<s>[INST] How many (decimal) digits does $2^{3021 377}$ have? I was wondering, how many (decimal) digits does $2^{3021377}$ have?
We have $2^4=16,\, 2^5=32,\, 2^6=64$ and $2^7=128,\, 2^8=256, \, 2^9=512$ but $2^{10}=1024,\, 2^{11}=2048, \, 2^{12}=4096, \, 2^{13}=8192$, so there is no periodicity in the power.
On the other hand,
\begin{align}
2^{3021377}=& 2^{3000000 + 20000 + 1000 + 300 + 70 +7}  \\
=& (2^{10^6})\times{2^3}  \times  (2^{10^4})\times{2^2} \times  2^{10^3} \times ( 2^{10^2})\times{2^3} \times (2^{10})\times{2^7}  \times2^7 \\
\end{align}
using the fact $2^{10}=1024$. Can we deduce the number of digits from this? Mathematica gives:
IntegerLength[2^{3021377}]
{909526}

 [/INST] $$\left\lfloor 3021377\log_{10}2\right\rfloor +1 $$
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