127 is probably the LAST / LARGEST prime number p such that $p^2$ mod q has an odd residue, where q is the previous prime number right before p. I have checked it up to $10^6$ , and it turned out to have been checked up to 4×10^18 by Charles R Greathouse(?). Is 127 the last prime with this property? But WHY ? Is there a reason for this phenomenon?
Let $q=p-d$, where $d$ is the prime gap between $p$ and the previous prime $q$. Then $p^2-d^2 =(p-d)(p+d) \equiv 0 \bmod q$, so $p^2 \equiv d^2 \bmod q$. If $d^2<q$, then the residue is $d^2$, which is even because prime gaps are even.
The only way to get an odd residue is to have a prime gap $d$ with $d^2>q$. So we need a prime $q$ so that the next prime is greater than $q+\sqrt q$.
In general, the prime gap grows as $\ln q$, so to have one as exceptionally large as $\sqrt q$ is improbable, and I think it gets more improbable for larger $q$.