Narcissistic / Armstrong Numbers Checker

Narcissistic (Armstrong) numbers explained

A narcissistic number equals the sum of its own digits when each digit is raised to the power of however many digits the number has. You'll also see them called Armstrong numbers, after Michael F. Armstrong, who used them in a programming course. The stricter names are plus perfect digital invariants (PPDI) or pluperfect digital invariants. Where does "narcissistic" come from? The number builds itself back up out of its own digits.

How it works

Take an n-digit number with digits d₁ d₂ … dₙ. The test is d₁ⁿ + d₂ⁿ + … + dₙⁿ = N. Since the exponent tracks the digit count, the whole property depends on which base you're working in. Every single-digit number 0–9 passes for free, because raising any digit to the power 1 leaves it unchanged. There are no two-digit narcissistic numbers at all. In base 10 the set turns out to be finite: there are exactly 88 narcissistic numbers, and the biggest one runs to 39 digits — 115132219018763992565095597973971522401. None can exceed 60 digits, since past that point the largest digit-power sum you could build is already too small to match an n-digit number. There's a cousin worth knowing, the perfect digit-to-digit invariants (Münchhausen numbers), where each digit is raised to itself instead of to n.

Worked example

Start with 153. It has 3 digits, so you cube each one: 1³ + 5³ + 3³ = 1 + 125 + 27 = 153. That checks out, which makes 153 narcissistic. A 4-digit example is 9474: 9⁴ + 4⁴ + 7⁴ + 4⁴ = 6561 + 256 + 2401 + 256 = 9474. Now look at 154, which gives 1³ + 5³ + 4³ = 1 + 125 + 64 = 190 ≠ 154. No match, so it fails.

FAQ

Why does the tool only list "known" ones up to a small range? Brute-force listing only stays practical over small ranges. In a window from 1 to about 4.6 billion you'll turn up just a few dozen. The full base-10 set of 88 numbers reaches into giants of up to 39 digits, and no scan over small integers will ever get there.

Are 153 and 9474 the only famous ones? Not even close. The 3-digit list also has 370, 371 and 407, and among 5-digit numbers you'll find 54748, 92727 and 93084. The 0, 1 and 2 cases count too, though they're just trivial single-digit invariants.