<s>[INST] Is $1111111111111111111111111111111111111111111111111111111$ ($55$ $1$'s) a composite number? This is an exercise from a sequence and series book that I am solving.
I tried manipulating the number to make it easier to work with:
$$111...1 = \frac{1}9(999...) = \frac{1}9(10^{55} - 1)$$
as the number of $1$'s is $55$.
The exercise was under Geometric Progression and Geometric Mean. However, I am unable to think of a way to solve this problem using GP.
How do I proceed from here?
 [/INST] More explicitly, 
$$\begin{align*}
\frac{10^{55} - 1}{9} 
&= \frac{(10^5)^{11} - 1}{9} \\
&= \frac{(10^5 - 1)(10^{50} + 10^{45} + 10^{40} + \cdots + 10 + 1)}{9} \\
&= \frac{(10 - 1)(10^4 + 10^3 + 10^2 + 10^1 + 1)(10^{50} + 10^{45} + \cdots + 1)}{9} \\
&= (10^4 + 10^3 + \cdots + 1)(10^{50} + 10^{45} + \cdots + 1).
\end{align*}$$
The first factor is $11111$, which in turn is $41 \cdot 271$, and the second factor has as its smallest prime factor $1321$.
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