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A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (III)
As in Question 319254, for an odd prime $p$ and integers $c,d$ we define
$$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right),$$
where $(\frac{\cdot}p)$ is the Legendre symbol.
In my ...
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