Armstrong Number

An Armstrong number (also called a narcissistic number or pluperfect number) is a non-negative natural number expressed in base 10 with $ n $ digits and equal to the sum of the nth powers of each of its digits.

In other words, if a number $ N $ can be written with the digits $ d_1, d_2, \dots, d_n $, then $ N = d_1^n + d_2^n + \dots + d_n^n $

To check if a number is an Armstrong number:

— Count the number of digits $ n $ in the number

— Extract each digit and calculate its nth power

— Calculate the sum of these powers

— Compare the sum to the original number

If the two values are equal, the number is an Armstrong number!

The complete list of Armstrong numbers is finite and known; it contains 89 terms:

The largest therefore has 39 digits.

The list is referenced on OEIS A005188 here

A rigorous mathematical proof can be provided using the theorem of comparative growth.

An n-digit number becomes very large very quickly as n increases.

The sum of the powers of its digits increases much more slowly.

Beyond a certain number of digits, the sum can no longer catch up with the number itself.

Here is an example of source code in Python:// Python

def is_armstrong(n):

m = len(str(n))

sum = sum(int(d)**m for d in str(n))

return sum == n

These numbers were named in reference to Michael F. Armstrong, a mathematician who popularized these numbers in the 1960s.