Powerful Numbers

[jatill]

1024 is a "powerful number" because 1024 is
1) expressed in simplest form (i.e. no leading 0's, etc),
2) can be represented as x^y, where x and y are integers, and
3) the number of characters in the number 1024 is the same as are the number of characters in x^y (i.e. 2^10 => 4 characters).

a) What is the highest powerful number?
b) What is the lowest powerful number?
c) What is the powerful number closest to 0?

[Zag]

For the lowest, how about: 4^2 or 2^4 = 16

Of course, you probably should have said the lowest in absolute value. There are negative powerful numbers, I'm sure. (-27, for instance = (-3)^3, which, I think, is the largest negative powerful number.)

[bonanova]

Zag, does your LPN candidate obey (3)? Maybe I don't understand (3).
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[Trojan Horse]

For the one closest to 0: .000001=1000^-2 (with 7 characters in each) is the best I've been able to come up with so far. (Assuming that's not cheating.)

[Zag]

Oh. I didn't read the rule carefully enough to realize we were supposed to count the caret. Of course, when you write it in normal math style, there isn't one.

[Jake2*]

For the largest, I get 99856 (316^2).

[jatill]

[Trojan Horse]

[jatill]

[Death Mage]

[jatill]

[Trojan Horse]

Oh, duh... [998001=(-999)^2]

I'm assuming, based on the last comment, that parentheses don't count. If they did count, we could do this: [9999800001=(-99999)^2]

[jatill]

[Trojan Horse]

*sigh*

Probably still not the best, but [9998244=-3162^2] works.

I await your reply that there is still a better answer out there.

[jatill]

You finally nailed part (a)