\(abc\) allows us to count squarefrees.

*(English)*Zbl 0924.11018For given any polynomial \(f(x)\in \mathbb{Z}[x]\), let \(B\) be the greatest common divisor of \(f(n)\) for all integers \(n\), and let \(B'\) be the smallest divisor of \(B\) such that \(B/B'\) is squarefree. Then \(f(n)/B'\) can feasibly be squarefree for various integers \(n\). It was P. Erdős [J. Lond. Math. Soc. 28, 416-425 (1953; Zbl 0051.27703)] who showed that if \(f(x)\) has degree \(\leq 3\) and \(B=1\), then there are infinitely many integers \(n\) for which \(f(n)\) is squarefree.

In this paper the author assumes the well-known \(abc\)-conjecture to investigate this problem together with some others in related topics. He proves that under the \(abc\)-conjecture if \(f(x)\) has no repeated roots then there are \(\sim c_fN\) positive integers \(n\leq N\) for which \(f(n)/ B'\) is squarefree where \(c_f>0\) is a constant. An explicit formula to determine \(c_f\) is provided in the paper. Similar results are obtained for homogeneous \(f(x,y)\in \mathbb{Z}[x,y]\) without any repeated linear factors.

Related topics are also discussed such as lower bounds for the products \(\prod_{p\mid f(m,n)}p\) and \(\prod_{p\mid g(m)}p\) where \(f(x,y)\in \mathbb{Z}[x,y]\) and \(g(x)\in \mathbb{Z}[x]\); and upper bounds for \(s_{n+1}-s_n\) and an asymptotic expression for its average moments where \(s_n\) is the increasing sequence of squarefree positive integers. These give more general or better results than those by several authors, e.g. N. Elkies [Int. Math. Res. Not. 1991, 99-109 (1991; Zbl 0763.11016)], M. Langevin [Sémin. Théorie des Nombres 1993-94, Publ. Math. Univ. Caen], M. Filaseta and O. Trifonov [Proc. Lond. Math. Soc. (3) 73, 241-278 (1996; Zbl 0867.11053)].

In this paper the author assumes the well-known \(abc\)-conjecture to investigate this problem together with some others in related topics. He proves that under the \(abc\)-conjecture if \(f(x)\) has no repeated roots then there are \(\sim c_fN\) positive integers \(n\leq N\) for which \(f(n)/ B'\) is squarefree where \(c_f>0\) is a constant. An explicit formula to determine \(c_f\) is provided in the paper. Similar results are obtained for homogeneous \(f(x,y)\in \mathbb{Z}[x,y]\) without any repeated linear factors.

Related topics are also discussed such as lower bounds for the products \(\prod_{p\mid f(m,n)}p\) and \(\prod_{p\mid g(m)}p\) where \(f(x,y)\in \mathbb{Z}[x,y]\) and \(g(x)\in \mathbb{Z}[x]\); and upper bounds for \(s_{n+1}-s_n\) and an asymptotic expression for its average moments where \(s_n\) is the increasing sequence of squarefree positive integers. These give more general or better results than those by several authors, e.g. N. Elkies [Int. Math. Res. Not. 1991, 99-109 (1991; Zbl 0763.11016)], M. Langevin [Sémin. Théorie des Nombres 1993-94, Publ. Math. Univ. Caen], M. Filaseta and O. Trifonov [Proc. Lond. Math. Soc. (3) 73, 241-278 (1996; Zbl 0867.11053)].

Reviewer: Ming-Chit Liu (Hongkong)