prime or not?

**noteiler** · January 31st 2010, 05:34 AM

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.

**PaulRS** · January 31st 2010, 05:42 AM

Well, note that $1111$ is not prime -ineed, it's divisible by 11-

Now: $11...1 = \frac{10^{n}-1}{9}$ where $n$ is the number of 1s.

If $n$ is composite, then $n=a\cdot b$ for some $a,b\geq 2$ then $(10^a-1)$ will divide $10^n - 1$ or equivalently $\frac{10^a-1}{9}$ will divide $\frac{10^n - 1}{9}$ . $(*)$

Hence, if the number of 1s in your number is composite, so will the number . You have a composite number there.

$(*)$ Since $\frac{x^b-1}{x-1}=1+x+x^2+...+x^{b-1}$ letting $x=10^a$ we prove that claim.

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