Let f is an element of Z[X] be a quadratic or cubic polynomial. We prove that there exists an integer G(f) >= 2 such that for every integer k >= G(f) one can find infinitely many integers n >= 0 with the property that none of f(n+1),f(n+2),...,f(n+k) is coprime to all the others. This extends previous results on linear polynomials and, in particular, on consecutive integers.
A coprimality condition on consecutive values of polynomials
SANNA, CARLO;
2017-01-01
Abstract
Let f is an element of Z[X] be a quadratic or cubic polynomial. We prove that there exists an integer G(f) >= 2 such that for every integer k >= G(f) one can find infinitely many integers n >= 0 with the property that none of f(n+1),f(n+2),...,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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