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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- Oct 24th 2009, 09:10 AM #1

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- Oct 24th 2009, 12:39 PM #2

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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?

- Oct 25th 2009, 12:51 AM #3

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