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**knguyen2005** · October 24th, 2009, 09:10 AM

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?

**clic-clac** · October 24th, 2009, 12:39 PM

Hi

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

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

**knguyen2005** · October 25th, 2009, 12:51 AM

Thanks clic-clac.

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

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