Narcissistic Number Checker

n = Σdᵢᵏ

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About Narcissistic Number Checker

A narcissistic (Armstrong) number checker testing if n equals the sum of its digits each raised to the number of digits. Example: 153 = 1³+5³+3³. Shows digital decomposition and lists all narcissistic numbers for given digit counts. All client-side.

Narcissistic Number Checker Features

  • Check
  • Decomposition
  • Digit power
  • List
  • Range scan
Narcissistic/Armstrong number: n equals sum of its digits each raised to the power k (number of digits). 153 = 1³+5³+3³. 370 = 3³+7³+0³. 9474 = 9⁴+4⁴+7⁴+4⁴. There are exactly 88 narcissistic numbers in base 10.

How to Use

Enter n:

  • Check: Is n narcissistic?
  • Digits: k = number of digits
  • Sum: Σdᵢᵏ

All Narcissistic Numbers

1-digit: 1-9. 3-digit: 153,370,371,407. 4-digit: 1634,8208,9474. There are 88 total in base 10, the largest being a 39-digit number.

Other Bases

In base 2: only 1. In base 3: 1,2,12,22,122. The concept extends to any base, and the count is always finite because digit powers grow slower than the number itself beyond a threshold.

Step-by-Step Instructions

  1. 1Enter n.
  2. 2Check narcissistic.
  3. 3View digit decomposition.
  4. 4See power sum.
  5. 5Find all for k digits.

Narcissistic Number Checker — Frequently Asked Questions

Why are there only 88?+

For k-digit numbers, the max digit power sum is k·9ᵏ, while k-digit numbers go up to 10ᵏ−1. Since 9ᵏ grows slower than 10ᵏ/k, eventually k·9ᵏ < 10ᵏ⁻¹ (no k-digit number can be narcissistic). This happens at k=60, giving a finite search space. Exhaustive search finds exactly 88.

What is the largest narcissistic number?+

The largest is the 39-digit number 115132219018763992565095597973971522401. It was found by exhaustive search. There are no narcissistic numbers with 40+ digits in base 10.

Is this the same as Armstrong numbers?+

Yes! Armstrong numbers, narcissistic numbers, and pluperfect digital invariants (PPDI) all refer to the same concept. The name varies by region and textbook. Michael F. Armstrong is often credited, though Hardy and Wright also studied them.

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