Proving for fractional parts

[idiotinabox]

Essentially the question asks if x^3 = [(x + 1)^3] then find all real solutions of x. The square brackets in this case represent the fractional part of the number they enclose.

So far I have figured out that [z] is the number minus the largest integer smaller than itself. I have no idea how to go about finding all real solutions. Any help would be appreciated

 

[idiotinabox]

Sorry about a mistake in my writing of the question. It is actually [(x+1)^3]. The other case is easy, yes, but that was incorrectly written on my part. So now i remain stuck

 

[idiotinabox]

Thanks a lot for that. I'm sure working on these isn't the most enjoy full activity. I'll see what I can do. I think I may have a paper outlining something similar. I'll keep battering away. Cheers

 

[pka]

I am now a little uncertain what the problem is: \(\displaystyle x^3 = [(x + 1)^3]\)?
If so, I can see that it is a harder problem, and I am not going to work it out. If you get stuck on my suggestion or it simply does not work, come back and ask for help from someone smarter than me.

I separated that simpler problem into a negative and non-negative part because I am not quite sure how [z] is defined if z < 0. You will have to look at the definition to see whether that separation is necessary.

Here is a rather good web page on factional parts. Please note that the fractional is denoted by \(\displaystyle \{x\}\), which is excepted notation by most.
Note that \(\displaystyle \{-2.7\}=0.3\) because \(\displaystyle \left\lfloor { - 2.7} \right\rfloor = - 3~.\) Thus \(\displaystyle \left\{ { - 2.7} \right\} = - 2.7 - \left\lfloor { - 2.7} \right\rfloor = - 2.7 - ( - 3) = 0.3\).

 

[pka]

Now I am even less sure what the OP meant because he does not seem to be using the standard notation.

I think the OP is clear. What \(\displaystyle x\) is it true that \(\displaystyle x^3=\{(x+1)^3\}\)
Now we know that \(\displaystyle 0\le\{x\}<1\) and we know that \(\displaystyle 0<x^3<1\text{ in and only if }0<x<1.\)
So any solution to the OP must belong to \(\displaystyle [0,1]\).

 

[idiotinabox]

So the fractional part is between 0 and 1. X must be between 0 and 1 because x^3 is between 0 and 1. In turn, that means that x+1 is between 1 and 2 and (x+1)^3 is between 1 and 8. From there I looked at the cases where x^3 = (x+1)^3 -1, -2,-3 etc... (coming back to {x} = x - [x]). What do I do from here to actually solve it?