Narcissistic Number Calculator
Check if a number is narcissistic (Armstrong number) by verifying if it equals the sum of its digits each raised to the power of the digit count.
Enter any non-negative integer (up to 100 digits) to check if it is a narcissistic number
A narcissistic number (also known as an Armstrong number or pluperfect digital invariant) is a number that equals the sum of its own digits, where each digit is raised to the power of the total number of digits.
Definition:
A k-digit number n is narcissistic if: n = d₁ᵏ + d₂ᵏ + ... + dₖᵏ
where d₁, d₂, ..., dₖ are the k digits of n.
Algorithm:
- Count the number of digits (d) in the input number
- Extract each digit from the number
- Raise each digit to the power d
- Sum all the digit powers
- Compare the sum to the original number
Example: Is 153 a narcissistic number?
- 153 has 3 digits, so d = 3
- Digits are: 1, 5, 3
- 1³ + 5³ + 3³ = 1 + 125 + 27 = 153
- 153 = 153, so YES, it is narcissistic!
Example: Is 123 a narcissistic number?
- 123 has 3 digits, so d = 3
- Digits are: 1, 2, 3
- 1³ + 2³ + 3³ = 1 + 8 + 27 = 36
- 36 ≠ 123, so NO, it is not narcissistic
All 3-digit Narcissistic Numbers:
153, 370, 371, 407
All 4-digit Narcissistic Numbers:
1634, 8208, 9474
Mathematical Property: There are only 88 narcissistic numbers in base 10. The largest is a 39-digit number. This finiteness occurs because the maximum sum of digit powers (d × 9ᵈ) eventually grows slower than the minimum d-digit number (10ᵈ⁻¹).
Applications: Narcissistic numbers are primarily studied in recreational mathematics and number theory. They serve as interesting examples for teaching digit manipulation, exponentiation, and mathematical proofs about finite sets.
What is a narcissistic number?
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 is narcissistic because 1³ + 5³ + 3³ = 1 + 125 + 27 = 153.
What are all the narcissistic numbers?
There are exactly 88 narcissistic numbers in base 10. The single-digit narcissistic numbers are 0-9 (trivially). The multi-digit ones start with 153, 370, 371, 407 (3-digit), then 1634, 8208, 9474 (4-digit), and continue up to the largest: 115,132,219,018,763,992,565,095,597,973,971,522,401 (39 digits).
Why are they called narcissistic numbers?
They are called narcissistic numbers because, like the mythological Narcissus who fell in love with his own reflection, these numbers are 'in love with themselves' - they can be reconstructed from their own digits. The term was coined by Joseph S. Madachy in 1966.
What is the difference between narcissistic numbers and Armstrong numbers?
There is no difference - they are the same thing. 'Armstrong number' is another common name for narcissistic numbers, named after Michael F. Armstrong who studied them. Other names include 'pluperfect digital invariant' (PPDI) and 'plus perfect number'.
Are there infinitely many narcissistic numbers?
No, there are only finitely many narcissistic numbers in any base. In base 10, there are exactly 88 narcissistic numbers. This is because for a d-digit number, the maximum possible sum of digit powers is d × 9^d, which eventually grows slower than the minimum d-digit number (10^(d-1)), making it impossible for larger numbers to be narcissistic.