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\begin{theorem} \label{tmn_0_0} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^0}{(2\cos(4\theta)^0},
\beta  = \frac{(2\cos(4\theta)^0}{(2\cos(8\theta)^0},
\gamma = \frac{(2\cos(8\theta)^0}{(2\cos(2\theta)^0}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_0_0} 
x^3 - 3 x^2 + 3 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -3 $. 
And 
\begin{equation} \label{fmn_0_0} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{0} 
\end{equation}  
\begin{equation} \label{gmn_0_0} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{0} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_0_3} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^0}{(2\cos(4\theta)^3},
\beta  = \frac{(2\cos(4\theta)^0}{(2\cos(8\theta)^3},
\gamma = \frac{(2\cos(8\theta)^0}{(2\cos(2\theta)^3}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_0_3} 
x^3 + 4 x^2 - 11 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -1 $. 
And 
\begin{equation} \label{fmn_0_3} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{1} 
\end{equation}  
\begin{equation} \label{gmn_0_3} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{8} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_0_6} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^0}{(2\cos(4\theta)^6},
\beta  = \frac{(2\cos(4\theta)^0}{(2\cos(8\theta)^6},
\gamma = \frac{(2\cos(8\theta)^0}{(2\cos(2\theta)^6}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_0_6} 
x^3 - 38 x^2 + 129 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 27 $. 
And 
\begin{equation} \label{fmn_0_6} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{125} 
\end{equation}  
\begin{equation} \label{gmn_0_6} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{216} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_0_9} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^0}{(2\cos(4\theta)^9},
\beta  = \frac{(2\cos(4\theta)^0}{(2\cos(8\theta)^9},
\gamma = \frac{(2\cos(8\theta)^0}{(2\cos(2\theta)^9}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_0_9} 
x^3 + 193 x^2 - 1460 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 41 $. 
And 
\begin{equation} \label{fmn_0_9} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{64} 
\end{equation}  
\begin{equation} \label{gmn_0_9} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{1331} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_0_12} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^0}{(2\cos(4\theta)^12},
\beta  = \frac{(2\cos(4\theta)^0}{(2\cos(8\theta)^12},
\gamma = \frac{(2\cos(8\theta)^0}{(2\cos(2\theta)^12}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_0_12} 
x^3 - 1186 x^2 + 16565 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 335 $. 
And 
\begin{equation} \label{fmn_0_12} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{2197} 
\end{equation}  
\begin{equation} \label{gmn_0_12} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{17576} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_0_15} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^0}{(2\cos(4\theta)^15},
\beta  = \frac{(2\cos(4\theta)^0}{(2\cos(8\theta)^15},
\gamma = \frac{(2\cos(8\theta)^0}{(2\cos(2\theta)^15}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_0_15} 
x^3 + 6829 x^2 - 187926 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 909 $. 
And 
\begin{equation} \label{fmn_0_15} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{4096} 
\end{equation}  
\begin{equation} \label{gmn_0_15} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{185193} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_1_2} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^1}{(2\cos(4\theta)^2},
\beta  = \frac{(2\cos(4\theta)^1}{(2\cos(8\theta)^2},
\gamma = \frac{(2\cos(8\theta)^1}{(2\cos(2\theta)^2}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_1_2} 
x^3 - 3 x^2 - 4 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -3 $. 
And 
\begin{equation} \label{fmn_1_2} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{0} 
\end{equation}  
\begin{equation} \label{gmn_1_2} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{7} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_1_5} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^1}{(2\cos(4\theta)^5},
\beta  = \frac{(2\cos(4\theta)^1}{(2\cos(8\theta)^5},
\gamma = \frac{(2\cos(8\theta)^1}{(2\cos(2\theta)^5}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_1_5} 
x^3 + 25 x^2 + 31 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -10 $. 
And 
\begin{equation} \label{fmn_1_5} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{49} 
\end{equation}  
\begin{equation} \label{gmn_1_5} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{7} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_1_8} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^1}{(2\cos(4\theta)^8},
\beta  = \frac{(2\cos(4\theta)^1}{(2\cos(8\theta)^8},
\gamma = \frac{(2\cos(8\theta)^1}{(2\cos(2\theta)^8}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_1_8} 
x^3 - 136 x^2 - 361 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -31 $. 
And 
\begin{equation} \label{fmn_1_8} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{49} 
\end{equation}  
\begin{equation} \label{gmn_1_8} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{448} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_1_11} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^1}{(2\cos(4\theta)^11},
\beta  = \frac{(2\cos(4\theta)^1}{(2\cos(8\theta)^11},
\gamma = \frac{(2\cos(8\theta)^1}{(2\cos(2\theta)^11}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_1_11} 
x^3 + 816 x^2 + 4091 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -171 $. 
And 
\begin{equation} \label{fmn_1_11} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{1323} 
\end{equation}  
\begin{equation} \label{gmn_1_11} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{3584} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_1_14} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^1}{(2\cos(4\theta)^14},
\beta  = \frac{(2\cos(4\theta)^1}{(2\cos(8\theta)^14},
\gamma = \frac{(2\cos(8\theta)^1}{(2\cos(2\theta)^14}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_1_14} 
x^3 - 4735 x^2 - 46414 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -535 $. 
And 
\begin{equation} \label{fmn_1_14} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{3136} 
\end{equation}  
\begin{equation} \label{gmn_1_14} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{48013} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_2_1} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^2}{(2\cos(4\theta)^1},
\beta  = \frac{(2\cos(4\theta)^2}{(2\cos(8\theta)^1},
\gamma = \frac{(2\cos(8\theta)^2}{(2\cos(2\theta)^1}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_2_1} 
x^3 + 4 x^2 + 3 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -3 $. 
