For a polynomial \(P\) with rational integral coefficients, denote by \(| P| \) the sum of the absolute values of its coefficients. For \(n=1,2,\dots\) let \(c_n\) be the smallest integer with the property that for every polynomial \(P\in {\mathbb Z}[X]\) of degree \(n\) there exists an irreducible polynomial \(Q\in {\mathbb Z}[X]\) of degree \(\leq n\) such that \(| P-Q| \leq c_n\). P. Turán asked whether the sequence \(c_1,c_2,\dots\) is bounded. The authors show that \(c_1=0\), \(c_2=1\), for \(3\leq n\leq 6\) one has \(c_n=2\), for \(7\leq n\leq 12\) one has \(c_n\leq 3\) and for \(13\leq n\leq 24\) one has \(c_n\leq 4\).
For the entire collection see [Zbl 0887.00013].