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Thread: prime or not?

  1. #1
    Junior Member
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    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.
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  2. #2
    Super Member PaulRS's Avatar
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    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.
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