Narcissistic (Armstrong) numbers

The definition

Let n have d digits. Raise each digit to the power d and add the results. If you get n back, n is narcissistic.

Every single-digit number qualifies trivially, because d = 1 and each number equals itself.

The complete list up to five digits

There are no two-digit examples at all. This list was produced by checking every number up to 100,000.

Why there are only 88

Adding digits cannot keep up with growing numbers forever. A d-digit number is at least 10ᵈ⁻¹, but the largest possible sum of d digits raised to the power d is d · 9ᵈ. Once d · 9ᵈ is smaller than 10ᵈ⁻¹, no d-digit number can be narcissistic. That happens from d = 61 onwards.

So the search is finite, and it has been completed: there are exactly 88 narcissistic numbers in base ten. The largest has 39 digits:

Related families

• Münchhausen numbers raise each digit to its own power: 3435 = 3³ + 4⁴ + 3³ + 5⁵. Apart from 1, it is the only such number in base ten under the usual convention.
• Dudeney numbers equal the digit sum of their cube. There are exactly six: 1, 8, 17, 18, 26 and 27. For example 17³ = 4913 and 4 + 9 + 1 + 3 = 17.
• If you repeatedly replace a number by the sum of the cubes of its digits, it ends in 1, 153, 370, 371, 407 or one of a few short loops. This is the cube version of happy numbers.

Searching efficiently

The sum of digit powers ignores the order of the digits. So instead of testing every number, you can test every multiset of digits once, compute the power sum and check whether its digits are that same multiset. This reduces the work enormously and is how the complete list of 88 was found. The same trick makes it practical to analyse Kaprekar’s routine for longer numbers.

Serious mathematics or entertainment?

In A Mathematician’s Apology (1940), G. H. Hardy mentioned 153, 370, 371 and 407 as examples of facts that are curious but not important, because they depend on base ten rather than on any deep property of numbers. Kaprekar’s constant 6174 has the same character: it belongs to the way we write numbers, not to the numbers themselves.

That does not make such patterns worthless. They are excellent for practising powers, for learning to search systematically and for understanding why some searches must end, which is a key idea in proofs about happy numbers and Kaprekar’s routine.