Posers and Puzzles

Posers and Puzzles

  1. Joined
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    01 Jan '10 19:511 edit
    Prove:

    If a, b, c are integers such that |a| < 10^32, |b| < 10^32, |c| < 10^32, and a, b, c are not all zero, then

    |a*sqrt(2) + b*sqrt(3) + c*sqrt(5)| > 10^(-100)
  2. Standard memberPBE6
    Bananarama
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    05 Jan '10 18:26
    Originally posted by David113
    Prove:

    If a, b, c are integers such that |a| < 10^32, |b| < 10^32, |c| < 10^32, and a, b, c are not all zero, then

    |a*sqrt(2) + b*sqrt(3) + c*sqrt(5)| > 10^(-100)
    Quite an interesting problem! I've had a few ideas so far, but none of them any good . 😕 Any hints? Here's what I have so far.

    The form of the problem reminds me of the triangle inequality, which when coupled with some sort of squeezing operation was my first guess on how to solve it. However, the inequality signs are pointing the wrong way, so I don't think the triangle inequality is going to help much. 😕

    My next idea was to look at the problem as a Diophantine equation, and see if any Diophantine solutions came pre-packaged for this problem. I didn't find any, but I did stumble across the concept of Diophantine approximations (approximating irrational numbers using rational ones) and methods used to guage just how "good" the approximations can be. I figure that if I can find some expression for approximating 0, and then proving that the approximation is only good enough to estimate the solution with an error greater than 10^(-100) given the bounds of the problem then I will have a fully formed solution. The trick is figuring out how to do all that. 😕

    My latest idea was to use calculus to determine the stationary points on a real valued function matching the given one, evaluate the function at those stationary points, and see if I could exhaust all the values at the lattice points surrounding these values to determine by brute force that the given inequality must hold. However, my thoughts are a bit muddled on this one at present because the gradient is constant (with different constant values in each "orthant" - had to look up that word! 🙂) which doesn't help to narrow down the search one bit. 😕

    Of course, this problem may end up being solved by a much simpler method by manipulating the inequality appropriately. If so, I'm all ears. Any help is appreciated! 🙂
  3. Standard memberrandolph
    the walrus
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    05 Jan '10 19:35
    I really wish I could understand what you were saying
  4. Joined
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    06 Jan '10 12:20
    Originally posted by PBE6
    Quite an interesting problem! I've had a few ideas so far, but none of them any good . 😕 Any hints? Here's what I have so far.

    The form of the problem reminds me of the triangle inequality, which when coupled with some sort of squeezing operation was my first guess on how to solve it. However, the inequality signs are pointing the wrong way, so I don't ...[text shortened]... ating the inequality appropriately. If so, I'm all ears. Any help is appreciated! 🙂
    Hint:

    Let P=a*sqrt(2)+b*sqrt(3)+c*sqrt(5). Think about
    Q=a*sqrt(2)+b*sqrt(3)-c*sqrt(5),
    R=a*sqrt(2)-b*sqrt(3)+c*sqrt(5) and
    S=a*sqrt(2)-b*sqrt(3)-c*sqrt(5).
  5. Standard membersonhouse
    Fast and Curious
    slatington, pa, usa
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    06 Jan '10 20:24
    Originally posted by David113
    Prove:

    If a, b, c are integers such that |a| < 10^32, |b| < 10^32, |c| < 10^32, and a, b, c are not all zero, then

    |a*sqrt(2) + b*sqrt(3) + c*sqrt(5)| > 10^(-100)
    Are those exponents written correctly? 10 to the +32 in the first one and 10 to the minus 100 power in the second? 132 orders of magnitude difference?
  6. SubscriberAThousandYoung
    iViva la Hispanidad!
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    07 Jan '10 05:311 edit
    Originally posted by David113
    Prove:

    If a, b, c are integers such that |a| < 10^32, |b| < 10^32, |c| < 10^32, and a, b, c are not all zero, then

    |a*sqrt(2) + b*sqrt(3) + c*sqrt(5)| > 10^(-100)
    nvm
  7. Joined
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    07 Jan '10 05:45
    Originally posted by sonhouse
    Are those exponents written correctly? 10 to the +32 in the first one and 10 to the minus 100 power in the second? 132 orders of magnitude difference?
    I don't think the exponents are very important.

    Solve it as if "> 10^(-100)" and "< 10^32" means positive.
    Just check the solution so the numbers are not outside these limits.

    That's what I think.
  8. Standard membersonhouse
    Fast and Curious
    slatington, pa, usa
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    11 Jan '10 00:44
    Originally posted by FabianFnas
    I don't think the exponents are very important.

    Solve it as if "> 10^(-100)" and "< 10^32" means positive.
    Just check the solution so the numbers are not outside these limits.

    That's what I think.
    Yes, I realize that, just wondered if he made a typo, which would change the character of the problem.
  9. Germany
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    11 Jan '10 07:191 edit
    The physicists' extremely rigid proof: since the square roots are irrational, you cannot cancel them out using integers.

    It's actually quite hard to prove it holds specifically for these values but I won't bother attempting that.
  10. Joined
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    12 Jan '10 10:412 edits
    Originally posted by KazetNagorra
    The physicists' extremely rigid proof: since the square roots are irrational, you cannot cancel them out using integers.
    It's not asking you to cancel them out, though, it's asking you to almost cancel them out. There's no inherent reason that can't be possible - irrational numbers can be approximated by rational ones.
  11. Standard memberPalynka
    Upward Spiral
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    14 Jan '10 09:49
    Originally posted by mtthw
    It's not asking you to cancel them out, though, it's asking you to almost cancel them out. There's no inherent reason that can't be possible - irrational numbers can be approximated by rational ones.
    This get me thinking about the size of the exponents... Maybe it isn't possible because the maximum size of a,b,c, limits that ability.
  12. Germany
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    17 Jan '10 15:51
    Originally posted by mtthw
    It's not asking you to cancel them out, though, it's asking you to almost cancel them out. There's no inherent reason that can't be possible - irrational numbers can be approximated by rational ones.
    Well, I'm guessing that the fact that 1/(10^32)^3 > 10^-100 plays a role.
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