Büchi’s Problem in Modular Arithmetic for Arbitrary Quadratic Polynomials

Given a prime $p\geqslant 5$ and an integer $s\geqslant 1$ , we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^{s}$ , such that $f$ is not a square, if a sequence $(f(1),\ldots ,f(N))$ is a sequence of squares, then $N$ is at most $M$ . We also provide some explicit formulas for the optimal $M$ .