Is every polynomial with integer coefficients near an irreducible polynomial? (Q2515078)

The following problem was proposed by P. Turán. Let \(f(x)\) be a polynomial with integer coefficients and of degree \(n\geq 1\). Is it true that with some absolute constant \(C\), there exists a polynomial \(g(x)\) with integer coefficients of the form \(\sum_{i=0}^n a_ix^i\) such that \(\sum_{i=0}^n |a_i|<C\) and \(f(x)+g(x)\) is irreducible? The problem is still open. The author gives an excellent review of the topic. On one hand, an approach for giving lower bounds for \(C\) is outlined. It is easy to see that if \(C\) exists, then \(C>1\) must hold. It is explained that the problem of establishing \(C>2\) is strongly related to covering congruence systems, and by a classical result of Schinzel and a problem of Erdős it is probably very hard. On the other hand, a theorem of Schinzel is recalled, supporting the plausibility that \(C\) does exist, and \(C\leq 3\) should be valid. Further, some numerical approach and results of Bérczes and Hajdu, and of Filaseta and Mossinghoff are presented, yielding strong numerical support for that \(C\leq 5\) should be valid.