Armstrong and Munchausen Numbers and Other Neat Stuff
Palindromes and palindromic dates are one thing. It takes a daring intellect to sally forth into the realm of pure numbers. But when you get to this promised land you will not be disappointed. Numbers with incredible properties abound.
Twenty Six is a Simple and Unique Number
My favorite number is 26. An innocuous little fellow, not a prime, not a square, so why would anyone but its mother love it? Just a simple composite of 2 times 13, and yet 26 is one of a kind. It is the only integer that lies between the square of an integer and the cube of an integer. 5 squared is 25 and 3 cubed is 27. It seems odd, but there is no other combination like it, even if you search numbers with a googolplex of digits. 26 lives in its own little world. And that's just the start for interesting integers.
The Armstrong Numbers
Ah, yes, now we come to more complicated stuff. An Armstrong number is best defined by what you do to get one. Take each digit of a number and raise it to the power of the number of digits. Add them together and if the sum equals the original number you have an Armstrong number. An example will make it clear. All the numbers from one to 9 are Armstrong numbers: for example, 5 raised to the power of one (there's one digit) is 5. Not terribly exciting. The next Armstrong number after 9 is 153. Let's check it out. There are 3 digits. One to the third power is 1; 5 to the third power is 125; 3 to the third power is 27; 1 + 125 + 27 = 153. Armstrong numbers are also known as narcissistic numbers (no comment); but word wonks prefer the names 'pluperfect digital invariant' or 'plus perfect number.'
It can be proved that there are only 88 Armstrong numbers, the largest of which has 39 digits. This alone makes 153 special, but there's more. 153 is a triangular number, which means it is the sum of a sequence starting with one. 153 is the sum of 1, 2, 3, ......, 17. And that's not all. 153 is also a sum of the factorials from 1 to 5. Pretty good for a number that really gets no attention in the popular press.
The Munchausen Numbers
Here's a real beauty. A Munchausen number is one that if you take each digit and raise it to the power of that digit, the sum equals the original number. Again there's a boring case: one. One to the first power is one. And now here's the surprise: in the whole entire universe there is only one other Munchausen number. That number is 3435. Go ahead and try it. Add 3 to the third power, 4 to the fourth power, 3 to the third power, and 5 to the fifth power and see if you don't get 3435. And search as you may, you'll never find another one like it. Armstrong and Munchausen numbers just scratch the surface of the incredible properties of pure numbers. Sometimes it's just nice to step away form the workaday world and contemplate matters that have no particular meaning for, or relationship to, anything real.
Sources:
https://pages.mtu.edu/~shene/COURSES/cs201/NOTES/chap04/arms.html
