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2^a + 1 not divisible by 2^b - 1

  1. Jun 21, 2012 #1
    1. The problem statement, all variables and given/known data
    If a, b are positive integers, b > 2, prove that

    2^a + 1 is not divisible by 2^b - 1.


    2. Relevant equations
    prime factorization of integers


    3. The attempt at a solution
    Suppose (2^a + 1)/(2^b - 1) = x, x an integer.

    Then x + 1 = (2^a + 2^b)/(2^b - 1).

    Write x + 1 = m2^r, m odd, r non-negative. Then

    (2^b - 1)m2^r = 2^a + 2^b.

    Let k = min{a,b,r}. Then

    (1) (2^b - 1)m2^(r - k) = 2^(a - k) + 2^(b - k)

    For most choices of k, (1) leads to contradiction. For example, if k = r, r < a, r < b, then LHS of (1) is odd while the RHS is even. The tricky cases are k = r = a with a < b, and k = r = b with b < a. In both these cases both LHS and RHS of (1) are odd.
     
  2. jcsd
  3. Jun 22, 2012 #2
    Hmm. Try looking at the values of ##(2^a+1)## mod ##(2^b-1)## as ##a## increases. Starting with ##a=b##.
     
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