And 
\begin{equation} \label{fmn_2_1} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{7} 
\end{equation}  
\begin{equation} \label{gmn_2_1} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{0} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_2_4} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^2}{(2\cos(4\theta)^4},
\beta  = \frac{(2\cos(4\theta)^2}{(2\cos(8\theta)^4},
\gamma = \frac{(2\cos(8\theta)^2}{(2\cos(2\theta)^4}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_2_4} 
x^3 - 17 x^2 + 10 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 11 $. 
And 
\begin{equation} \label{fmn_2_4} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{56} 
\end{equation}  
\begin{equation} \label{gmn_2_4} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{49} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_2_7} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^2}{(2\cos(4\theta)^7},
\beta  = \frac{(2\cos(4\theta)^2}{(2\cos(8\theta)^7},
\gamma = \frac{(2\cos(8\theta)^2}{(2\cos(2\theta)^7}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_2_7} 
x^3 + 95 x^2 - 88 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 11 $. 
And 
\begin{equation} \label{fmn_2_7} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{56} 
\end{equation}  
\begin{equation} \label{gmn_2_7} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{49} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_2_10} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^2}{(2\cos(4\theta)^10},
\beta  = \frac{(2\cos(4\theta)^2}{(2\cos(8\theta)^10},
\gamma = \frac{(2\cos(8\theta)^2}{(2\cos(2\theta)^10}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_2_10} 
x^3 - 563 x^2 + 1011 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 102 $. 
And 
\begin{equation} \label{fmn_2_10} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{875} 
\end{equation}  
\begin{equation} \label{gmn_2_10} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{1323} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_2_13} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^2}{(2\cos(4\theta)^13},
\beta  = \frac{(2\cos(4\theta)^2}{(2\cos(8\theta)^13},
\gamma = \frac{(2\cos(8\theta)^2}{(2\cos(2\theta)^13}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_2_13} 
x^3 + 3280 x^2 - 11463 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 291 $. 
And 
\begin{equation} \label{fmn_2_13} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{2401} 
\end{equation}  
\begin{equation} \label{gmn_2_13} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{10584} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_3_0} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^3}{(2\cos(4\theta)^0},
\beta  = \frac{(2\cos(4\theta)^3}{(2\cos(8\theta)^0},
\gamma = \frac{(2\cos(8\theta)^3}{(2\cos(2\theta)^0}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_3_0} 
x^3 + 4 x^2 - 11 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -1 $. 
And 
\begin{equation} \label{fmn_3_0} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{1} 
\end{equation}  
\begin{equation} \label{gmn_3_0} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{8} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_3_3} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^3}{(2\cos(4\theta)^3},
\beta  = \frac{(2\cos(4\theta)^3}{(2\cos(8\theta)^3},
\gamma = \frac{(2\cos(8\theta)^3}{(2\cos(2\theta)^3}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_3_3} 
x^3 + 11 x^2 - 4 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -1 $. 
And 
\begin{equation} \label{fmn_3_3} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{8} 
\end{equation}  
\begin{equation} \label{gmn_3_3} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{1} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_3_6} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^3}{(2\cos(4\theta)^6},
\beta  = \frac{(2\cos(4\theta)^3}{(2\cos(8\theta)^6},
\gamma = \frac{(2\cos(8\theta)^3}{(2\cos(2\theta)^6}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_3_6} 
x^3 - 66 x^2 - 25 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -15 $. 
And 
\begin{equation} \label{fmn_3_6} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{27} 
\end{equation}  
\begin{equation} \label{gmn_3_6} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{64} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_3_9} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^3}{(2\cos(4\theta)^9},
\beta  = \frac{(2\cos(4\theta)^3}{(2\cos(8\theta)^9},
\gamma = \frac{(2\cos(8\theta)^3}{(2\cos(2\theta)^9}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_3_9} 
x^3 + 389 x^2 + 248 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -43 $. 
And 
\begin{equation} \label{fmn_3_9} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{512} 
\end{equation}  
\begin{equation} \label{gmn_3_9} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{125} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_3_12} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^3}{(2\cos(4\theta)^12},
\beta  = \frac{(2\cos(4\theta)^3}{(2\cos(8\theta)^12},
\gamma = \frac{(2\cos(8\theta)^3}{(2\cos(2\theta)^12}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_3_12} 
x^3 - 2271 x^2 - 2832 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -183 $. 
And 
\begin{equation} \label{fmn_3_12} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{1728} 
\end{equation}  
\begin{equation} \label{gmn_3_12} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{3375} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_3_15} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^3}{(2\cos(4\theta)^15},
\beta  = \frac{(2\cos(4\theta)^3}{(2\cos(8\theta)^15},
\gamma = \frac{(2\cos(8\theta)^3}{(2\cos(2\theta)^15}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_3_15} 
x^3 + 13297 x^2 + 32119 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -778 $. 
And 
\begin{equation} \label{fmn_3_15} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{15625} 
\end{equation}  
\begin{equation} \label{gmn_3_15} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{29791} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_4_2} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^4}{(2\cos(4\theta)^2},
\beta  = \frac{(2\cos(4\theta)^4}{(2\cos(8\theta)^2},
\gamma = \frac{(2\cos(8\theta)^4}{(2\cos(2\theta)^2}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_4_2} 
x^3 - 10 x^2 + 17 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 11 $. 
