A

number, with a length of

`n` digits, which has the property of being the

sum of the

`n`^{th} powers of its constituent digits.

Eh?

Take a trivial example: 1.

Now, 1 is obviously 1 digit long, so the `n`^{th} power is
1^{1} = 1.

This works for all 10 single digit integers (in base 10), so the first 10 narcissistic numbers are

0,1,2,3,4,5,6,7,8 and, eh ... 9

*Boring!* You say. Well, that's what the great mathematician G.H. Hardy said, but it gets a little more interesting when we tackle larger numbers. There are no solutions for two-digit numbers; the next narcissistic numbers are the three-digit

153 = 1^{3} + 5^{3} + 3^{3}

370 = 3^{3} + 7^{3} + 0^{3}

371 = 3^{3} + 7^{3} + 1^{3}

407 = 4^{3} + 0^{3} + 7^{3}

Four-digit numbers with the same property are 1634, 8208 and 9474; five digits gives us 54748, 92727 and 93084. The list goes on for a bit but there are only 88 narcissistic numbers in base 10^{{1}}, the largest being the lovely 39-digit

115132219018763992565095597973971522401

These numbers are sometimes also called Armstrong numbers or perfect digital invariants, but I prefer the term narcissistic, as in

“Excessive love or admiration of oneself”

-dictionary.com

Which leads to

some sources stating that narcissism in numbers is any expression using the same digits as the number, saying for instance that the expression

36 = 3! * 6

is narcissistic. I guess the jury's still out on this one.

^{{1}} As proved by D. Winter in 1985.

credit: mathworld.wolfram.com