Enter a positive integer to check if it is an Armstrong (narcissistic) number
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Digit Count
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Sum of Powers
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Detailed Derivation
Perfect Digital Invariant Definition
n = sum(digit_i^d) where d = number of digits
Also called: Armstrong, narcissistic, pluperfect
153 = 1^3 + 5^3 + 3^3 = 1 + 125 + 27
88 known PDI numbers in base 10
A perfect digital invariant (PDI) is a number that equals the sum of its digits raised to the power of the number of digits. These self-descriptive numbers are also called Armstrong numbers after Michael F. Armstrong.
⚠All single-digit numbers 0-9 are trivially PDI. There are no 2-digit PDIs in base 10. The largest has 39 digits.
What Is a Perfect Digital Invariant?
A perfect digital invariant is a number that equals the sum of its digits raised to the power of its digit length. These numbers are sometimes called narcissistic because they are self-referential. They are a fascinating class of numbers in recreational mathematics.
3-Digit PDIs
153, 370, 371, 407. These are the only four 3-digit Armstrong numbers. 153 is the most famous (the biblical number of fish).
4-Digit PDIs
1634, 8208, 9474. Only three 4-digit Armstrong numbers exist. 1634 = 1^4+6^4+3^4+4^4.
Naming History
Called Armstrong numbers after Michael Armstrong who used them as programming exercises. Also called pluperfect digital invariants or narcissistic numbers.
Base Dependence
PDI property is base-dependent. In base 2, only 0 and 1. In base 4: 0,1,2,3,130. Each base has a finite set of PDIs.
Teaching Example: Test 153. Digits: 1,5,3. Count = 3. Compute: 1^3=1, 5^3=125, 3^3=27. Sum = 1+125+27 = 153. Matches! 153 is a narcissistic number. Test 1634: 4 digits. 1^4+6^4+3^4+4^4 = 1+1296+81+256 = 1634. Also narcissistic!
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