Problem 063

[mrlinx]

I've stumble upon some problems with hard descriptions, but Problem 63 seems to make a stand.

If possible, I would appreciate if anyone could clarify the Problem 63.

Its my understanding that there's a infinity of numbers that satisfy the property of having a power to represent the value.
I've tried with the base and exponent from 1 to 9, but doesn't seems to get accepted.

[rayfil]

Its my understanding that there's a infinity of numbers that satisfy the property of having a power to represent the value.

I would agree with that statement. However, there is a finite number which will satisfy the parameters of the problem. For example, a number represented by 1000000ⁿ can never have n digits, regardless of the value of n.
Similarly, x^1000000 can definitely never have exactly 1000000 digits, regardless of the value of x.

Have another good look at the examples provided in the problem description.

[StormXI]

Just a few quick questions on the problem.

Consider the form A^N = B where length of B = N...

Q1 - How can I find the limit/max for N?
Q2 - Can A be negative as well, as long as B is positive?

Thanks.

[daniel.is.fischer]

Question 1: Well, that's your task, so no hints here.
Question 2: If A is negative and N odd, then A^N is negative, if N is even, A^N = (-A)^N, so the sign of A doesn't matter, but you need only consider positive A.

[Tommy137]

StormXI's additional information aren't correctly displayed (http://projecteuler.net/index.php?secti ... try=Jordan). Maybe, Euler could take a look at it.


To Problem 63: Try to find an upper bound for A first, then find a limit for N (even possible with pen & paper)

[StormXI]

I solved the problem. The thing was I didn't know was that each n-digit number needed to be unique.

I figured it out now. Thanks anyway.

[euler]

StormXI's additional information aren't correctly displayed (http://projecteuler.net/index.php?secti ... try=Jordan). Maybe, Euler could take a look at it.

Maybe he's changed it since you made this post, but it looks okay to me?

[Tommy137]

StormXI's additional information aren't correctly displayed (http://projecteuler.net/index.php?secti ... try=Jordan). Maybe, Euler could take a look at it.

Maybe he's changed it since you made this post, but it looks okay to me?

Ah yes, he changed it from

"NAME >> ..."
to
"NAME // ..."

The > were not displayed, but something like &...;

[MrTuring]

So the problem wants to know:
"How many n-digit positive integers exist which are also an nth power?" Problem 63 (View Problem)

Unless I'm missing something 9^n for all n (>0 of course) is an n-digit number. I tried it out briefly in Scala, this isn't exactly what I wrote but it hopefully gets the idea across:

powers = for(i <- 1 to 20) yield {BigInt(9).pow(i)}
powers.map(_.length) = 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20

Basically there is always an n-digit number that is the nth power of 9. Am I missing something? Please help, it's driving me crazy

[jaap]

9²² = 9,84770,90218,36112,32881

[MrTuring]

Oh wow, don't I feel like an idiot hah I stopped my naive first attempt with some trial-and-error checking just before it would've failed...thank you very much!

[oenone]

I assumed that 0¹ = 0 counts, too. But seems to have been false.

[hk]

I assumed that 0¹ = 0 counts, too. But seems to have been false.

How many n-digit positive integers exist which are also an nth power?

Is 0 a positive integer?

[oenone]

I assumed that 0¹ = 0 counts, too. But seems to have been false.

How many n-digit positive integers exist which are also an nth power?

Is 0 a positive integer?

I knew I missed something...

[JMW1994]

bⁿ(n = nth-digit and b = base). Can the base can exceed nine?

[TripleM]

The problem doesn't put any limits on the base, so yes.