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Suppose $n$ is a natural number. Prove that $\sqrt{n^2 + 2}$ is irrational.

From looking at the expression, it seems quite obvious to me that $\sqrt{n^2 + 2}$ will be irrational, since $n^2$ will be a natural number, and after adding $2$ to it, $n^2 + 2$ will no longer be a perfect square.

From that logic, I tried letting $n^2 + 2$ = ${a^2\over b^2}$ to find a contradiction, but I wasn't able to formulate a proof that worked. Any suggestions as to how I should approach this?

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It can be easily proven that $\sqrt k$ is irrational if $k\in$ N is not a perfect square. At the same time, the difference between two consecutive perfect squares is $(n+1)^2-n^2=2n+1>2$ for $n>0$. So $n^2+2$ lies in between two consecutive perfect squares, meaning it is not one. Therefore, its radical is irrational.

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When $n$ is even, show that $n^2+2$ will be $2$ times an odd number. Then show that such a number doesn't have a rational square root (you can modify the usual proof that $\sqrt{2}$ is irrational).

When $n$ is odd, show that $n^2+2\equiv 3$ (mod $4$). Then show that any integer squared is congruent to $0$ or $1$ mod $4$, so that $n^2+2$ is not a perfect square. As in Jaoa's comment, finish up by arguing the square root of a non-square is irrational, which again can be done by modifying the $\sqrt{2}$ proof.

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Without loss of generality we can assume $x,y\in \mathbb{N}\ (?)$.

$$n^2+2=\dfrac{x^2}{y^2} \implies x^2=y^2\left(n^2+2\right)$$

For odd or even $n$, $x,y$ can be both odd, or exactly one of them should be even.

Case 1 ($n$ odd)

Case 1.1 (both odd)

$\left(x^2\equiv1\pmod4, y^2\equiv1\pmod4\right)\land \left(x^2=y^2\left(n^2+2\right)\right) \implies 1\equiv3\pmod 4$

Case 1.2 ($x$ odd, $y$ even)

$\left(x^2\equiv1\pmod4, y^2\equiv0\pmod4\right)\land \left(x^2=y^2\left(n^2+2\right)\right) \implies 1\equiv0\pmod 4$

Case 1.3 ($x$ even, $y$ odd)

$\left(x^2\equiv0\pmod4, y^2\equiv1\pmod4\right)\land \left(x^2=y^2\left(n^2+2\right)\right) \implies 0\equiv3\pmod 4$

Case 2 ($n$ even)

Case 2.1 (both odd)

$\left(x^2\equiv1\pmod4, y^2\equiv1\pmod4\right)\land \left(x^2=y^2\left(n^2+2\right)\right) \implies 1\equiv2\pmod 4$

Case 2.2 ($x$ odd, $y$ even)

$\left(x^2\equiv1\pmod4, y^2\equiv0\pmod4\right)\land \left(x^2=y^2\left(n^2+2\right)\right) \implies 1\equiv0\pmod 4$

Case 2.3 ($x$ even, $y$ odd)

$\left(x^2\equiv0\pmod4, y^2\equiv1\pmod4\right)\land \left(x^2=y^2\left(n^2+2\right)\right) \implies 0\equiv2\pmod 4$

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  • $\begingroup$ Why the "?" is this really an answer? $\endgroup$ – John Smith Dec 10 '14 at 6:32
  • $\begingroup$ @JohnSmith: It is not an answer, just a hint. $\endgroup$ – user170039 Dec 10 '14 at 6:33
  • $\begingroup$ Your second line is wrong. The $2$ should be $2y^2$. $\endgroup$ – Bungo Dec 10 '14 at 6:33
  • $\begingroup$ @Bungo: It's now edited. Is it ok now? $\endgroup$ – user170039 Dec 10 '14 at 6:46
  • $\begingroup$ I think it's OK now. $\endgroup$ – Bungo Dec 10 '14 at 6:51
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$$\begin{array}{l} \sqrt {2 + n^2 } = \frac{p}{q};\quad and\quad p \wedge q = 1 \\ \sqrt {2 + n^2 } = \frac{p}{q} \Leftrightarrow 2 + n^2 = \left( {\frac{p}{q}} \right)^2 \\ \Leftrightarrow 2 = \left( {\frac{p}{q}} \right)^2 - n^2 \\ \Leftrightarrow 2 = \left( {\frac{p}{q} - n} \right)\left( {\frac{p}{q} + n} \right) \\ \Leftrightarrow 2q^2 = \left( {p - nq} \right)\left( {p - nq} \right) \cdots \left( * \right) \\ \left( * \right) \Rightarrow \left\{ \begin{array}{l} \exists m_0 \in\mathbb{ N} :p - nq = m_0 q \\ or \\ \exists m_1 \in \mathbb{N} :p + nq = m_1 q \\ \end{array} \right. \\ \Rightarrow \left\{ \begin{array}{l} \exists m_0 \in \mathbb{N} :p = \left( {n + m_0 } \right)q \\ or \\ \exists m_1 \in \mathbb{N} :p = \left( { - n + m_1 } \right)q \\ \end{array} \right. \\ \left\{ \begin{array}{l} p = \left( {n + m_0 } \right)q \Rightarrow q/p \\ or \\ p = \left( { - n + m_1 } \right)q \Rightarrow q/p \\ \end{array} \right. \\ \end{array}$$

contradiction with $\quad p \wedge q = 1$

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If $n\equiv0\pmod{2}$, then $n^2+2\equiv2\pmod{4}$.
If $n\equiv1\pmod{2}$, then $n^2+2\equiv3\pmod{4}$.

Thus, $n^2+2$ cannot be a perfect square since neither $2$ nor $3$ is a quadratic residue mod $4$.

As shown in this answer or using the Rational Root Theorem, if $n^2+2$ is not a perfect square, then $\sqrt{n^2+2}$ is irrational.

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