Armstrong number

I just came across the post on the Facebook page Art of Mathematics, which says:

13+53+33=153\textcolor{green}{1}^3 + \textcolor{orange}{5}^3 + \textcolor{blue}{3}^3 = \textcolor{green}{1}\textcolor{orange}{5}\textcolor{blue}{3}

163+503+333=165033\textcolor{green}{1}\textcolor{red}{6}^3 + \textcolor{orange}{5}0^3+ \textcolor{blue}{33}^3=\textcolor{green}{1}\textcolor{red}{6}\textcolor{orange}{5}0\textcolor{blue}{33}

1663+5003+3333=166500333\textcolor{green}{1}\textcolor{red}{66}^3 + \textcolor{orange}{5}00^3+ \textcolor{blue}{333}^3=\textcolor{green}{1}\textcolor{red}{66}\textcolor{orange}{5}00\textcolor{blue}{333}

16663+50003+33333=166650003333\textcolor{green}{1}\textcolor{red}{666}^3 + \textcolor{orange}{5}000^3+ \textcolor{blue}{3333}^3=\textcolor{green}{1}\textcolor{red}{666}\textcolor{orange}{5}000\textcolor{blue}{3333}

and so on.

And here is the proof:

Let A=1...6n 6s3+5...0n 0s3+3...3n+1 3s3A=\underbrace{1...6}_{n\ 6's}^{3} +\underbrace{5...0}_{n\ 0's}^{3} +\underbrace{3...3}_{n+1 \ 3's}^{3}

We have:

1...63 =[10n +23(10n1)]35...03=53×103n=125×103n3...33=(10n+113)31...6^{3} \ =\left[ 10^{n} \ +\dfrac{2}{3}\left( 10^{n} -1\right)\right]^{3}\\ 5...0^{3} =5^{3} \times 10^{3n} =125\times 10^{3n}\\ 3...3^{3} =\left(\dfrac{10^{n+1} -1}{3}\right)^{3}

Then

A=[10n +23(10n1)]3+125×103n+(10n+113)3=5003103n503102n+10310n13A=\left[ 10^{n} \ +\dfrac{2}{3}\left( 10^{n} -1\right)\right]^{3} +125\times 10^{3n} +\left(\dfrac{10^{n+1} -1}{3}\right)^{3}\\ =\dfrac{500}{3} 10^{3n} -\dfrac{50}{3} 10^{2} n+\dfrac{10}{3} 10^{n} -\dfrac{1}{3}

By multiplying 5003\dfrac {500} {3} with 103n10^{3n} using decimal places, we get:

5003103n=166.6×103n=166 6...6n 6s6...6n 6s6...6n 6s.6\dfrac{500}{3} 10^{3n} =166.\overline{6} \times 10^{3n} =166\ \underbrace{6...6}_{n\ 6's}\underbrace{6...6}_{n\ 6's}\underbrace{6...6}_{n\ 6's} .\overline{6}\\

Likewise,

503102n=16.6×102n=166...6n 6s6...6n 6s.6\dfrac{50}{3} 10^{2n} =16.\overline{6} \times 10^{2n} =16\underbrace{6...6}_{n\ 6's}\underbrace{6...6}_{n\ 6's} .\overline{6}\\

5003103n503102n=16...6n 6s50...0n 0s0...0n 0s\Rightarrow \dfrac{500}{3} 10^{3n} -\dfrac{50}{3} 10^{2n} =1\underbrace{6...6}_{n\ 6's} 5\underbrace{0...0}_{n\ 0's}\underbrace{0...0}_{n\ 0's}\\

For the last term of the polynomial:

10310n13=3.3×10n0.3=3...3n+1 3s\dfrac{10}{3} 10^{n} -\dfrac{1}{3} =3.\overline{3} \times 10^{n} -0.\overline{3} =\underbrace{3...3}_{n+1\ 3's}\\

Therefore,

A=5003103n503102n+10310n13=16...6n 6s50...0n 0s3...3n+1 3sA=\dfrac{500}{3} 10^{3n} -\dfrac{50}{3} 10^{2n} +\dfrac{10}{3} 10^{n} -\dfrac{1}{3} =1\underbrace{6...6}_{n\ 6's} 5\underbrace{0...0}_{n\ 0's}\underbrace{3...3}_{n+1\ 3's} \blacksquare

Note by Tín Phạm Nguyễn
9 months ago

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This is very interesting

Abha Vishwakarma - 8 months, 3 weeks ago

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Where does the name "Armstrong Number" come from?

Blan Morrison - 9 months ago

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I think the word "Armstrong" means repeated.

Ram Mohith - 8 months, 4 weeks ago

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An Armstrong number is an integer of nn digits; the sum of its digits each one raised to the power of nn equals the integer itself.

For example, 153153 is an Armstrong number because 153=13+53+33153 = 1^3 + 5^3 + 3^3

Tín Phạm Nguyễn - 8 months, 4 weeks ago

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