What is the probability that three pairs $(a,b) $ , $(c,d) $ and $(e,f) $ of integers generate $\mathbb Z^2$? As usual the probability is the limit as $n\to \infty$ of the same probability for the $n\times n$ square. It is well known that for $\mathbb Z $ the probability of two numbers to generate is $6/\pi^2$.



Probability of generation of ${\mathbb Z}^2$
According to Proposition 1 in the paper 


G. Maze, Gérard, J. Rosenthal, U. Wagner: [Natural density of rectangular unimodular integer matrices](http://dx.doi.org/10.1016/j.laa.2010.11.015), *Linear Algebra Appl.* **434**, No. 5 (2011), 1319-1324, [ZBL1211.15044](https://zbmath.org/?q=an:1211.15044),


the probability that $n$ random vectors generate $\mathbb{Z}^{n-1}$ is $$p\_n = \prod\_{j=2}^n \zeta(j)^{-1}.$$


For $n=2$ this gives $p\_2=\zeta(2)^{-1}=6/\pi^2$, whereas for $n=3$ we obtain $$p\_3= \zeta(2)^{-1} \zeta(3)^{-1} \simeq 0.505739038$$


