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A coprimality condition on consecutive values of polynomials

Abstract

Let fZ[X]f\in\mathbb{Z}[X] be quadratic or cubic polynomial. We prove that there exists an integer Gf2G_f\geq 2 such that for every integer kGfk\geq G_f one can find infinitely many integers n0n\geq 0 with the property that none of f(n+1),f(n+2),,f(n+k)f(n+1),f(n+2),\dots,f(n+k) is coprime to all the others. This extends previous results on linear polynomials and, in particular, on consecutive integers

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