The Beauty of Narcissistic Numbers

If you are a reader of Greek mythology, then you are probably familiar with Narcissus.  He was  a hunter of exceptional beauty that he died because he was unable to leave a pool after falling in love with his own reflection. That’s why I keep myself away from pools these days (kidding).

narcissus

Narcissus gazing at his own reflection (via Wikipedia)

In mathematics, he has kins by the name of narcissistic numbers — numbers that can’t get enough of themselves.  In particular, they are numbers that are the sum of their digits when raised to the power of the number of digits.

For example, 371 is a narcissistic number; it has three digits, and if we cube each digits 33 + 73 + 13 the sum is 371. Other 3-digit narcissistic numbers are

153 = 13 + 53 + 33
370 = 33 + 73 + 03
407 = 43 + 03 + 73.

There are also 4-digit narcissistic numbers, some of which are 1634, 8208, 9474 since

1634 = 14+64+34+44
8208 = 84+24+04+84
9474 = 94+44+74+44

 It has been proven that there are only 88 narcissistic numbers (in the decimal system) and that the largest of which is

115,132,219,018,763,992,565,095,597,973,971,522,401

has 39 digits.

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