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Math Help - diopantine equation

  1. #1
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    diopantine equation

    Which one is greater:2^100+3^100 or 4^80?
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by dila View Post
    Which one is greater:2^100+3^100 or 4^80?
    I can't think of an elegant way of doing this.

    4^{80}=2^{160}

    2^{160}-2^{100}=2^{100}(2^{60}-1)

    ln 2^{100}(2^{60}-1)=ln 2^{100}+ln (2^{60}-1)=100 ln 2+41.59=69.31+41.59=110.9

    ln 3^{100}=100ln 3=109.86

    Thus, ln 3^{100}<ln 2^{100}(2^{60}-1)

    3^{100}<2^{100}(2^{60}-1)

    2^{100}+3^{100}<2^{160}=4^{80}
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by alexmahone View Post
    I can't think of an elegant way of doing this.

    4^{80}=2^{160}

    2^{160}-2^{100}=2^{100}(2^{60}-1)

    ln 2^{100}(2^{60}-1)=ln 2^{100}+ln (2^{60}-1)=100 ln 2+41.59=69.31+41.59=110.9

    ln 3^{100}=100ln 3=109.86

    Thus, ln 3^{100}<ln 2^{100}(2^{60}-1)

    3^{100}<2^{100}(2^{60}-1)

    2^{100}+3^{100}<2^{160}=4^{80}
    ...and without unnecessary calculation...?
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  4. #4
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    thnks alots..
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  5. #5
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    Mine is not so beautiful but you can take a look :

    Consider

     2^8 = 256 = 243 + 13 = 3^5 + 13 so we have

     2^{160} = (3^5 + 13)^{20} > 3^{100} + 20(3^{95})(13)

     = 3^{100} + (260)3^{95} > 3^{100} + (243)3^{95} = 3^{100} + 3^{100} > 3^{100} + 2^{100}
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