prime or not?

• Jan 31st 2010, 05:34 AM
noteiler
prime or not?
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.
• Jan 31st 2010, 05:42 AM
PaulRS
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 $\displaystyle (10^a-1)$ will divide $\displaystyle 10^n - 1$ or equivalently $\displaystyle \frac{10^a-1}{9}$ 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 $\displaystyle \frac{x^b-1}{x-1}=1+x+x^2+...+x^{b-1}$ letting $\displaystyle x=10^a$ we prove that claim.