And 
\begin{equation} \label{fmn_4_2} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{49} 
\end{equation}  
\begin{equation} \label{gmn_4_2} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{56} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_4_5} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^4}{(2\cos(4\theta)^5},
\beta  = \frac{(2\cos(4\theta)^4}{(2\cos(8\theta)^5},
\gamma = \frac{(2\cos(8\theta)^4}{(2\cos(2\theta)^5}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_4_5} 
x^3 + 46 x^2 + 3 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -3 $. 
And 
\begin{equation} \label{fmn_4_5} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{49} 
\end{equation}  
\begin{equation} \label{gmn_4_5} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{0} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_4_8} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^4}{(2\cos(4\theta)^8},
\beta  = \frac{(2\cos(4\theta)^4}{(2\cos(8\theta)^8},
\gamma = \frac{(2\cos(8\theta)^4}{(2\cos(2\theta)^8}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_4_8} 
x^3 - 269 x^2 + 66 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 39 $. 
And 
\begin{equation} \label{fmn_4_8} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{392} 
\end{equation}  
\begin{equation} \label{gmn_4_8} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{189} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_4_11} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^4}{(2\cos(4\theta)^11},
\beta  = \frac{(2\cos(4\theta)^4}{(2\cos(8\theta)^11},
\gamma = \frac{(2\cos(8\theta)^4}{(2\cos(2\theta)^11}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_4_11} 
x^3 + 1572 x^2 - 697 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 81 $. 
And 
\begin{equation} \label{fmn_4_11} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{1323} 
\end{equation}  
\begin{equation} \label{gmn_4_11} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{448} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_4_14} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^4}{(2\cos(4\theta)^14},
\beta  = \frac{(2\cos(4\theta)^4}{(2\cos(8\theta)^14},
\gamma = \frac{(2\cos(8\theta)^4}{(2\cos(2\theta)^14}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_4_14} 
x^3 - 9201 x^2 + 7934 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 459 $. 
And 
\begin{equation} \label{fmn_4_14} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{10584} 
\end{equation}  
\begin{equation} \label{gmn_4_14} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{9317} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_5_1} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^5}{(2\cos(4\theta)^1},
\beta  = \frac{(2\cos(4\theta)^5}{(2\cos(8\theta)^1},
\gamma = \frac{(2\cos(8\theta)^5}{(2\cos(2\theta)^1}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_5_1} 
x^3 - 3 x^2 - 46 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -3 $. 
And 
\begin{equation} \label{fmn_5_1} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{0} 
\end{equation}  
\begin{equation} \label{gmn_5_1} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{49} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_5_4} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^5}{(2\cos(4\theta)^4},
\beta  = \frac{(2\cos(4\theta)^5}{(2\cos(8\theta)^4},
\gamma = \frac{(2\cos(8\theta)^5}{(2\cos(2\theta)^4}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_5_4} 
x^3 - 31 x^2 - 25 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -10 $. 
And 
\begin{equation} \label{fmn_5_4} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{7} 
\end{equation}  
\begin{equation} \label{gmn_5_4} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{49} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_5_7} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^5}{(2\cos(4\theta)^7},
\beta  = \frac{(2\cos(4\theta)^5}{(2\cos(8\theta)^7},
\gamma = \frac{(2\cos(8\theta)^5}{(2\cos(2\theta)^7}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_5_7} 
x^3 + 186 x^2 + 3 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -3 $. 
And 
\begin{equation} \label{fmn_5_7} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{189} 
\end{equation}  
\begin{equation} \label{gmn_5_7} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{0} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_5_10} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^5}{(2\cos(4\theta)^10},
\beta  = \frac{(2\cos(4\theta)^5}{(2\cos(8\theta)^10},
\gamma = \frac{(2\cos(8\theta)^5}{(2\cos(2\theta)^10}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_5_10} 
x^3 - 1088 x^2 - 179 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -73 $. 
And 
\begin{equation} \label{fmn_5_10} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{875} 
\end{equation}  
\begin{equation} \label{gmn_5_10} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{392} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_5_13} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^5}{(2\cos(4\theta)^13},
\beta  = \frac{(2\cos(4\theta)^5}{(2\cos(8\theta)^13},
\gamma = \frac{(2\cos(8\theta)^5}{(2\cos(2\theta)^13}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_5_13} 
x^3 + 6367 x^2 + 1956 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -213 $. 
And 
\begin{equation} \label{fmn_5_13} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{7000} 
\end{equation}  
\begin{equation} \label{gmn_5_13} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{1323} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_5_16} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^5}{(2\cos(4\theta)^16},
\beta  = \frac{(2\cos(4\theta)^5}{(2\cos(8\theta)^16},
\gamma = \frac{(2\cos(8\theta)^5}{(2\cos(2\theta)^16}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_5_16} 
x^3 - 37250 x^2 - 22229 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -955 $. 
And 
\begin{equation} \label{fmn_5_16} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{34391} 
\end{equation}  
\begin{equation} \label{gmn_5_16} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{25088} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_6_0} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^6}{(2\cos(4\theta)^0},
\beta  = \frac{(2\cos(4\theta)^6}{(2\cos(8\theta)^0},
\gamma = \frac{(2\cos(8\theta)^6}{(2\cos(2\theta)^0}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_6_0} 
x^3 - 38 x^2 + 129 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 27 $. 
And 
\begin{equation} \label{fmn_6_0} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{125} 
\end{equation}  
\begin{equation} \label{gmn_6_0} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{216} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_6_3} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^6}{(2\cos(4\theta)^3},
\beta  = \frac{(2\cos(4\theta)^6}{(2\cos(8\theta)^3},
\gamma = \frac{(2\cos(8\theta)^6}{(2\cos(2\theta)^3}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_6_3} 
x^3 + 25 x^2 + 66 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -15 $. 
