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The Distance to an Irreducible Polynomial

An old problem of P. Turán asks if every polynomial with integer coefficients lies close to an irreducible polynomial of the same degree or less, where the distance between two polynomials f and g is measured as the sum of the absolute values of the coefficients of f − g. We develop some algorithms to answer this question in the affirmative for monic polynomials of degree at most 34, showing that an irreducible polynomial can always be found within distance 4 in this case, and in fact distance 3 suffices for degree at most 18. This improves some results of Bérczes and Hajdu. We also employ a probabilistic model to make some predictions for polynomials with larger degree, and conjecture that distance 4 suffices in general.