Find $\log_{256}N$ if $N=(2+1)(2^2+1)(2^4+1)\cdots(2^{32}+1)+1$

Question

Find $\log_{256}N$ if $N=(2+1)(2^2+1)(2^4+1)\cdots(2^{32}+1)+1$.

I can't think of any way to begin solving this question. Can someone help me with this question? A hint to help me obtain the right approach to solve this question would be really useful.

Thank you :)

Answer 1

HINT:

Using $a^2-b^2=(a-b)(a+b)$

$$N-1=2^{64}-1$$

Now use $\log_ba=\dfrac{\log a}{\log b}$ and $\log(c^m)=m\log c$