Narcissistic Number — Definition, Formula & Examples

A narcissistic number (also called an Armstrong number) is a number that equals the sum of each of its digits raised to the power of how many digits it has. For example, 153 is narcissistic because

1^3 + 5^3 + 3^3 = 153

An $n$ -digit number $N$ in base 10 is a narcissistic number if $N = d_1^n + d_2^n + \cdots + d_n^n$ , where $d_1, d_2, \ldots, d_n$ are the individual decimal digits of $N$ .

Key Formula

N = \sum_{i=1}^{n} d_i^{\,n}

Where:

$N$ = The number being tested
$d_i$ = The $i$-th digit of $N$
$n$ = The total number of digits in $N$

How It Works

To check whether a number is narcissistic, first count how many digits it has — call that count

n

. Then raise each digit to the

n

th power and add the results. If the sum equals the original number, it is narcissistic. All single-digit numbers (0 through 9) are trivially narcissistic since any digit raised to the first power equals itself. The interesting cases start at three digits: the narcissistic numbers with three digits are 153, 370, 371, and 407.

Worked Example

Problem: Determine whether 370 is a narcissistic number.

Count the digits: 370 has three digits, so

n = 3

Cube each digit: Raise each digit to the 3rd power.

3^3 = 27, \quad 7^3 = 343, \quad 0^3 = 0

Sum the results: Add the cubes together.

27 + 343 + 0 = 370

Answer: The sum equals the original number, so 370 is a narcissistic number.

Why It Matters

Narcissistic numbers are a classic recreational math topic that builds fluency with exponents and digit manipulation. They appear frequently in programming exercises and math competitions, making them a great way to practice both arithmetic skills and algorithmic thinking.

Common Mistakes

Mistake: Using a fixed exponent (like always cubing) instead of raising to the power of the digit count.

Correction: The exponent must equal the total number of digits. For a 4-digit number, raise each digit to the 4th power, not the 3rd.