Tìm `a,b \in NN` tm `\sqrt{a+\sqrt{2021}}+\sqrt{b-\sqrt{2021}}\in QQ`

Question

Tìm `a,b \in NN` tm `\sqrt{a+\sqrt{2021}}+\sqrt{b-\sqrt{2021}}\in QQ`

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Isabelle 1 năm 2021-07-05T17:23:26+00:00 1 Answers 10 views 0

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    2021-07-05T17:24:51+00:00

    Đáp án: `a=b=1011`

    Giải thích các bước giải:

    Đặt `A=\sqrt{a+\sqrt{2021}}+\sqrt{b-\sqrt{2021}}\in QQ`

    `=>A^2=a+sqrt2021+b-sqrt2021+2sqrt((a+2021)(b-sqrt(2021)))\in QQ“

    `=>sqrt((a+2021)(b-sqrt(2021)))\inQQ(1)`

    `=>(a+sqrt2021)(b-sqrt2021)\inQQ`

    `=>ab-2021-sqrt2011(a-b)\inQQ`

    `=>sqrt2011(a-b)\inQQ`

    `=>a-b=0=>a=b`

    Từ `(1)` có `sqrt((a+2021)(b-sqrt(2021)))\inQQ` 

    `=>sqrt(a^2-2021)\inQQ`

    Mà `a\inNN` nên `a^2-2021\inNN`

    `=>sqrt(a^2-2021)\inNN`

    `=>a^2-2021=p^2(p>=1)`

    `=>` `a^2-p^2=2021`

    `=>` `(a-p)(a+p)=2021`

    `=>{(a-p=1),(a+p=2021):}=>a=1011=>b=1011`

    Vậy `a=b=1011`

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