I am looking for a proof of the equality in the title (where $q\in\mathbb N$, ($q\ge2$)).
Does anyone know such a proof?


Thanks in advance



$LCM(q-1,\cdots,q^n-1)=q^{\frac{3}{\pi^2} n^2+o(n)}$
Niven, Fermat's theorem for matrices, Duke Math. J. 15 (1948) 823–826, MR0026672 (10,183e), proved, for any positive integers $a$ and $r$, $${\rm lcm}[x-1,x^2-1,\dots,x^r-1]\_{x=a}={\rm lcm}[a-1,a^2-1,\dots,a^r-1]$$ Now follow Greg Martin's comment. 


