Numbers too can be ‘narcissistic’

What is the commonality between – 153, 370, 371, and 407?

We can observe that,

1³ + 5³ + 3³ = 1 + 125 + 27 = 153

3³ + 7³ + 0³ = 27 + 343 + 0 = 370

3³ + 7³ + 1³ = 27 + 343 + 1 = 371

4³ + 0³ + 7³ = 64 + 0 + 343 = 407

Similarly, we have three such four-digit numbers, 1634, 8208 and 9474.

1⁴ + 6⁴ + 3⁴ + 4⁴ = 1 + 1296 + 81 + 256 = 1634

8⁴ + 2⁴ + 0⁴ + 8⁴ = 4096 + 16 + 0 + 4096 = 8208

9⁴ + 4⁴ + 7⁴ + 4⁴ = 6561 + 256 + 2401 + 256 = 9474

There are three such five-digit numbers, 54748, 92727 and 93084 and one six-digit number, 548834. The readers may verify. Such numbers are called Narcissistic Numbers. The highest possible number of digits for which we can get a Narcissistic Numbers is 39. There are two such numbers –

115132219018763992565095597973971522400 and 115132219018763992565095597973971522401