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Thread: TA’s Challenge Problem #2

  1. #1
    Senior Member TheAbstractionist's Avatar
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    TA’s Challenge Problem #2

    This is the second Challenge Problem I’ve made up.

    Prove that $\displaystyle n^{4k}+4m^4$ is not prime for all positive integers $\displaystyle n,\,k,\,m$ with $\displaystyle m>1.$
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    Quote Originally Posted by TheAbstractionist View Post
    This is the second Challenge Problem I’ve made up.

    Prove that $\displaystyle n^{4k}+4m^4$ is not prime for all positive integers $\displaystyle n,\,k,\,m$ with $\displaystyle m>1.$
    $\displaystyle n^{4k}+4m^4=((n^k - m)^2 + m^2)((n^k + m)^2+m^2)$ and both factors are > 1 because m > 1.
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  3. #3
    Lord of certain Rings
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    NCA's identity has a name, its called Sophie Germain's Identity.
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    Senior Member TheAbstractionist's Avatar
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    Wow, that was really quick!
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    Grand Panjandrum
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    Quote Originally Posted by NonCommAlg View Post
    $\displaystyle n^{4k}+4m^4=((n^k - m)^2 + m^2)((n^k + m)^2+m^2)$ and both factors are > 1 because m > 1.
    For real $\displaystyle m$ and $\displaystyle n$ and integer $\displaystyle k$:

    $\displaystyle n^{4k}+4m^4=|n|^{4k}+4|m|^4$

    So the restriction to positive integers for $\displaystyle n$ and $\displaystyle m$ is unnecessary.

    Also if $\displaystyle m=0$ we have $\displaystyle n^{4k}+4m^4=n^{4k}$ which can never be prime for any integer $\displaystyle n$ (including $\displaystyle 0$).

    So the problem can be rewordrd to for all positive integers $\displaystyle k$ and integers $\displaystyle n,m$ show that

    $\displaystyle n^{4k}+4m^4$

    is never prime.

    (we may have to be explicit about what we want $\displaystyle 0^0$ to mean here or exclude the case where one or both of $\displaystyle n$ and $\displaystyle m$ are zero and $\displaystyle k$ is zero)

    CB
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    Quote Originally Posted by CaptainBlack View Post

    So the problem can be rewordrd to for all positive integers $\displaystyle k$ and integers $\displaystyle n,m$ show that

    $\displaystyle n^{4k}+4m^4$

    is never prime.
    this is not true. for example if n = m = 1, we'll get the prime number 5. it's true that m, n need not be positive though.
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    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    For real $\displaystyle m$ and $\displaystyle n$ and integer $\displaystyle k$:

    $\displaystyle n^{4k}+4m^4=|n|^{4k}+4|m|^4$

    So the restriction to positive integers for $\displaystyle n$ and $\displaystyle m$ is unnecessary.

    Also if $\displaystyle m=0$ we have $\displaystyle n^{4k}+4m^4=n^{4k}$ which can never be prime for any integer $\displaystyle n$ (including $\displaystyle 0$).

    So the problem can be rewordrd to for all positive integers $\displaystyle k$ and integers $\displaystyle n,m$ show that

    $\displaystyle n^{4k}+4m^4$

    is never prime.

    (we may have to be explicit about what we want $\displaystyle 0^0$ to mean here or exclude the case where one or both of $\displaystyle n$ and $\displaystyle m$ are zero and $\displaystyle k$ is zero)

    CB
    That’s a very good point. However, the condition $\displaystyle m>1$ (or $\displaystyle |m|>1$ if you like) is absolutely vital, otherwise you could find a counterexample as NonCommAlg pointed out.

    The way I originally did it was to factorize $\displaystyle n^{4k}+4m^4$ as $\displaystyle \left(n^{2k}+2mn^k+2m^2\right)\left(n^{2k}-2mn^k+2m^2\right)$ instead of writing the factors the way NonCommAlg did – and then it isn’t so clear that both factors are actually greater than 1. (One of them certainly is, but the other is iffy.) I then proceeded by considering the expressions $\displaystyle n^{2k}\pm2mn^k+2m^2-1$ as quadratics in $\displaystyle n^k.$ Their determinant is $\displaystyle 4m^2-8m^2+4=4(1-m^2)$ and the vital condition $\displaystyle m>1$ ensures that the determinant is negative and therefore that the expressions are positive.
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  8. #8
    Grand Panjandrum
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    Quote Originally Posted by NonCommAlg View Post
    this is not true. for example if n = m = 1, we'll get the prime number 5. it's true that m, n need not be positive though.
    Aghh.. I had missed that |x|>1 excluded x=+/-1 and so had not considered those cases.

    CB
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