Floor Function

Algebra Level 5

How many positive integers N N less than 1000 are there such that it can be expressed as x x \large x^{\lfloor x \rfloor} for some real x x ?


Notation: \lfloor \cdot \rfloor denotes the floor function .


The answer is 412.

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1 solution

Hana Wehbi
Mar 28, 2017

First x x must be less than 5 5 , since otherwise x x x^{\lfloor x \rfloor} would be 3125 3125 which is greater than 1000 1000 .

Because x \lfloor x \rfloor must be an integer, we have to do the following:

For x = 0 \lfloor x\rfloor =0 , N = 1 N=1 , as long as x 0 x\ne 0 . This gives us 1 1 value of N N .

For x = 1 \lfloor x \rfloor = 1 , N N can be anything between 1 1 1^1 and 2 1 2^1 excluding 2 1 2^1 .

Therefore, N = 1 N=1 . However, we got N = 1 N=1 in case 1 1 so it got counted twice.

For x = 2 \lfloor x\rfloor =2 , N N can be anything between 2 2 2^2 to 3 2 3^2 excluding 3 2 3^2 . This gives us 3 2 2 2 = 5 N s 3^2-2^2= 5N's

For x = 3 \lfloor x \rfloor=3 , N N can be anything between 3 3 3^3 to 4 3 4^3 excluding 4 3 4^3 . This gives us 4 3 3 3 = 37 N s 4^3-3^3=37 N's .

For x = 4 \lfloor x \rfloor=4 , N N can be anything between 4 4 4^4 to 5 4 5^4 excluding 5 4 5^4 . This gives us

5 4 4 4 = 369 N s 5^4-4^4=369 N's . We stop here since x < 5 x<5 . Thus the possible answers for N = 1 + 5 + 37 + 369 = 412 N= 1+5+37+369=\boxed{412} .

For [x]=2 how you will get any N between 4 and 9 ???

Kushal Bose - 4 years, 2 months ago

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2 x 2^{\lfloor x \rfloor} = 2 2 2^2 anything between 4 and 9.

Hana Wehbi - 4 years, 2 months ago

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Suppose x = 2.5 x=2.5 so ( 2.5 ) [ 2.5 ] = 2. 5 2 (2.5)^{[2.5]}=2.5^2 which is not an integer

Kushal Bose - 4 years, 2 months ago

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@Kushal Bose But x = 2.5 x= 2.5 is not an integer either. The question asks for how many positive integers.

Hana Wehbi - 4 years, 2 months ago

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@Hana Wehbi Then how all integers between 4 to 9 can be achieved ?

Kushal Bose - 4 years, 2 months ago

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@Kushal Bose 4 , 5 , 6 , 7 , 8 4,5,6,7,8 aren't these are 5 positive integers between 2 2 and 3 2 2^2 \text{and }\ 3^2 excluding 3 2 3^2 .

Hana Wehbi - 4 years, 2 months ago

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