Armstrong numbers

Three of the four interesting base-10 examples were already a curiosity in the 1930s. G. H. Hardy, in A Mathematician's Apology (1940), used 153, 370, 371 and 407 — the only numbers above 1 that equal the sum of the cubes of their digits — as his specimen of a mathematical fact that goes nowhere: a coincidence of decimal notation with no generalisation behind it, fit for a puzzle column rather than a research paper. He meant it as a put-down, and he was right about the mathematics, which is exactly why the family has thrived in recreational and teaching contexts ever since.

The name "narcissistic number" is generally traced to Joseph S. Madachy's Mathematics on Vacation (1966), which collected the pattern alongside other self-referential digit games. "Pluperfect digital invariant" and "plus perfect number" are the terms used in the more systematic literature on digital invariants. "Armstrong number" — overwhelmingly the name used in programming courses — is attributed to one Michael F. Armstrong, said to have set the problem as a class exercise, but the attribution is repeated far more often than it is documented, and no primary source for it is easy to find. Treat it as folklore with a plausible name attached.

The one genuinely satisfying result about them is that there are finitely many. A k-digit number is at least 10^(k−1), while the sum of the k-th powers of its digits is at most k · 9^k. For k = 61 the second quantity is already smaller than the first, so no narcissistic number can have 61 or more digits. That bounds the search, and the full base-10 enumeration comes to 88 numbers, ending at a 39-digit term.

Key properties

• There are exactly 88 narcissistic numbers in base 10; OEIS A005188 lists 89 terms because it includes 0.
• The largest is 115,132,219,018,763,992,565,095,597,973,971,522,401, which has 39 digits.
• No narcissistic number has 61 or more digits, because k · 9^k < 10^(k−1) for every k ≥ 61 — so the whole sequence sits below 10^60.
• The three-digit members are exactly 153, 370, 371 and 407; the four-digit members are exactly 1634, 8208 and 9474.
• There are no two-digit narcissistic numbers, and none with 12 or 13 digits either.
• Each of 1 through 9 qualifies trivially, being a one-digit number equal to itself raised to the first power.
• 153 is also the 17th triangular number and equals 1! + 2! + 3! + 4! + 5!.

Where they turn up

• Writing a program to find the three-digit Armstrong numbers is one of the most common introductory exercises in programming courses, which is why the "Armstrong" name dominates outside mathematics.
• G. H. Hardy cited 153, 370, 371 and 407 in A Mathematician's Apology (1940) as examples of facts that are curious but mathematically sterile.
• 153 appears in the Gospel of John 21:11 as the number of fish in the miraculous catch, and has attracted a long tradition of numerological commentary — an interpretive tradition rather than a mathematical result.
• Narcissistic numbers are the digit-count-matched case of the broader family of perfect digital invariants, where the exponent is fixed independently of the number of digits.
• The sequence is a standard test case for arbitrary-precision arithmetic libraries, since verifying the 39-digit maximum exceeds 64-bit integers.