A generalized ec-number (named by Enzo Creti) is a number that emerges when we concatenate two arbitary Mersenne-numbers ($2^1-1=1$ is allowed as well). For example, $77$ , $637$ or $13$ are generalized ec-numbers. The general formula is $$cm(m,n)=(2^m-1)\cdot 10^d+2^n-1$$ where $d$ is the number of digits in the decimal expansion of $2^n-1$. The name cm is because of "concatenated Mersennes".
Can distinct pairs $(m,n)$ generate the same cm-number ?
I don't think it is possible and I checked it for $1\le m,n\le 200$. But how can we prove it ?