Let $R={\left(8+3\sqrt{7}\right)}^{20}$  and $\left[R\right]=$  the greater integer less than or equal to $R$ . Then

$\begin{array}{c}0<F<1\\ 0<f^1<1\end{array}\}\begin{array}{c}0<F+f^1<2\\ F+f^1=1\end{array}$

$f^1={\left(8-3\sqrt{7}\right)}^{20}$

$I+F+f=2k$

$I=2k-1$

$\left[R\right]=$  is odd integer

Show that $R-\left[R\right]=1-\frac{1}{{\left(8+3\sqrt{7}\right)}^{20}}$

$F+f=1$

$F=1-f$

$R-I=1-f$

$R-\left[R\right]=1-f$

$R-\left[R\right]=1-{\left(8-3\sqrt{7}\right)}^{20}$

$R-\left[R\right]=1-\frac{1}{{\left(8+3\sqrt{7}\right)}^{20}}$

$R-\left[R\right]=1-\frac{1}{R}$

$R^2-R\left[R\right]=R-1$

$R+R\left[R\right]=1+R^2$