#### Vol. 6, No. 5, 2012

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Squareful numbers in hyperplanes

### Karl Van Valckenborgh

Vol. 6 (2012), No. 5, 1019–1041
##### Abstract

Let $n\ge 4$. In this article, we will determine the asymptotic behavior of the size of the set of integral points $\left({a}_{0}:\cdots :{a}_{n}\right)$ on the hyperplane ${\sum }_{i=0}^{n}{X}_{i}=0$ in ${ℙ}^{n}$ such that ${a}_{i}$ is squareful (an integer $a$ is called squareful if the exponent of each prime divisor of $a$ is at least two) and $|{a}_{i}|\le B$ for each $i\in \left\{0,\dots ,n\right\}$, when $B$ goes to infinity. For this, we will use the classical Hardy–Littlewood method. The result obtained supports a possible generalization of the Batyrev–Manin program to Fano orbifolds.

##### Keywords
squareful, Campana, asymptotic behavior
##### Mathematical Subject Classification 2010
Primary: 11D45
Secondary: 14G05, 11D72, 11P55