
prime or not?
hi, can you please help me with solving this problem.
the number is given: 11....111 where "1" is written 1111 times. is this number a prime or not?
the Wilson's theorem didn't help me, and i have checked out all primes up to 29, but none of them are the factors.

Well, note that $\displaystyle 1111$ is not prime ineed, it's divisible by 11
Now: $\displaystyle 11...1 = \frac{10^{n}1}{9}$ where $\displaystyle n$ is the number of 1s.
If $\displaystyle n$ is composite, then $\displaystyle n=a\cdot b$ for some $\displaystyle a,b\geq 2$ then $\displaystyle (10^a1)$ will divide $\displaystyle 10^n  1$ or equivalently $\displaystyle \frac{10^a1}{9}$ will divide $\displaystyle \frac{10^n  1}{9}$. $\displaystyle (*)$
Hence, if the number of 1s in your number is composite, so will the number . You have a composite number there.
$\displaystyle (*)$ Since $\displaystyle \frac{x^b1}{x1}=1+x+x^2+...+x^{b1}$ letting $\displaystyle x=10^a$ we prove that claim.