These are the only numbers that have this property!

Level 2

A certain property $P$ is shared by the numbers $0, 1, 512, 4913, 5832,$ and $17576$ .

This property only holds for one other whole number $n$ .

What is $n$ ?

Hint: It has probably got something to do with the sum of the digits.

2 solutions

Pranjal Jain
Dec 9, 2014

It is clear from initial 3 numbers that first step is to check cube roots which come out to be $0,1,8,17,18,26$

Now next step is to notice that sum of digits of respective cubes are the numbers itself

Now the remaining number is 27 $27^{3}=19683,\ 1+9+6+8+3=27$

So our answer is $\boxed{19683}$

@ArielGershon No need to provide the max limit of n as $10^{10}$ . These are the only numbers possible

Pranjal Jain - 6 years, 6 months ago

How do you know that this is all the possible numbers? That makes this question really interesting.

Calvin Lin Staff - 6 years, 6 months ago

http://hostilefork.com/2009/12/24/six-dudeney-numbers-proof/

I found this computer program on a blog that proves there are only 6 numbers. I cannot see any mathematical way of proving it.

Satyen Nabar - 6 years, 6 months ago

@Satyen Nabar – If the cube root of n has x digits, the n has at most (3x) digits, sum of digits of n is at most 27x.

$(27x)^{3} \geq n \geq 10^{3x-3}$

This already gives a bound of at most 3 digits.

Joel Tan - 6 years, 6 months ago

@Joel Tan – Right, and that is why the original version of "numbers up to $10 ^ {10}$ is almost immediately equivalent to the unbounded version.

This is one way that CS can be used to prove math theorems. First we find a (ridiculously) large bound, and then we check all cases with a computer.

Calvin Lin Staff - 6 years, 6 months ago

@Satyen Nabar – Thanks for the link! Thats something nice to spend time upon!

Pranjal Jain - 6 years, 6 months ago

OEIS link says! Though it would be interesting to prove this!

Pranjal Jain - 6 years, 6 months ago

Yeah you're right, I'll take that part out. Good job solving it, by the way :)

Ariel Gershon - 6 years, 6 months ago

Dudeny Numbers !

Venkata Karthik Bandaru - 6 years, 2 months ago

Ashu Gupta
Dec 9, 2014

512=8^3 and 5+1+2=8 , and similar 4913=17^3 and 4+9+1+3=17 so next no is 19683=27^3 AND 1+9+6+8+3=27