Find all real number(s) $x$ satisfying the equation $\{(x +1)^3\}$ = $x^3$ , where $\{y\}$ denotes the fractional part of $y$

Question

Find all real number(s) $x$ satisfying the equation $\{(x +1)^3\}$ = $x^3$ , where $\{y\}$ denotes the fractional part of $y$ , for example $\{3.1416\ldots\}=0.1416\ldots$.

I am trying all positive real numbers from $1,2,\dots$ but I didn't get any decimals.

Is there a smarter way to solve this problem? ... Please advise.

Answer 1

Fractional part is always in $[0,1)$. So the domain of $g(x)=x^3$ you are looking for is such that $R_g\in[0,1)$, which happens to be $[0,1)$.

$y=\{(x+1)^3\}$ is basically $y=(x+1)^3$ chopped into appropriate pieces and translated down by some $k\in\mathbb{Z}$ where $k=\lfloor(x+1)^3\rfloor$

So we solve $$(x+1)^3-k=x^3$$ which gives $$k=3x^2+3x+1$$

Since for $[0,1)$, $0\le(x+1)^3<8$, we check $$3x^2+3x+1=0,1,2,3,4,5,6,7$$ whose positive roots give the answers.

Answer 2

Hints:

• $0 \le \{y\} \lt 1$
• so any solution to $\{(x +1)^3\} = x^3$ has $0 \le x^3 \lt 1$ and thus $0 \le x \lt 1$
• so $1 \le x+1 \lt 2$ and $1 \le (x+1)^3 \lt 8$
• any solution has $(x+1)^3 = x^3 +n$ for $n \in \{1,2,3,4,5,6,7\}$, which gives you seven quadratic equations to check
• for example, $x=0$ is a solution when $n=1$

Answer 3

Well, first of all you're checking only 'all' integers and not all reals which include also $1/2$ or $\pi$ for example. I will try to give you an outline of how I would approach it.

Second of all, note that the left-hand side is always between $0$ and $1$. So, what is implied about the range $x$ might be in?
Once, you determined that you should be splitting up the range of possible $x$ into smaller segments such that for each segment you can find a natural number $n$ such that $$\{(x+1)^3\} = (x+1)^3 - n.$$ Hint: there are $7$ segments that you should consider. Then for each segment it boils down in solving $$ (x+1)^3 - n = x^3.$$ This looks difficult at first sight but I guarantee it will simplify to a quadratic polynomial (which you can solve). Please don't forget that the solutions of this equation should still be in the range on which we consider solutions to be part of.

I hope this helps and if you have any questions about any of these steps, feel free to ask them.