And 
\begin{equation} \label{fmn_6_3} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{64} 
\end{equation}  
\begin{equation} \label{gmn_6_3} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{27} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_6_6} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^6}{(2\cos(4\theta)^6},
\beta  = \frac{(2\cos(4\theta)^6}{(2\cos(8\theta)^6},
\gamma = \frac{(2\cos(8\theta)^6}{(2\cos(2\theta)^6}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_6_6} 
x^3 - 129 x^2 + 38 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 27 $. 
And 
\begin{equation} \label{fmn_6_6} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{216} 
\end{equation}  
\begin{equation} \label{gmn_6_6} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{125} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_6_9} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^6}{(2\cos(4\theta)^9},
\beta  = \frac{(2\cos(4\theta)^6}{(2\cos(8\theta)^9},
\gamma = \frac{(2\cos(8\theta)^6}{(2\cos(2\theta)^9}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_6_9} 
x^3 + 753 x^2 - 25 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 6 $. 
And 
\begin{equation} \label{fmn_6_9} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{729} 
\end{equation}  
\begin{equation} \label{gmn_6_9} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{1} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_6_12} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^6}{(2\cos(4\theta)^12},
\beta  = \frac{(2\cos(4\theta)^6}{(2\cos(8\theta)^12},
\gamma = \frac{(2\cos(8\theta)^6}{(2\cos(2\theta)^12}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_6_12} 
x^3 - 4406 x^2 + 493 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 167 $. 
And 
\begin{equation} \label{fmn_6_12} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{4913} 
\end{equation}  
\begin{equation} \label{gmn_6_12} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{1000} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_6_15} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^6}{(2\cos(4\theta)^15},
\beta  = \frac{(2\cos(4\theta)^6}{(2\cos(8\theta)^15},
\gamma = \frac{(2\cos(8\theta)^6}{(2\cos(2\theta)^15}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_6_15} 
x^3 + 25778 x^2 - 5485 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 461 $. 
And 
\begin{equation} \label{fmn_6_15} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{24389} 
\end{equation}  
\begin{equation} \label{gmn_6_15} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{4096} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_7_2} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^7}{(2\cos(4\theta)^2},
\beta  = \frac{(2\cos(4\theta)^7}{(2\cos(8\theta)^2},
\gamma = \frac{(2\cos(8\theta)^7}{(2\cos(2\theta)^2}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_7_2} 
x^3 - 3 x^2 - 186 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -3 $. 
And 
\begin{equation} \label{fmn_7_2} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{0} 
\end{equation}  
\begin{equation} \label{gmn_7_2} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{189} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_7_5} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^7}{(2\cos(4\theta)^5},
\beta  = \frac{(2\cos(4\theta)^7}{(2\cos(8\theta)^5},
\gamma = \frac{(2\cos(8\theta)^7}{(2\cos(2\theta)^5}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_7_5} 
x^3 + 88 x^2 - 95 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 11 $. 
And 
\begin{equation} \label{fmn_7_5} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{49} 
\end{equation}  
\begin{equation} \label{gmn_7_5} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{56} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_7_8} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^7}{(2\cos(4\theta)^8},
\beta  = \frac{(2\cos(4\theta)^7}{(2\cos(8\theta)^8},
\gamma = \frac{(2\cos(8\theta)^7}{(2\cos(2\theta)^8}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_7_8} 
x^3 - 521 x^2 - 60 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -45 $. 
And 
\begin{equation} \label{fmn_7_8} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{392} 
\end{equation}  
\begin{equation} \label{gmn_7_8} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{189} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_7_11} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^7}{(2\cos(4\theta)^11},
\beta  = \frac{(2\cos(4\theta)^7}{(2\cos(8\theta)^11},
\gamma = \frac{(2\cos(8\theta)^7}{(2\cos(2\theta)^11}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_7_11} 
x^3 + 3049 x^2 + 94 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -31 $. 
And 
\begin{equation} \label{fmn_7_11} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{3136} 
\end{equation}  
\begin{equation} \label{gmn_7_11} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{7} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_7_14} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^7}{(2\cos(4\theta)^14},
\beta  = \frac{(2\cos(4\theta)^7}{(2\cos(8\theta)^14},
\gamma = \frac{(2\cos(8\theta)^7}{(2\cos(2\theta)^14}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_7_14} 
x^3 - 17839 x^2 - 1369 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -346 $. 
And 
\begin{equation} \label{fmn_7_14} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{16807} 
\end{equation}  
\begin{equation} \label{gmn_7_14} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{2401} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_8_1} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^8}{(2\cos(4\theta)^1},
\beta  = \frac{(2\cos(4\theta)^8}{(2\cos(8\theta)^1},
\gamma = \frac{(2\cos(8\theta)^8}{(2\cos(2\theta)^1}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_8_1} 
x^3 + 60 x^2 + 521 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -45 $. 
And 
\begin{equation} \label{fmn_8_1} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{189} 
\end{equation}  
\begin{equation} \label{gmn_8_1} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{392} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_8_4} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^8}{(2\cos(4\theta)^4},
\beta  = \frac{(2\cos(4\theta)^8}{(2\cos(8\theta)^4},
\gamma = \frac{(2\cos(8\theta)^8}{(2\cos(2\theta)^4}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_8_4} 
x^3 - 66 x^2 + 269 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 39 $. 
