Let $${f \in \mathbb{Z}[x]}$$ , $${\deg f =3}$$ . Assume that$f$does not have repeated roots. Assume as well that, for every prime$q$, $${f(x)\not\equiv 0}$$ mod$q$^{2}has at least one solution in $${(\mathbb{Z}/q^2 \mathbb{Z})^*}$$ . Then, under these two necessary conditions, there are infinitely many primes$p$such that$f$($p$) is square-free.

For any natural number$m$(>1) let$P$($m$) denote the greatest prime divisor of$m$. By the problem of Erdős in the title of the present paper we mean the general version of his problem, that is, his conjecture from 1965 that $$\frac{P(2^n-1)}{n} \to \infty \quad {\rm as}\, n \to \infty$$ (see Erdős [10]) and its far-reaching generalization to Lucas and Lehmer numbers. In the present paper the author delivers three refinements upon Yu [40] required by C. L. Stewart for solving completely the problem of Erdős (see Stewart [25]). The author gives also some remarks on the solution of this problem, aiming to be more streamlined with respect to the$p$-adic theory of logarithmic forms.

Let$u$_{$n$}be the$n$th term of a Lucas sequence or a Lehmer sequence. In this article we shall establish an estimate from below for the greatest prime factor of$u$_{$n$}which is of the form$n$exp(log$n$/104 log log$n$). In doing so, we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdős from 1965. In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.

In our previous paper “The primes contain arbitrarily long polynomial progressions” [$Acta Math$., 201 (2008), 213–305] there were some minor errors in our definition of the polynomial forms and polynomial correlation conditions, which is unreasonably strong as written due to the overly loose bound on the degree of the polynomials involved. In this erratum we repair this issue by capping the degree of the polynomials more strongly, and adjusting some other relevant components of the argument accordingly.

In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we derive the universality of eigenvalue gap distribution and$k$-point correlation, and many other statistics (under some mild assumptions) for both Wigner Hermitian matrices and Wigner real symmetric matrices.