# Density of twin square-free numbers

It is well-known how to compute the density of square-free numbers, to get $$\lim_{N\to\infty} \frac{\#\{ n \leq N : n \text{ square-free}\}}{N} = \frac{6}{\pi^2}.$$

What is the density of numbers such that both $$n$$ and $$n+1$$ are square-free? In other words, what is $$\lim_{N\to\infty} \frac{\#\{ n \leq N : n(n+1) \text{ square-free}\}}{N}$$ (if the limit exists)? I'm guessing this has been studied before. Does anyone have a textbook or paper reference?

## 1 Answer

See NOTE ON AN ASYMPTOTIC FORMULA CONNECTED WITH r-FREE INTEGERS by L. MIRSKY, The Quarterly Journal of Mathematics, Volume os-18, Issue 1, 1947, Pages 178–182, https://doi.org/10.1093/qmath/os-18.1.178

This paper is more general, i.e., for $$r$$ tuples of square free numbers with fixed gap sizes. The number of such integer pairs $$\leq x$$ is given by $$Ax+O( x^{\frac{2}{3}+\epsilon}(\log x)^{\frac{4}{3}}),$$ where $$A$$ is a constant. See also here where the constant $$A$$ is evaluated in terms of Euler products.