What are Narcissistic Numbers?

Numbers can be many things: they can be odd, they can be irrational, and they can be friendly. Today on the Center of Math Blog, we'll be talking about a different type of number: the narcissistic number!

What is a Narcissistic Number?

A narcissistic number is a number that, as its name might suggest, is full of itself. An n-digit number is narcissistic if the number is equal to the sum of each of its digits to the nth power. 153, for example is a narcissistic number. This is because it is a 3-digit number, and we can observe that 1³ + 5³ + 3³ = 153. We can very easily name 9 narcissistic numbers: the first nine positive integers are all narcissistic numbers - any number to the first power is equal to itself. There are no two digit narcissistic numbers, but there are a few three digit narcissistic numbers. In addition to 153, the three narcissistic numbers are 370 (3³ + 7³ + 0³ = 370), 371 (3³ + 7³ +1 ³ = 371), and 407 (4³ + 0³ + 7³ = 407).

How many Narcissistic Numbers are there?

You might expect there to be an infinite number of narcissistic numbers. However, there is a finite number of narcissistic numbers, and it might not be as many as you think there are. As proven by D. Winter, and verified by D. Hoey, there exist only 88 Narcissistic Numbers. How were they able to prove it? They were able to prove that a narcissistic number could not exist even in theory for a number with greater than 60 digits, as n*9ⁿ will be less than 10ⁿ  for any n > 60. Thus, there has to be a finite number of narcissistic numbers. From there, it was simply a matter of plugging and chugging. The largest of the narcissistic numbers is 39 digits long, and narcissistic numbers with 38, 37, and 35 digits!

Are there Narcissistic Numbers in different bases?

The counting system we've talked about above is base-10 (where there are only 10 digits), but narcissistic numbers exist in other bases as well. For example in base 3, 5 is a narcissistic number. 5 written in base 3 is 12, and 1+4=5, making it a narcissistic number. For a full list of narcissistic numbers in base 10 and different bases, check out this link from Wolfram MathWorld: http://mathworld.wolfram.com/NarcissisticNumber.html