1000 Followers problem!

If we have polynomial $x^3+ax^2+bx+c$ we denote $\sigma_1=p+q+r$ and $\sigma_2=pq+qr+pr$ where $p,q,r$ are its roots.

If you consider all polynomials of kind $\large\alpha^{\eta}-\alpha^2-\alpha-\eta=0$ ,where $\large5\leq\eta\leq 1000$ , we notice that : $\large\boxed{\sigma_1=\sigma_2=0}$ .

Since $\beta_{(1,\eta)}$ is a root of the given polynomial , we have :

$\Large{\beta_{(1,\eta)}^{\eta} - \beta_{(1,\eta)}^2 - \beta_{(1,\eta)} -\eta = 0 \\ \Rightarrow\beta_{(1,\eta)}^{\eta} = \beta_{(1,\eta)}^2 + \beta_{(1,\eta)} + \eta}$

Similarly we have for $\large\beta_{(2,\eta)},\beta_{(3,\eta)} \dots \beta_{(\eta,\eta)}$ .Adding them all we get :

$\Large{\displaystyle\sum_{i=1}^{\eta} \beta_{(i,\eta)}^{\eta} = \displaystyle\sum_{i=1}^{\eta} \beta_{(i,\eta)}^2 + \displaystyle\sum_{i=1}^{\eta} \beta_{(i,\eta)} + \eta^2 \\ \Rightarrow \displaystyle\sum_{i=1}^{\eta} \beta_{(i,\eta)}^{\eta} = \sigma_1^2 - 2\sigma_2 + \sigma_1 + \eta^2 \\ \Rightarrow \displaystyle\sum_{i=1}^{\eta} \beta_{(i,\eta)}^{\eta} = \eta^2 = S_{\eta}}$

Since $5\leq\eta\leq 1000$ , we have :

$\displaystyle\sum_{j=5}^{1000} S_j = \text{(sum of squares from 1 to 1000)} - \text{(sum of squares form 1 to 4)} \\ \large{\Rightarrow \displaystyle\sum_{j=5}^{1000} S_j = \dfrac{(1000)(1001)(2001)}{6} - \dfrac{(4)(5)(9)}{6} \\ \Rightarrow \displaystyle\sum_{j=5}^{1000} S_j = 333833500 - 30 = \boxed{333833470}}$

Let the $\eta = y$ and $\alpha = x$ .
We know the fact that
$a_0 S_1 + a_1 = 0 \\ a_0 S_2 + a_1 S_1 + 2a_2 = 0 \\ a_0S_r + a_1S_{r-1} + ...... + a_{r-1}S_1 + ra_r = 0$ .

Now the Polynomial is $P(x) = x^y - x^2 - x - y = 0 \\ \Rightarrow a_0 = 1 , a_1 = a_2 = ... = a_{y-3} = 0 , a_{y-2} = -1 , a_{y-1} = -1 , a_y = -y$

So if we apply the above results we'll get $S_1 = S_2 = S_3 = ... = S_{y-3} = 0$ .
Further applying the result we'll get $S_{y-2} = y-2 , S_{y-1} = y-1 , S_{y} = y^2$ .

So we now have to evaluate $\displaystyle \sum_{y=5}^{1000} {S_y}$ . Which we know is equal to $\displaystyle \sum_{y=5}^{1000}{y^2} = \boxed{333833470}$