TITLE: $\{x^2\} = \{x\}^2$, how many solutions in interval $[1, 10]$
QUESTION [9 upvotes]: 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)

REPLY [2 votes]: Wow it's have so many solutions...
$\left\{{x}^{\mathrm{2}} \right\}={x}^{\mathrm{2}} −\left[{x}^{\mathrm{2}} \right]=\left\{{x}\right\}^{\mathrm{2}}  \\ $
${x}^{\mathrm{2}} −\left\{{x}\right\}^{\mathrm{2}} =\left[\left(\left[{x}\right]+\left\{{x}\right\}\right)^{\mathrm{2}} \right] \\ $
$\left[{x}\right]\left(\left[{x}\right]+\mathrm{2}\left\{{x}\right\}\right)=\left[{x}\right]^{\mathrm{2}} +\left[\mathrm{2}\left[{x}\right]\left\{{x}\right\}\right] \\ $
$\mathrm{2}\left[{x}\right]\left\{{x}\right\}=\left[\mathrm{2}\left[{x}\right]\left\{{x}\right\}\right] \\ $
$\left(\mathrm{2}\left[{x}\right]\left\{{x}\right\}\right)\in{Z} \\ $
${x}\in\left\{\mathrm{1},\mathrm{1}\frac{\mathrm{1}}{\mathrm{2}},\mathrm{2},\mathrm{2}\frac{\mathrm{1}}{\mathrm{4}},\mathrm{2}\frac{\mathrm{2}}{\mathrm{4}},\mathrm{2}\frac{\mathrm{3}}{\mathrm{4}},\right. \\ $
$\:\:\:\:\:\:\:\:\mathrm{3},\mathrm{3}\frac{\mathrm{1}}{\mathrm{6}},\mathrm{3}\frac{\mathrm{2}}{\mathrm{6}},\mathrm{3}\frac{\mathrm{3}}{\mathrm{6}},\mathrm{3}\frac{\mathrm{4}}{\mathrm{6}},\mathrm{3}\frac{\mathrm{5}}{\mathrm{6}}, \\ $
$\:\:\:\:\:\:\:\:\mathrm{4},\mathrm{4}\frac{\mathrm{1}}{\mathrm{8}},\mathrm{4}\frac{\mathrm{2}}{\mathrm{8}},\mathrm{4}\frac{\mathrm{3}}{\mathrm{8}},\mathrm{4}\frac{\mathrm{4}}{\mathrm{8}},\mathrm{4}\frac{\mathrm{5}}{\mathrm{8}}, \\ $
$\:\:\:\:\:\:\:\:\mathrm{4}\frac{\mathrm{6}}{\mathrm{8}},\mathrm{4}\frac{\mathrm{7}}{\mathrm{8}},\mathrm{5},\mathrm{5}\frac{\mathrm{1}}{\mathrm{10}},\mathrm{5}\frac{\mathrm{2}}{\mathrm{10}},\mathrm{5}\frac{\mathrm{3}}{\mathrm{10}}, \\ $
$\:\:\:\:\:\:\:\:\mathrm{5}\frac{\mathrm{4}}{\mathrm{10}},\mathrm{5}\frac{\mathrm{5}}{\mathrm{10}},\mathrm{5}\frac{\mathrm{6}}{\mathrm{10}},\mathrm{5}\frac{\mathrm{7}}{\mathrm{10}},\mathrm{5}\frac{\mathrm{8}}{\mathrm{10}},\mathrm{5}\frac{\mathrm{9}}{\mathrm{10}}, \\ $
$\:\:\:\:\:\:\:\:\mathrm{6}\frac{\mathrm{1}}{\mathrm{12}},\mathrm{6}\frac{\mathrm{2}}{\mathrm{12}},\mathrm{6}\frac{\mathrm{3}}{\mathrm{12}},\mathrm{6}\frac{\mathrm{4}}{\mathrm{12}},\mathrm{6}\frac{\mathrm{5}}{\mathrm{12}},\mathrm{6}\frac{\mathrm{6}}{\mathrm{12}}, \\ $
$\:\:\:\:\:\:\:\:\mathrm{6}\frac{\mathrm{7}}{\mathrm{12}},\mathrm{6}\frac{\mathrm{8}}{\mathrm{12}},\mathrm{6}\frac{\mathrm{9}}{\mathrm{12}},\mathrm{6}\frac{\mathrm{10}}{\mathrm{12}},\mathrm{6}\frac{\mathrm{11}}{\mathrm{12}},\mathrm{7},\mathrm{7}\frac{\mathrm{1}}{\mathrm{14}} \\ $
