Find how many solutions there are in the interval $[1, 10]$ to the fractional part equation:
$$\left\{x\right\}^2 = \left\{x^2\right\}$$
Where $\{\cdot\}$ is the fractional part function, meaning that:
$$\left\{a\right\} = a – \left\lfloor a \right\rfloor$$
Some research about the problem:
I graphed both functions on a Graphing Calculator:
And the problem was looking like it had a tremendous amount of solutions!
Approach
The equation is equivalent to:
$$(x – \lfloor x\rfloor)^2 = (x^2 – \lfloor x^2\rfloor)$$
So:
$$x^2 – 2x\lfloor x\rfloor + \lfloor x\rfloor^2 = x^2 – \lfloor x^2\rfloor$$
Further investigations lead to:
$$2x\lfloor x \rfloor \in \mathbb{Z}$$
In which I hope I could find a clue for solving, using divisibility arguments, however no information appeared obvious to me from this.
(References: the graph was made using GeoGebra Graphing)
Best Answer
Let $x=i+f, 0\le f<1$ (integer plus fractional parts).
The equation turns to
$$\{(i+f)^2\}=f^2$$ which simplifies to $$2if=n$$ for some $n$.
Hence the solutions come with all fractions $$f=\frac{n}{2i}$$ with $0\le n <2i$.
Now count the possible values of $n$ for $i\in[1,10]$.