# nt.quantity idea – On the determinant \$det[gcd(i-j,n)]_{1le i,jle n}\$ retort

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## nt.quantity idea – On the determinant \$det[gcd(i-j,n)]_{1le i,jle n}\$

In Sept. 2013, I investigated the determinant
$$D_n=det[gcd(i-j,n)]_{1le i,jle n}$$
and computed the values $$D_1,ldots,D_{100}$$ (cf. http://oeis.org/A228884). To my astonish, they’re all optimistic!

Question. Does $$D_n>0$$ maintain for all $$n=1,2,3,ldots$$?

I endure that $$D_n$$ is all the time optimistic. How to show this?

It is unostentatious to graze that $$D_n$$ is divisible by $$sum_{ok=1}^ngcd(ok,n)=sum_{dmid n}varphi(d)frac nd$$.
It appears that $$varphi(n)^{varphi(n)}sum_{ok=1}^ngcd(ok,n)$$ divides $$D_n$$. Maybe there’s a unostentatious rationalization for this.