PlanetMath: Armstrong number

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Armstrong number

(Definition)

$\displaystyle n = \sum_{i = 1}^k d_ib^{i - 1}$

where

is the most significant, if it's also the case that for some power

$\displaystyle n = \sum_{i = 1}^k {d_i}^m$

also holds, then

is an Armstrong number or narcissistic number or plus perfect number or perfect digital invariant.

In any given base there is a finite amount of Armstrong numbers, since the inequality $k(b - 1)^m > b^{k - 1}$ is false after a certain threshold.

"Armstrong number" is owned by CompositeFan. [ owner history (1) ]

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Other names:

narcissistic number, plus perfect number, perfect digital invariant

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Cross-references: inequality, finite, equality, least significant digit, integer, base
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This is version 1 of Armstrong number, born on 2006-07-06.
Object id is 8123, canonical name is ArmstrongNumber.
Accessed 3321 times total.

Classification:

11A63 (Number theory :: Elementary number theory :: Radix representation; digital problems)

Pending Errata and Addenda

None.

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