## narcissistic numberAlso 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
3^{3} + 7^{3} + 1^{3} = 371, and 9474 is narcissistic
because 9^{4} + 4^{4} + 7^{4} + 4^{4} =
9474. The smallest narcissistic number of more than one digit is 153 = 1^{3}
+ 5^{3} + 3^{3}. 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, 10^{70}-1
is a number consisting of 70 nines in a row, and the sum of the 70th powers
of its digits is 70 × 9^{70} is approximately equal to 4.386051
× 10^{68}, 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 | |||||

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