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  1. #1
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    question

    hello everyone, I dont know how to do this question
    (i) For n belongs to IN+, show that n^2 + n + 1 is never a square in IN+

    (ii) Find all r in Z (integers) such that r^2 + r +1 is a squre in Z(integers)

    Can you give me hints please?
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  2. #2
    Senior Member
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    Hi

    i) Assume $\displaystyle n\geq 1,$ then $\displaystyle n^2<n^2+n+1<n^2+2n+1.$ Can you conclude?

    ii) Thanks to i), we know the only $\displaystyle r\in\mathbb{Z}$ such that $\displaystyle r^2+r+1$ is a square are among the non positive integers. So let $\displaystyle r$ be a non positive integer, and define $\displaystyle s=-r.$
    If $\displaystyle r=s=0,$ then $\displaystyle r^2+r+1=1$ is a square.
    Assume $\displaystyle r<0,\ \text{i.e.}\ s\geq 1.$
    Then since $\displaystyle r^2+r+1=s^2-s+1=(s-1)^2+s\ ,$ we have $\displaystyle (s-1)^2<r^2+r+1<(s-1)^2+2s=(s+1)^2$.
    Therefore, when $\displaystyle r<0,\ r^2+r+1$ is a square iff $\displaystyle r^2+r+1=s^2=r^2.$ Can you conclude?
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  3. #3
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    Thanks clic-clac.


    It makes sense to me now but in part (ii) you said
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