Minimum logarithmic pair

For any two integers $m>1$ and $n>1$, we call an integer pair $(m,n)$ a minimum logarithmic pair if there does not exist two integers $a$ and $b$ with $1<a<m$ and $1<b<n$ such that ${\text{log}}_a(b)={\text{log}}_m(n)$.

For example, $(4,9)$ is not a minimum logarithmic pair because ${\text{log}}_4(9)={\text{log}}_2(3)$. Also $(3,9)$ is not a minimum logarithmic pair since ${\text{log}}_3(9)={\text{log}}_2(4)$. However $(2,3)$,$(3,4)$, and$(2,4)$ are all minimum logarithmic pairs.

Let $D(N)$ be the sum of $m+n$ over all minimum logarithmic pairs $(m,n)$ for $1<m,n\le N$

You are given $D(10)=818$, $D(100)=982052$, and $D(10000)=999830459302$

Find $D({11}^{111})$ and give your answer modulo ${10}^{18}+3$.