And 
\begin{equation} \label{fmn_8_4} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{189} 
\end{equation}  
\begin{equation} \label{gmn_8_4} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{392} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_8_7} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^8}{(2\cos(4\theta)^7},
\beta  = \frac{(2\cos(4\theta)^8}{(2\cos(8\theta)^7},
\gamma = \frac{(2\cos(8\theta)^8}{(2\cos(2\theta)^7}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_8_7} 
x^3 + 361 x^2 + 136 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -31 $. 
And 
\begin{equation} \label{fmn_8_7} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{448} 
\end{equation}  
\begin{equation} \label{gmn_8_7} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{49} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_8_10} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^8}{(2\cos(4\theta)^10},
\beta  = \frac{(2\cos(4\theta)^8}{(2\cos(8\theta)^10},
\gamma = \frac{(2\cos(8\theta)^8}{(2\cos(2\theta)^10}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_8_10} 
x^3 - 2110 x^2 + 101 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 95 $. 
And 
\begin{equation} \label{fmn_8_10} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{2401} 
\end{equation}  
\begin{equation} \label{gmn_8_10} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{392} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_8_13} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^8}{(2\cos(4\theta)^13},
\beta  = \frac{(2\cos(4\theta)^8}{(2\cos(8\theta)^13},
\gamma = \frac{(2\cos(8\theta)^8}{(2\cos(2\theta)^13}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_8_13} 
x^3 + 12345 x^2 - 298 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 81 $. 
And 
\begin{equation} \label{fmn_8_13} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{12096} 
\end{equation}  
\begin{equation} \label{gmn_8_13} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{49} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_8_16} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^8}{(2\cos(4\theta)^16},
\beta  = \frac{(2\cos(4\theta)^8}{(2\cos(8\theta)^16},
\gamma = \frac{(2\cos(8\theta)^8}{(2\cos(2\theta)^16}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_8_16} 
x^3 - 72229 x^2 + 3818 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 767 $. 
And 
\begin{equation} \label{fmn_8_16} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{74536} 
\end{equation}  
\begin{equation} \label{gmn_8_16} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{6125} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_9_0} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^9}{(2\cos(4\theta)^0},
\beta  = \frac{(2\cos(4\theta)^9}{(2\cos(8\theta)^0},
\gamma = \frac{(2\cos(8\theta)^9}{(2\cos(2\theta)^0}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_9_0} 
x^3 + 193 x^2 - 1460 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 41 $. 
And 
\begin{equation} \label{fmn_9_0} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{64} 
\end{equation}  
\begin{equation} \label{gmn_9_0} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{1331} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_9_3} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^9}{(2\cos(4\theta)^3},
\beta  = \frac{(2\cos(4\theta)^9}{(2\cos(8\theta)^3},
\gamma = \frac{(2\cos(8\theta)^9}{(2\cos(2\theta)^3}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_9_3} 
x^3 + 25 x^2 - 753 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 6 $. 
And 
\begin{equation} \label{fmn_9_3} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{1} 
\end{equation}  
\begin{equation} \label{gmn_9_3} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{729} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_9_6} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^9}{(2\cos(4\theta)^6},
\beta  = \frac{(2\cos(4\theta)^9}{(2\cos(8\theta)^6},
\gamma = \frac{(2\cos(8\theta)^9}{(2\cos(2\theta)^6}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_9_6} 
x^3 - 248 x^2 - 389 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -43 $. 
And 
\begin{equation} \label{fmn_9_6} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{125} 
\end{equation}  
\begin{equation} \label{gmn_9_6} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{512} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_9_9} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^9}{(2\cos(4\theta)^9},
\beta  = \frac{(2\cos(4\theta)^9}{(2\cos(8\theta)^9},
\gamma = \frac{(2\cos(8\theta)^9}{(2\cos(2\theta)^9}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_9_9} 
x^3 + 1460 x^2 - 193 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 41 $. 
And 
\begin{equation} \label{fmn_9_9} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{1331} 
\end{equation}  
\begin{equation} \label{gmn_9_9} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{64} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_9_12} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^9}{(2\cos(4\theta)^12},
\beta  = \frac{(2\cos(4\theta)^9}{(2\cos(8\theta)^12},
\gamma = \frac{(2\cos(8\theta)^9}{(2\cos(2\theta)^12}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_9_12} 
x^3 - 8543 x^2 - 186 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -183 $. 
And 
\begin{equation} \label{fmn_9_12} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{8000} 
\end{equation}  
\begin{equation} \label{gmn_9_12} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{729} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_9_15} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^9}{(2\cos(4\theta)^15},
\beta  = \frac{(2\cos(4\theta)^9}{(2\cos(8\theta)^15},
\gamma = \frac{(2\cos(8\theta)^9}{(2\cos(2\theta)^15}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_9_15} 
x^3 + 49984 x^2 + 885 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -225 $. 
And 
\begin{equation} \label{fmn_9_15} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{50653} 
\end{equation}  
\begin{equation} \label{gmn_9_15} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{216} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_10_2} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^10}{(2\cos(4\theta)^2},
\beta  = \frac{(2\cos(4\theta)^10}{(2\cos(8\theta)^2},
\gamma = \frac{(2\cos(8\theta)^10}{(2\cos(2\theta)^2}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_10_2} 
x^3 - 101 x^2 + 2110 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 95 $. 
And 
\begin{equation} \label{fmn_10_2} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{392} 
\end{equation}  
\begin{equation} \label{gmn_10_2} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{2401} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_10_5} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^10}{(2\cos(4\theta)^5},
\beta  = \frac{(2\cos(4\theta)^10}{(2\cos(8\theta)^5},
\gamma = \frac{(2\cos(8\theta)^10}{(2\cos(2\theta)^5}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_10_5} 
x^3 + 179 x^2 + 1088 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -73 $. 
