For all polynomials $q\in\mathbb Z[X]$ and all $n\in \mathbb N$ $$\exists k\in\mathbb N: q(p_n)\equiv q(-2k)\pmod{p_{n+1}}$$ Where $p_n$ is the $n$-th prime.
This conjecture is tested for random polynomials and seems to hold. Are there counterexamples? Is it a known conjecture? Is it trivial?