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    $\begingroup$ Your definition of $D_p$ does not require $p$ to be prime, but just odd and at least 3. The following more specific suggestion is based on data up to 400: for odd $n \geq 3$, $D_n \not= 0$ if and only if $n=7$, $n=25$, or or $n$ is a prime that's $1 \bmod 4$ other than $13$ ($D_7 = -1$, $D_{25} = 9$, and $D_{13} = 0$). $\endgroup$ Commented Nov 19, 2018 at 16:58
  • $\begingroup$ @KConrad: I've added a few words on this case into my answer. Generally, the only interesting case in this setup is $p\equiv1\pmod4$, and I think it should contain more counterexamples. Can you check that for larger $p$? $\endgroup$ Commented Nov 19, 2018 at 22:40
  • $\begingroup$ @IlyaBogdanov I asked Alvaro Lozano-Robledo to check primes $p \equiv 1 \bmod 4$ up to 2000 and he didn't find any example with $D_p = 0$ other than $p = 13$. $\endgroup$ Commented Nov 20, 2018 at 4:38
  • $\begingroup$ I've checked now up to 5000, and the only example is $p=13$. $\endgroup$ Commented Nov 20, 2018 at 5:53