The seemingly trivial, innocent number 8208 actually has something special. It is a member of a niche class of numbers called narcissistic numbers.
What’s so special?
First note it has 4 digits. If we raise every digit 8, 2, 0, 8 to the 4th power, and then add them together:
84 + 24 + 04 +84 = 8208
(To find 84, just think 84 = 212 = 4 x 210 = 4096)
Now read the four digits of the total. It’s 8, 2, 0, 8 again!
See the point? All the digits of the number are applied to generate that same number. Repeat the same procedure once more, you still get the same number. Seems this number loves itself so its ego is repeated. — That speaks for its name: narcissistic.
The 3-digit number 153 has similar property.
Since it has 3 digits, we raise every digit 1, 5, 3 to the 3rd power, and add them together!
13 + 53 + 33 = 153
Another time to witness the magic!
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Any one-digit number belongs to the class of narcissistic numbers. Reason like this: (if you raise everything to exponent one, it is just itself); this is plain yet may be too trivial to be interesting.
And tell you another secret. There is NOT ANY 2-digit number that is narcissistic (we assume that tens-digit cannot be zero: it has to be between 10 and 99). Some may be close (almost narcissistic!): for example
32 + 52 = 34; 72 + 52 = 74 (note 34 is just one off from 35)
That’s it. Happy reading!