Find all odd natural numbers $n$ for which there is a unique perfect square strictly between $n^2$ and $2n^2$. I considered some numerical examples $3$ is and odd number and between $3^2$ and $2(3)^2$. i.e., between 9 and 18 the unique square is 16. It is true. How should I begin the proof
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asked Jul 20, 2014 at 17:57
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1$\begingroup$ In terms of $n$, what is this perfect square that should lie between $n^2$ and $2n^2$? $\endgroup$– Andrés E. CaicedoJul 20, 2014 at 17:59
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1$\begingroup$ You are trying to find unique perfect squares between $n^2$ and $2n^2 = (\sqrt{2}n)^2$. So it is the same as finding a unique integer between $n$ and $\sqrt{2}n$. Does that help? $\endgroup$– JamesJul 20, 2014 at 18:00
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$\begingroup$ What is the next perfect square after $n^2$? If there is any squares between $n^2$ and $2n^2$, it would have include the next highest square. $\endgroup$– Thomas AndrewsJul 20, 2014 at 18:04
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$\begingroup$ And if there were more than one, it would include the next two. $\endgroup$– John HughesJul 20, 2014 at 18:07
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1 Answer
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basically this question asks for what odd value of $n$ do we have $2n^2\leq(n+2)^2=n^2+4n+4$
So we first solve the inequality: $2n^2\leq n^2+4n+4 \iff n^2<4n+4\iff n\leq4+\frac{4}{n}\implies n\leq4$
so the only values can be $1$ or $3$. Check there is no perfect square between 1 and 2, and check $3$ works.
answered Jul 20, 2014 at 18:06
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$\begingroup$ if $(n+1)^2>2n^2$ we get $0$ perfect squares between $n^2$ and $2n^2$ $\endgroup$– AsinomásJul 20, 2014 at 18:35
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$\begingroup$ Oh, he wants a unique perfect square. I can't read. $\endgroup$ Jul 20, 2014 at 18:39
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