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Thread: Help with proof

  1. #1
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    Help with proof

    If $\displaystyle n^2+1$ is prime, then $\displaystyle n^2+1$ can be expressed as $\displaystyle 4k+1$

    I was thinking you probably take $\displaystyle n^2+1$ and maybe do $\displaystyle n^2+1=p$ and manipulate it from there?
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  2. #2
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    Quote Originally Posted by JSB1917 View Post
    If $\displaystyle n^2+1$ is prime, then $\displaystyle n^2+1$ can be expressed as $\displaystyle 4k+1$

    I was thinking you probably take $\displaystyle n^2+1$ and maybe do $\displaystyle n^2+1=p$ and manipulate it from there?
    Unless I am missing something the statement is false.

    let $\displaystyle n=1 \implies 1^2+1=2$ is prime but there isn't an integer such that

    $\displaystyle 4k+1=2$
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  3. #3
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    Quote Originally Posted by JSB1917 View Post
    If $\displaystyle n^2+1$ is prime, then $\displaystyle n^2+1$ can be expressed as $\displaystyle 4k+1$
    That is not true if $\displaystyle n=1$. Is it?
    2 is prime.
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  4. #4
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    sorry, I forgot to put n =/= 1.
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  5. #5
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    If $\displaystyle n>0~\&~n^2+1$ is a prime number, then it is odd.
    So $\displaystyle n^2$ is even. WHY?
    That means $\displaystyle n$ is even, or $\displaystyle n=2k$ for some $\displaystyle k$
    Therefore, $\displaystyle n^2+1=4j+1$ where $\displaystyle j=k^2.$
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