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Math Help - 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
    Joined
    Oct 2007
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    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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