Narcissisitic Numbers .... The History and Etymology of Math Terms

 Narcissisitic Numbers  The term seems to come from the pen of Joseph S. Madachy in his Mathematics on Vacation, 1966.  His definition is broader than the current use (at least as I know it) . "A Narcissistic number is one which can be represented as some function of its digits ." He includes examples like 145 = 1! + 4! + 5!. It also would include things like n=(sum of digits of n)^(number of digits in n).  

Today the term is used for numbers n (in a particular base) in which the digits are each raised to the power of the number of digits n has and then summed and found to be equal to n.  An early example is 153 = 1^3 + 5^3 + 3^3.  It has been proven that there are only 89 numbers in base ten that have this quality. 

G H Hardy's A Mathematician's Apology mentions the four three-digit solutions, (without using any particular name for them) but dismisses them with, "There is nothing in these odd facts which appeals to the mathematician." 

Another related problem is is there a number A in n digits where the sum of the digits of A raised to the power n = some Other number B and so that the same function of B, gives A again. Are there any that form three-cycles A-> B -> C ->A etc. (Pssst, the answer is yes)

Other names that have been used for these numbers are Armstrong Numbers and PluPerfect digital invariants, which is usually abbreviated PPDI, and is from the Latin past tense for "more than perfect".  

Armstrong numbers is from the name of  Michael F. Armstrong, a computer science teacher who died in 2020.  It seems to have been created shortly after the Narcissistic name, and grew in popularity as one finding these numbers became a popular task for programming classes.

There is actually a little more to the story of 153.   

Pick any old number you want, and multiply by three (or just pick a number that is a multiple of three).

Now take all the digits and cube them and add the cubes together.

For example, if you picked 231, you would add 2^3 + 3^3 + 1^3 to get 36. 

Yeah, So what you're probably thinking... but take that new number and do the same thing... cube the digits and add them up... Nothing?  Keep going... eventually you get to 153, and then when you  do it again, you get 153 forever.

In slightly more formal language, it seems that 153 is the fixed point attractor of any multiple of three under the process of summing the cubes of the digits.