$\:\:\:\:\:\:\:\:\mathrm{7}\frac{\mathrm{2}}{\mathrm{14}},\mathrm{7}\frac{\mathrm{3}}{\mathrm{14}},\mathrm{7}\frac{\mathrm{4}}{\mathrm{14}},\mathrm{7}\frac{\mathrm{5}}{\mathrm{14}},\mathrm{7}\frac{\mathrm{6}}{\mathrm{14}},\mathrm{7}\frac{\mathrm{7}}{\mathrm{14}}, \\ $
$\:\:\:\:\:\:\:\:\mathrm{7}\frac{\mathrm{8}}{\mathrm{14}},\mathrm{7}\frac{\mathrm{9}}{\mathrm{14}},\mathrm{7}\frac{\mathrm{10}}{\mathrm{14}},\mathrm{7}\frac{\mathrm{11}}{\mathrm{14}},\mathrm{7}\frac{\mathrm{12}}{\mathrm{14}},\mathrm{7}\frac{\mathrm{13}}{\mathrm{14}}, \\ $
$\:\:\:\:\:\:\:\:\mathrm{8},\mathrm{8}\frac{\mathrm{1}}{\mathrm{16}},\mathrm{8}\frac{\mathrm{2}}{\mathrm{16}},\mathrm{8}\frac{\mathrm{3}}{\mathrm{16}},\mathrm{8}\frac{\mathrm{4}}{\mathrm{16}},\mathrm{8}\frac{\mathrm{5}}{\mathrm{16}},\mathrm{8}\frac{\mathrm{6}}{\mathrm{16}}, \\ $
$\:\:\:\:\:\:\:\:\mathrm{8}\frac{\mathrm{7}}{\mathrm{16}},\mathrm{8}\frac{\mathrm{8}}{\mathrm{16}},\mathrm{8}\frac{\mathrm{9}}{\mathrm{16}},\mathrm{8}\frac{\mathrm{10}}{\mathrm{16}},\mathrm{8}\frac{\mathrm{11}}{\mathrm{16}},\mathrm{8}\frac{\mathrm{12}}{\mathrm{16}}, \\ $
$\:\:\:\:\:\:\:\:\mathrm{8}\frac{\mathrm{13}}{\mathrm{16}},\mathrm{8}\frac{\mathrm{14}}{\mathrm{16}},\mathrm{8}\frac{\mathrm{15}}{\mathrm{16}},\mathrm{9},\mathrm{9}\frac{\mathrm{1}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{2}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{3}}{\mathrm{18}}, \\ $
$\:\:\:\:\:\:\:\:\mathrm{9}\frac{\mathrm{4}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{5}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{6}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{7}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{8}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{9}}{\mathrm{18}}, \\ $
$\:\:\:\:\:\:\:\:\mathrm{9}\frac{\mathrm{10}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{11}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{12}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{13}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{14}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{14}}{\mathrm{18}}, \\ $
$\left.\:\:\:\:\:\:\:\:\mathrm{9}\frac{\mathrm{15}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{16}}{\mathrm{18}},\mathrm{9}\frac{\mathrm{17}}{\mathrm{18}},  \mathrm{10}\right\} \\ $