# How can I find the exponent $n$ efficiently?

Denote $$z=(2^{19}-1)\cdot10^6+2^{18}-1$$ $$a=ord_2(z)$$ $$b=ord_{10}(z)$$

The object is to find a positive integer of the form $$n=ka+19$$ with positive integer $$k$$ such that $$m=f(n)=\lceil(n-1)\cdot log_2(10)\rceil$$ is of the form $$m=lb+6$$

Motivation : An "ec-number" has the form $$ec(n)=(2^n-1)\cdot 10^m+2^{n-1}-1$$ where $$m$$ is the number of decimal digits of $$2^{n-1}$$. I want to find an exponent $$n>19$$ , such that $$ec(19)\mid ec(n)$$ If it helps, $$z=ec(19)$$ is a prime number.