A Diophantine equation and decimal digits Solutions of the Diophantine equation 
$a10^n+(a+1) = (2^{m+1}-1)*2^{m+1}$ 
are
12=3*4, 
56=7*8, 
67100672=8191*8192.
Are there more solutions/examples like that or a generalization of the equation? Does it describe a natural phenomenon?
 A: $1.$ Denote $t=2^{m+1},N=10^n.$ If $a<N$ then $1\leq a=\dfrac{t^2-t-1}{N+1}<N,$ 
hence $$N+1\leq t^2-t-1<N(N+1),$$with some discussion we will get 
$$\sqrt{N}\leq t<N,10^{\frac{n}{2}}<2^{m+1}<10^n,$$
$$\frac{n}{2}\log_2{10}<m+1<n\log_2{10}$$
So for every given $n$,we can try all the $m$ one-by-one.I did this for $0<n<3000$,but no other solution was found.
$2.$ If $a<10^n$ is not necessary, we can use the following method for some little $n$.
Denote $t=2^{m+1},d=10^n+1,$ then $t^2-t-1\equiv0 \pmod d,$$$(2t-1)^2\equiv5 \pmod d\tag1$$
hence if $(1)$ has no solution,then $n$ is not the solution for the original problem.
If $(1)$ has some solutions, such as $t$,then $$2^{m+1}\equiv t \pmod d \tag2$$
If $(2)$ has solutions for $m$,then we get a solution for the original problem, and $m^{'}=m+k\phi(d)$ is a solution,too.
For example: 
$n=5,m+1=13,a=671$
$n=2,m+1=64,a=3369132345751865974702256072852065939.$
$n=2,m+1=86,a=59270403034726518346161316806283592011628044701331.$
$n=5,m+1=4532,a=34346\cdots 204319.$
