Narcissistic Numbers

A narcissistic number (also called an Armstrong number or pluperfect digital invariant) is a number that equals the sum of its own digits, each raised to the power of the number of digits. For example, 153 = 1³ + 5³ + 3³ = 1 + 125 + 27 = 153. These self-referential numbers are rare and fascinating -- in base 10, only 88 of them exist.

How to Identify a Narcissistic Number

To check if a number is narcissistic: count its digits (call this n), raise each digit to the n-th power, and add the results. If the sum equals the original number, it is narcissistic. Here are some examples:

Complete List by Digit Count

Narcissistic numbers are rare. Here is the complete inventory organized by the number of digits:

In total, there are exactly 88 narcissistic numbers in base 10. The largest has 39 digits. Beyond that, the maximum possible digit-power sum cannot reach the number itself, so no more can exist.

Properties of Narcissistic Numbers

History and Etymology

The name "narcissistic number" was coined by mathematician D. H. Lehmer in reference to numbers that are fixated on themselves, like the mythological Narcissus. They are also called Armstrong numbers after Michael F. Armstrong, who introduced them in a 1969 assignment for computer science students at the University of Rochester. The OEIS (On-Line Encyclopedia of Integer Sequences) lists them as sequence A005188.

Narcissistic Numbers up to 10,000

Click on any narcissistic number to see its complete mathematical analysis.

Large Narcissistic Numbers

Beyond the small examples, narcissistic numbers grow quickly in size. Here are some notable larger specimens with their digit-power decompositions:

Did You Know?

• 153 is a narcissistic number: 1³ + 5³ + 3³ = 153
• All single-digit numbers are narcissistic: 1¹ = 1, 2¹ = 2, etc.
• In base 10, there are exactly 88 narcissistic numbers total
• The largest narcissistic number is 24678050 with 8 digits
• No narcissistic numbers with 9 or more digits exist in base 10