Representing prime numbers with elementary symmetric polynomials

Question

Consider an $s$ tuple of the integer $2$ $$t_s =\underbrace{(2,2,\ldots,2)}_{s\text{ times}}$$ I write $\sigma_1(t_s)$ and $\sigma_2(t_s)$ for the first and second elementary symmetric polynomials on $t_s$ respectively. If $q$ is any prime number greater than $3$ is it true that

$$q =\sqrt{{\sigma_2(t_s) - \sigma_1(t_s) \above 1.5pt 2}+1}$$

for some $s$. It appears that $s \in$ A087743. Surely $s \equiv 0 \text{ modulo $2$}$ and I think $$s=q+1$$ Not all even numbers work though and the first one to fail is $10$. The set of even fails appears to be A238204.

Answer 1

As discussed in the comments, $\sigma_2(t_s)=2s(s-1)$, $\sigma_1(t_s)=2s$. Thus $$q=s-1$$ Indeed any nonnegative integer can be expressed in this way by taking $s=q+1$.

Answer 2

I just wanted to show my work: $$\sqrt{{2s(s-1)-2s \above 1.5pt 2}+1} = \sqrt{{2s^2-2s-2s \above 1.5pt 2}+1} =\sqrt{{2s^2-4s \above 1.5pt 2}+1} =\sqrt{s^2-2s+1} = \sqrt{(s-1)(s-1)} =s-1$$ From a learning point I was not able to get an explicit formula for $\sigma_2(t_s)$ and make the necessary reduction.