725, 4210818, 9800817, 9926315, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153, 4679307774

The largest Armstrong number (in base 10) is the 39-digit beast:

115132219018763992565095597973971522401.

Other names for Armstrong numbers include "plus perfect numbers", "narcisstic numbers" and "perfect digital invariant numbers". Number theorists are so indecisive.