And 
\begin{equation} \label{fmn_10_5} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{392} 
\end{equation}  
\begin{equation} \label{gmn_10_5} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{875} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_10_8} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^10}{(2\cos(4\theta)^8},
\beta  = \frac{(2\cos(4\theta)^10}{(2\cos(8\theta)^8},
\gamma = \frac{(2\cos(8\theta)^10}{(2\cos(2\theta)^8}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_10_8} 
x^3 - 1011 x^2 + 563 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 102 $. 
And 
\begin{equation} \label{fmn_10_8} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{1323} 
\end{equation}  
\begin{equation} \label{gmn_10_8} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{875} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_10_11} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^10}{(2\cos(4\theta)^11},
\beta  = \frac{(2\cos(4\theta)^10}{(2\cos(8\theta)^11},
\gamma = \frac{(2\cos(8\theta)^10}{(2\cos(2\theta)^11}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_10_11} 
x^3 + 5912 x^2 + 269 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -73 $. 
And 
\begin{equation} \label{fmn_10_11} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{6125} 
\end{equation}  
\begin{equation} \label{gmn_10_11} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{56} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_10_14} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^10}{(2\cos(4\theta)^14},
\beta  = \frac{(2\cos(4\theta)^10}{(2\cos(8\theta)^14},
\gamma = \frac{(2\cos(8\theta)^10}{(2\cos(2\theta)^14}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_10_14} 
x^3 - 34590 x^2 + 381 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 375 $. 
And 
\begin{equation} \label{fmn_10_14} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{35721} 
\end{equation}  
\begin{equation} \label{gmn_10_14} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{1512} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_10_17} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^10}{(2\cos(4\theta)^17},
\beta  = \frac{(2\cos(4\theta)^10}{(2\cos(8\theta)^17},
\gamma = \frac{(2\cos(8\theta)^10}{(2\cos(2\theta)^17}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_10_17} 
x^3 + 202381 x^2 - 2552 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 557 $. 
And 
\begin{equation} \label{fmn_10_17} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{200704} 
\end{equation}  
\begin{equation} \label{gmn_10_17} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{875} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_11_1} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^11}{(2\cos(4\theta)^1},
\beta  = \frac{(2\cos(4\theta)^11}{(2\cos(8\theta)^1},
\gamma = \frac{(2\cos(8\theta)^11}{(2\cos(2\theta)^1}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_11_1} 
x^3 - 269 x^2 - 5912 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -73 $. 
And 
\begin{equation} \label{fmn_11_1} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{56} 
\end{equation}  
\begin{equation} \label{gmn_11_1} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{6125} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_11_4} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^11}{(2\cos(4\theta)^4},
\beta  = \frac{(2\cos(4\theta)^11}{(2\cos(8\theta)^4},
\gamma = \frac{(2\cos(8\theta)^11}{(2\cos(2\theta)^4}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_11_4} 
x^3 - 94 x^2 - 3049 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -31 $. 
And 
\begin{equation} \label{fmn_11_4} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{7} 
\end{equation}  
\begin{equation} \label{gmn_11_4} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{3136} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_11_7} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^11}{(2\cos(4\theta)^7},
\beta  = \frac{(2\cos(4\theta)^11}{(2\cos(8\theta)^7},
\gamma = \frac{(2\cos(8\theta)^11}{(2\cos(2\theta)^7}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_11_7} 
x^3 + 697 x^2 - 1572 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 81 $. 
And 
\begin{equation} \label{fmn_11_7} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{448} 
\end{equation}  
\begin{equation} \label{gmn_11_7} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{1323} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_11_10} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^11}{(2\cos(4\theta)^10},
\beta  = \frac{(2\cos(4\theta)^11}{(2\cos(8\theta)^10},
\gamma = \frac{(2\cos(8\theta)^11}{(2\cos(2\theta)^10}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_11_10} 
x^3 - 4091 x^2 - 816 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -171 $. 
And 
\begin{equation} \label{fmn_11_10} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{3584} 
\end{equation}  
\begin{equation} \label{gmn_11_10} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{1323} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_11_13} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^11}{(2\cos(4\theta)^13},
\beta  = \frac{(2\cos(4\theta)^11}{(2\cos(8\theta)^13},
\gamma = \frac{(2\cos(8\theta)^11}{(2\cos(2\theta)^13}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_11_13} 
x^3 + 23937 x^2 - 361 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 102 $. 
And 
\begin{equation} \label{fmn_11_13} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{23625} 
\end{equation}  
\begin{equation} \label{gmn_11_13} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{49} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_11_16} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^11}{(2\cos(4\theta)^16},
\beta  = \frac{(2\cos(4\theta)^11}{(2\cos(8\theta)^16},
\gamma = \frac{(2\cos(8\theta)^11}{(2\cos(2\theta)^16}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_11_16} 
x^3 - 140052 x^2 - 865 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -759 $. 
And 
\begin{equation} \label{fmn_11_16} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{137781} 
\end{equation}  
\begin{equation} \label{gmn_11_16} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{3136} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_12_0} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^12}{(2\cos(4\theta)^0},
\beta  = \frac{(2\cos(4\theta)^12}{(2\cos(8\theta)^0},
\gamma = \frac{(2\cos(8\theta)^12}{(2\cos(2\theta)^0}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_12_0} 
x^3 - 1186 x^2 + 16565 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 335 $. 
