No S(n)>S(n^5)?

Inspired by this and this, I tried the same thing with n^5 and couldn't find an x that worked (or it is much much larger). Is there an x such that the sum of the digits of x is greater than the sum of the digits of x^5?

#NumberTheory #ComputerScience

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

No output in 4 hours!! :(

Pranjal Jain - 6 years, 2 months ago

I crashed my computer doing this ... Nice question by the way

Soutrik Bandyopadhyay - 6 years, 2 months ago

Great question to ask!

Calvin Lin Staff - 6 years, 2 months ago

Well I tried the same thing with S(n)>S(n^4) and it didn't work either.

Ronak Agarwal - 6 years, 2 months ago

124499 works for the forth power. 1+2+4+4+9+9=29 124499**4=240250031031001002001. 2+4+2+5+3+1+3+1+1+2+1=25

David Holcer - 6 years, 2 months ago

Oh It seems my computer can't handle such large numbers.

Ronak Agarwal - 6 years, 2 months ago

@Ronak Agarwal – Depends on programming language you are working upon

Pranjal Jain - 6 years, 2 months ago

@Pranjal Jain – Do you think there is a way to prove whether or not there exists an x for x^5?

David Holcer - 6 years, 2 months ago

@David Holcer – Probably there exists some large $x$ for which the inequality is satisfied. Probably finite solutions exist only when you consider $x^{k}$ for very very large $k$ .

Jake Lai - 6 years, 2 months ago

@Jake Lai – Clearly if there is one such $x$ then there are infinitely many such $x$ ; just append zeroes at the end.

Ivan Koswara - 6 years, 2 months ago

@Ivan Koswara – Yes and this proves that either there are no solutions or there are infinite solutions.

Ronak Agarwal - 6 years, 2 months ago

@Ivan Koswara – Yeah, but I mean one is not a power of ten times of the other. Otherwise, it would be pretty boring, wouldn't it?

Jake Lai - 6 years, 2 months ago

@David Holcer – Not till now

Pranjal Jain - 6 years, 2 months ago

@Pranjal Jain – What, what number?!!

David Holcer - 6 years, 2 months ago

@David Holcer – I mean I don't know if there is any way to prove whether such n exists or not

Pranjal Jain - 6 years, 2 months ago

@Pranjal Jain – Oh, ok.Thnx.

David Holcer - 6 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$