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Published online by Cambridge University Press: 26 April 2019
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 $M$.
All three authors were partially supported by the grants “Fondecyt research projects 1130134 and 1170315, Chile” to author X. V., from Conicyt. Author M. V. was partially supported by the government of the Russian Federation (grant 14.Z50.31.0030).