And 
\begin{equation} \label{fmn_12_0} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{2197} 
\end{equation}  
\begin{equation} \label{gmn_12_0} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{17576} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_12_3} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^12}{(2\cos(4\theta)^3},
\beta  = \frac{(2\cos(4\theta)^12}{(2\cos(8\theta)^3},
\gamma = \frac{(2\cos(8\theta)^12}{(2\cos(2\theta)^3}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_12_3} 
x^3 + 186 x^2 + 8543 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -183 $. 
And 
\begin{equation} \label{fmn_12_3} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{729} 
\end{equation}  
\begin{equation} \label{gmn_12_3} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{8000} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_12_6} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^12}{(2\cos(4\theta)^6},
\beta  = \frac{(2\cos(4\theta)^12}{(2\cos(8\theta)^6},
\gamma = \frac{(2\cos(8\theta)^12}{(2\cos(2\theta)^6}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_12_6} 
x^3 - 493 x^2 + 4406 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 167 $. 
And 
\begin{equation} \label{fmn_12_6} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{1000} 
\end{equation}  
\begin{equation} \label{gmn_12_6} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{4913} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_12_9} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^12}{(2\cos(4\theta)^9},
\beta  = \frac{(2\cos(4\theta)^12}{(2\cos(8\theta)^9},
\gamma = \frac{(2\cos(8\theta)^12}{(2\cos(2\theta)^9}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_12_9} 
x^3 + 2832 x^2 + 2271 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -183 $. 
And 
\begin{equation} \label{fmn_12_9} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{3375} 
\end{equation}  
\begin{equation} \label{gmn_12_9} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{1728} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_12_12} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^12}{(2\cos(4\theta)^12},
\beta  = \frac{(2\cos(4\theta)^12}{(2\cos(8\theta)^12},
\gamma = \frac{(2\cos(8\theta)^12}{(2\cos(2\theta)^12}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_12_12} 
x^3 - 16565 x^2 + 1186 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 335 $. 
And 
\begin{equation} \label{fmn_12_12} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{17576} 
\end{equation}  
\begin{equation} \label{gmn_12_12} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{2197} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_12_15} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^12}{(2\cos(4\theta)^15},
\beta  = \frac{(2\cos(4\theta)^12}{(2\cos(8\theta)^15},
\gamma = \frac{(2\cos(8\theta)^12}{(2\cos(2\theta)^15}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_12_15} 
x^3 + 96919 x^2 + 444 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -141 $. 
And 
\begin{equation} \label{fmn_12_15} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{97336} 
\end{equation}  
\begin{equation} \label{gmn_12_15} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{27} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_13_2} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^13}{(2\cos(4\theta)^2},
\beta  = \frac{(2\cos(4\theta)^13}{(2\cos(8\theta)^2},
\gamma = \frac{(2\cos(8\theta)^13}{(2\cos(2\theta)^2}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_13_2} 
x^3 + 361 x^2 - 23937 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 102 $. 
And 
\begin{equation} \label{fmn_13_2} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{49} 
\end{equation}  
\begin{equation} \label{gmn_13_2} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{23625} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_13_5} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^13}{(2\cos(4\theta)^5},
\beta  = \frac{(2\cos(4\theta)^13}{(2\cos(8\theta)^5},
\gamma = \frac{(2\cos(8\theta)^13}{(2\cos(2\theta)^5}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_13_5} 
x^3 + 298 x^2 - 12345 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 81 $. 
And 
\begin{equation} \label{fmn_13_5} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{49} 
\end{equation}  
\begin{equation} \label{gmn_13_5} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{12096} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_13_8} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^13}{(2\cos(4\theta)^8},
\beta  = \frac{(2\cos(4\theta)^13}{(2\cos(8\theta)^8},
\gamma = \frac{(2\cos(8\theta)^13}{(2\cos(2\theta)^8}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_13_8} 
x^3 - 1956 x^2 - 6367 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -213 $. 
And 
\begin{equation} \label{fmn_13_8} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{1323} 
\end{equation}  
\begin{equation} \label{gmn_13_8} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{7000} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_13_11} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^13}{(2\cos(4\theta)^11},
\beta  = \frac{(2\cos(4\theta)^13}{(2\cos(8\theta)^11},
\gamma = \frac{(2\cos(8\theta)^13}{(2\cos(2\theta)^11}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_13_11} 
x^3 + 11463 x^2 - 3280 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 291 $. 
And 
\begin{equation} \label{fmn_13_11} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{10584} 
\end{equation}  
\begin{equation} \label{gmn_13_11} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{2401} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_13_14} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^13}{(2\cos(4\theta)^14},
\beta  = \frac{(2\cos(4\theta)^13}{(2\cos(8\theta)^14},
\gamma = \frac{(2\cos(8\theta)^13}{(2\cos(2\theta)^14}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_13_14} 
x^3 - 67070 x^2 - 1733 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -619 $. 
And 
\begin{equation} \label{fmn_13_14} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{65219} 
\end{equation}  
\begin{equation} \label{gmn_13_14} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{3584} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_13_17} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^13}{(2\cos(4\theta)^17},
\beta  = \frac{(2\cos(4\theta)^13}{(2\cos(8\theta)^17},
\gamma = \frac{(2\cos(8\theta)^13}{(2\cos(2\theta)^17}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_13_17} 
x^3 + 392417 x^2 - 424 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 137 $. 
