# narcissistic number

Also known as an Armstrong number or a plus perfect number, an n-digit number equal to the sum of its digits raised to the nth power. For instance, 371 is narcissistic because 33 + 73 + 13 = 371, and 9474 is narcissistic because 94 + 44 + 74 + 44 = 9474. The smallest narcissistic number of more than one digit is 153 = 13 + 53 + 33. The largest narcissistic number (in base 10) is 115132219018763992565095597973971522401, which is the sum of the 39th powers of its digits. The reason there are no larger numbers is related to the fact that, as the number of digits increases, more and more 9's are required to get a sum that has n digits. For example, 1070-1 is a number consisting of 70 nines in a row, and the sum of the 70th powers of its digits is 70 × 970 is approximately equal to 4.386051 × 1068, which is only 69 digits long. So there is no way any 70-digit number can be equal to the sum of the 70th powers of its digits. The reason we see the last number occur at 39 digits is because, as the limit is approached, the number of big digits like 8's and 9's has to increase to make sure the sum will be big enough, but this means that there are a lot fewer combinations of digits to choose from.

## Related category

• TYPES OF NUMBERS