[Math Lair] Armstrong Numbers

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An Armstrong number is a n-digit number that is equal to the sum of each of its digits taken to the nth power. For example, 153 is an armstrong number because 153 = 1³ + 5³ + 3³. Other than the numbers 1 through 9, it is the smallest Armstrong number; there are none with two digits. After 153, the next smallest Armstrong numbers are 370, 371, 407, 1,634, 8,208, and 9,474. There are only 89 Armstrong numbers in total. The largest Armstrong number is 115,132,219,018,763,992,565,095,597,973,971,522,401, which has 39 digits. It has been proven that there are no Armstrong numbers with more than 39 digits.

If the ones digit of an Armstrong number is 0, then the following number will also be an Armstrong number, because 1n is always equal to 1. Consecutive Armstrong numbers are quite rare; there are only 9 pairs in total. The smallest pair is 370 and 371; the largest is 115,132,219,018,763,992,565,095,597,973,971,522,400 and 115,132,219,018,763,992,565,095,597,973,971,522,401.

One related investigation that might be of interest is the following:

Another investigation that might be of interest is to find out what happens when you take a number and find the sum of its digits to different powers.

While we're discussing the number 153, that number has several other interesting properties. We've already mentioned that 153 = 1³ + 5³ + 3³. It is also a triangular number; 1 + 2 + 3 + ... + 17 = 153. Finally, 153 = 1! + 2! + 3! + 4! + 5!.

It's also interesting that the number 153 makes an appearance in the Bible: "So Simon Peter went aboard and hauled the net ashore, full of large fish, a hundred fifty-three of them; and though there were so many, the net was not torn." (John 21:11, NRSV). It isn't known whether the author of John had any of these properties in mind when writing this verse.

Sources used (see bibliography page for titles corresponding to numbers): 50.