Context: In his paper, Cameron Stewart (click here for the paper) shows that the largest prime factor of $2^n-1$ is at least $n \times \exp\Big( \frac{\log n}{104 \log \log n}\Big)$ , if $n$ is large enough.
Question: Is this result valid for $2^{m} \times r -1$ where we can write- $2^{m} \times r -1=2^{m} \times 2^{\log_2(r)} -1$.
If not, would you please explain why it is not valid or point out which part of the paper declines it?