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Thread: no positive integers

  1. #1
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    no positive integers

    Show no positive integers m for which m^4 + 2m^3 + 2m^2 + 2m + 1 is a perfect square. Are there any positive integers m for which m^4 +m^3 +m^2 +m+1 is a perfect square?

    If so, find all such m.
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  2. #2
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    Quote Originally Posted by Fayyaz View Post
    Show no positive integers m for which m^4 + 2m^3 + 2m^2 + 2m + 1 is a perfect square. Are there any positive integers m for which m^4 +m^3 +m^2 +m+1 is a perfect square?

    If so, find all such m.
    Hi

    $\displaystyle m^4 + 2m^3 + 2m^2 + 2m + 1 = m^4 + 2m^2 + 1 + 2m^3 + 2m
    = (m^2 + 1)^2 + 2m\m^2 + 1)$

    $\displaystyle m^4 + 2m^3 + 2m^2 + 2m + 1 = (m^2 + 1)\m^2 + 2m + 1) = (m^2 + 1)\m+1)^2$
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  3. #3
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    Hello, Fayyaz!

    running-gag has an excellent proof, but omitted the punchline.


    $\displaystyle N \;=\;m^4 + 2m^3 + 2m^2 + 2m + 1 \;=\; m^4 + 2m^2 + 1 + 2m^3 + 2m$

    . . $\displaystyle =\; (m^2 + 1)^2 + 2m\,(m^2 + 1) \;=\; (m^2 + 1)\,(m^2 + 2m + 1) \;=\; (m^2 + 1)\,(m+1)^2$

    If $\displaystyle N$ is a square, then $\displaystyle (m^2+1)$ must be a square.

    . . Hence, $\displaystyle m^2 + 1 \:=\:a^2\quad\hdots$ for some integer $\displaystyle a.$

    Then: .$\displaystyle a^2-m^2 \:=\:1 \quad\hdots $ two squares differ by 1.

    . . But the only two squares that differ by 1 are: .$\displaystyle a = \pm1,\text{ and }\, m = 0$


    Since $\displaystyle m$ must be a positive integer, no solution exists.

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  4. #4
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    thanks guys

    how should i proceed with the next bit:

    Are there any positive integers m for which m^4 +m^3 +m^2 +m+1 is a perfect square?
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  5. #5
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    Hello Fayyaz
    Quote Originally Posted by Fayyaz View Post
    thanks guys

    how should i proceed with the next bit:

    Are there any positive integers m for which m^4 +m^3 +m^2 +m+1 is a perfect square?
    When $\displaystyle m = 3, m^4 +m^3 +m^2 +m+1=121 = 11^2$. So there's certainly one value. Whether there are any more, I don't know.

    Grandad
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  6. #6
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    hi

    thanks, yeah i figured that out using excel spreadsheet, but anyway I can do an algebraic proof?
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