Question

How many positive integers \(N\) less than 1000 are there such that the equation \(x^{\lfloor x\rfloor} = N\) has a solution for \(x\)? (The notation \(\lfloor x\rfloor \) denotes the greatest integer that is less than or equal to \(x\).)

Answer 1

This is on a no-calculator problem...

Answer 2

Hint: notice that
\[4^4 = 256\]\[5^5 = 3125\]

Answer 3

So we can have x=1 (N=1), x=2 (N=4), x=3 (N=27), x=4 (N=256).
What about a non-integer value of x?

Answer 4

Without a calculator I would probably just start testing values by taking advantage of exponent properties, like
\[4^4.5 = 4^4 * 4^{(1/2)} = 512\]

Answer 5

Sorry I meant to say \[4^{4.5}\] at the beginning of that.

Answer 6

oh, ok. but the exponent must be an integer, not the base.

Answer 7

Oh, that makes it even easier then. You can just test values with long multiplication.

Answer 8

Would a decimal to an integer exponent ever equal an integer?
Like \(2.5^2=6.25\).

Answer 9

Oh, sorry I'm really butchering this, I was looking at parts of the question in isolation.

Answer 10

Square roots will always either be integers or irrationals.

Answer 11

like maybe \(\sqrt{5}\)? Cuz that's just over 2, and you would square that. So, I'd have root 5, root 6, root 7, root 8 ( more)

Answer 12

Yeah, I don't know why I was trying to find the greatest x that gave a value under 1000 initially.

Answer 13

it was a good start. So I have root 5-root8, then the cube roots of 28 thru 63, then the fourth roots of 257 to 624.
That gives me 4 integer values of x, 4 square root values of x, 36 cube root values of x, and 368 fourth root values of x. No fifth roots, though.
So, 4+4+36+368 = 412. Does that seem right?

Answer 14

It is right. (I just checked the answer...)
https://www.artofproblemsolving.com/Wiki/index.php/2009_AIME_I_Problems/Problem_6

Answer 15

Oh, an AIME problem. No wonder that was more interesting than usual.

Answer 16

yeah, I'm trying to prep. This is the first #6 I've gotten right. My goal is 4 or 5.