And 
\begin{equation} \label{fmn_13_17} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{392000} 
\end{equation}  
\begin{equation} \label{gmn_13_17} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{7} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_14_1} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^14}{(2\cos(4\theta)^1},
\beta  = \frac{(2\cos(4\theta)^14}{(2\cos(8\theta)^1},
\gamma = \frac{(2\cos(8\theta)^14}{(2\cos(2\theta)^1}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_14_1} 
x^3 + 1733 x^2 + 67070 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -619 $. 
And 
\begin{equation} \label{fmn_14_1} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{3584} 
\end{equation}  
\begin{equation} \label{gmn_14_1} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{65219} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_14_4} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^14}{(2\cos(4\theta)^4},
\beta  = \frac{(2\cos(4\theta)^14}{(2\cos(8\theta)^4},
\gamma = \frac{(2\cos(8\theta)^14}{(2\cos(2\theta)^4}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_14_4} 
x^3 - 381 x^2 + 34590 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 375 $. 
And 
\begin{equation} \label{fmn_14_4} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{1512} 
\end{equation}  
\begin{equation} \label{gmn_14_4} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{35721} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_14_7} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^14}{(2\cos(4\theta)^7},
\beta  = \frac{(2\cos(4\theta)^14}{(2\cos(8\theta)^7},
\gamma = \frac{(2\cos(8\theta)^14}{(2\cos(2\theta)^7}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_14_7} 
x^3 + 1369 x^2 + 17839 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -346 $. 
And 
\begin{equation} \label{fmn_14_7} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{2401} 
\end{equation}  
\begin{equation} \label{gmn_14_7} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{16807} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_14_10} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^14}{(2\cos(4\theta)^10},
\beta  = \frac{(2\cos(4\theta)^14}{(2\cos(8\theta)^10},
\gamma = \frac{(2\cos(8\theta)^14}{(2\cos(2\theta)^10}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_14_10} 
x^3 - 7934 x^2 + 9201 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 459 $. 
And 
\begin{equation} \label{fmn_14_10} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{9317} 
\end{equation}  
\begin{equation} \label{gmn_14_10} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{10584} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_14_13} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^14}{(2\cos(4\theta)^13},
\beta  = \frac{(2\cos(4\theta)^14}{(2\cos(8\theta)^13},
\gamma = \frac{(2\cos(8\theta)^14}{(2\cos(2\theta)^13}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_14_13} 
x^3 + 46414 x^2 + 4735 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -535 $. 
And 
\begin{equation} \label{fmn_14_13} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{48013} 
\end{equation}  
\begin{equation} \label{gmn_14_13} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{3136} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_14_19} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^14}{(2\cos(4\theta)^19},
\beta  = \frac{(2\cos(4\theta)^14}{(2\cos(8\theta)^19},
\gamma = \frac{(2\cos(8\theta)^14}{(2\cos(2\theta)^19}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_14_19} 
x^3 + 1588863 x^2 + 3 x^2 - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -3 $. 
And 
\begin{equation} \label{fmn_14_19} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{1588866} 
\end{equation}  
\begin{equation} \label{gmn_14_19} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
 \sqrt[3]{0} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_15_0} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^15}{(2\cos(4\theta)^0},
\beta  = \frac{(2\cos(4\theta)^15}{(2\cos(8\theta)^0},
\gamma = \frac{(2\cos(8\theta)^15}{(2\cos(2\theta)^0}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_15_0} 
x^3 + 6829 x^2 - 187926 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 909 $. 
And 
\begin{equation} \label{fmn_15_0} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{4096} 
\end{equation}  
\begin{equation} \label{gmn_15_0} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{185193} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_15_3} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^15}{(2\cos(4\theta)^3},
\beta  = \frac{(2\cos(4\theta)^15}{(2\cos(8\theta)^3},
\gamma = \frac{(2\cos(8\theta)^15}{(2\cos(2\theta)^3}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_15_3} 
x^3 - 444 x^2 - 96919 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -141 $. 
And 
\begin{equation} \label{fmn_15_3} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{27} 
\end{equation}  
\begin{equation} \label{gmn_15_3} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{97336} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_15_6} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^15}{(2\cos(4\theta)^6},
\beta  = \frac{(2\cos(4\theta)^15}{(2\cos(8\theta)^6},
\gamma = \frac{(2\cos(8\theta)^15}{(2\cos(2\theta)^6}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_15_6} 
x^3 - 885 x^2 - 49984 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -225 $. 
And 
\begin{equation} \label{fmn_15_6} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{216} 
\end{equation}  
\begin{equation} \label{gmn_15_6} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{50653} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_15_9} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^15}{(2\cos(4\theta)^9},
\beta  = \frac{(2\cos(4\theta)^15}{(2\cos(8\theta)^9},
\gamma = \frac{(2\cos(8\theta)^15}{(2\cos(2\theta)^9}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_15_9} 
x^3 + 5485 x^2 - 25778 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ 461 $. 
And 
\begin{equation} \label{fmn_15_9} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
-\sqrt[3]{4096} 
\end{equation}  
\begin{equation} \label{gmn_15_9} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{24389} 
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_15_12} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^15}{(2\cos(4\theta)^12},
\beta  = \frac{(2\cos(4\theta)^15}{(2\cos(8\theta)^12},
\gamma = \frac{(2\cos(8\theta)^15}{(2\cos(2\theta)^12}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_15_12} 
x^3 - 32119 x^2 - 13297 x - 1 = 0
\end{equation}  
The associatd Rammnujan equation has integer solution $ -778 $. 
And 
\begin{equation} \label{fmn_15_12} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{29791} 
\end{equation}  
\begin{equation} \label{gmn_15_12} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
-\sqrt[3]{15625} 
\end{equation}  
\end{