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What does this equation tell me 2^a=3^b-1

  1. Apr 29, 2012 #1
    1. The problem statement, all variables and given/known data
    what does this equation tell me 2^a=3^b-1?


    2. Relevant equations



    3. The attempt at a solution
    its meant to be telling me something but im not sure what
     
  2. jcsd
  3. Apr 29, 2012 #2
    Well more of the question would be useful, but it you want a solution of either variable in terms of the other take logs on both sides and solve.
     
  4. Apr 29, 2012 #3
    this is what i have


    We are looking for solutions of:
    2^x + 3^y = z^2 [1]
    where x, y, and z are non-negative integers.
    So z^2 ≡1 (mod 3) unless z^2=3m for any integer m, and if x is odd, say x=(2k+1) for some integer k. However when z=3, we have 2^3+3^0=3^2, so one solution is when x=3,y=0 and z=3 When x is odd 2^(2k+1)≡2(mod 3) and when x is even 2^x≡1 (mod3) so z^2 is congrgent to 2^x when x is even.
    We now know x is even, let x=2k. So we have:
    3^y=(z-2^k )(z+2^k )
    Now (z-2^k ) and (z+2^k ) should both be powers of three, so therefore (z+2^k )-(z-2^k ) should equal a power of three, so
    3^w=(z+2^k )-(z-2^k )
    3^w≠2^2k=2^x
    So 2^x should be divisible by three, but it isn’t. So let (z-2^k )=3^a and (z+2^k )=3^b then we have
    █((z+2^k )=3^b@(z-2^k )=3^a )/(2^(k+1)=3^b-3^a )
    If a>0 then 3^b-3^a will be a power of three and 2^(k+1) is not divisible by three, so let a=0. We are left with
    2^(k+1)=3^b-1
    Let a=k+1, then we have 2^a=3^b-1.So this means that b≥0, otherwise 2^a wont be an integer, which means a≥1. Also a≤3 because if k>2, then 3^b≡1mod(8) which is a contradiction, so therefore a has to be either 1,2 or 3, making k=0,1 or 2.
    When k=0 we have
    2^1=3^b-1 → 3=3^1 so therefore b=1 and
    When k=1 we have
    2^2≠3^b-1 so therefore k=1 doesn’t exist.
    When k=2 we have
    2^3=3^b-1 → 9=3^b so therefore b=2.

    We are looking for solutions of:
    2^x + 3^y = z^2 [1]
    where x, y, and z are non-negative integers.
    So z^2 ≡1 (mod 3) unless z^2=3m for any integer m, and if x is odd, say x=(2k+1) for some integer k. However when z=3, we have 2^3+3^0=3^2, so one solution is when x=3,y=0 and z=3 When x is odd 2^(2k+1)≡2(mod 3) and when x is even 2^x≡1 (mod3) so z^2 is congrgent to 2^x when x is even.
    We now know x is even, let x=2k. So we have:
    3^y=(z-2^k )(z+2^k )
    Now (z-2^k ) and (z+2^k ) should both be powers of three, so therefore (z+2^k )-(z-2^k ) should equal a power of three, so
    3^w=(z+2^k )-(z-2^k )
    3^w≠2^2k=2^x
    So 2^x should be divisible by three, but it isn’t. So let (z-2^k )=3^a and (z+2^k )=3^b then we have
    █((z+2^k )=3^b@(z-2^k )=3^a )/(2^(k+1)=3^b-3^a )
    If a>0 then 3^b-3^a will be a power of three and 2^(k+1) is not divisible by three, so let a=0. We are left with
    2^(k+1)=3^b-1
    Let a=k+1, then we have 2^a=3^b-1.So this means that b≥0, otherwise 2^a wont be an integer, which means a≥1. Also a≤3 because if k>2, then 3^b≡1mod(8) which is a contradiction, so therefore a has to be either 1,2 or 3, making k=0,1 or 2.
    When k=0 we have
    2^1=3^b-1 → 3=3^1 so therefore b=1 and
    When k=1 we have
    2^2≠3^b-1 so therefore k=1 doesn’t exist.
    When k=2 we have
    2^3=3^b-1 → 9=3^b so therefore b=2.
     
  5. Apr 29, 2012 #4
    its the bit where i get to a<=3, im not sure what this is meant to be telling me
     
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