# How to find all square solutions?

Given the equation

$f(k) = 4-7k$

what is the easiest way to find which integer values of $k$ makes $f(k)$ a perfect square $(-3, -11, ...)$

(I hope for something better than just testing values of $k$)

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Take all perfect squares and apply $f^{-1}$? Your $f$ is bijective so this will work for any perfefct square. If you want integer solutions of $k$, this will also narrow down the needed "tests". – AlexR Dec 6 '13 at 12:35
Ah yes, i need integer solutions.. should have specified that. sorry. – ChessMetrics Dec 6 '13 at 12:36
$k=\frac{-x^2+4}{7}$ where $x\in\mathbb{N}$ – Riccardo.Alestra Dec 6 '13 at 12:37
@Riccardo.Alestra note that in general $7 \not | -x^2 + 4$ and thus $f^{-1}(x^2)$ might not be in $\mathbb Z$. – AlexR Dec 6 '13 at 12:38
I am only interested in integer values of k. – ChessMetrics Dec 6 '13 at 12:42

Let's reduce the possibilities first by working modulo 7. The equation $x^2 = 4 - 7k$ implies that $\overline x^2 = 4 \in \mathbb{Z}/7\mathbb{Z}$. A quick check shows that $\overline 2$ and $\overline 5$ are the only numbers in that ring that square to 2. So we must have $x \equiv 2$ or $x \equiv 5$ modulo 7.

The second step is a trick. We are only interested in $x^2$, so we do not care if we use $x$ or $-x$. If $x \equiv 5$ modulo 7 we can replace $x$ by $-x$ and then obtain $x \equiv 2$ modulo 7.

So we know that there is an integer $m$ such that $x = 7m +2$. Let's see if all $m$ work.

\begin{align*} x^2 &= 4 - 7k\\ (7m+2)^2 &= 4 - 7k\\ 49m^2 + 14m + 4 &= 4 - 7k\\ 49m^2 + 14m &= -7k\\ -(7m^2 + 2m) &= k \end{align*}

So not only we must have $x = 7m +2$, also all integer values for $m$ work. The above values for $k$ are your final answer.

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Consider $f(k)=j^2$ whenever any value of $k$ gives a perfect square.

Now, $j^2-4=-7k$

or,$(j-2)(j+2)=-7k$

Since, $gcd(7,k)=1$

So,whenever, $7|(j-2)$ or $7|(j+2)$, we get a solution.

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This was very nice...Thanks – ChessMetrics Dec 6 '13 at 12:51
@ChessMetrics.Most welcome! – Hawk Dec 6 '13 at 12:52

As mentioned in the comments your function is bijective, so every square $x^2$ where $x\in \mathbb{Z}$ has exactly one inverse image given by $f^{-1}(x^2)=\frac{4-x^2}{7}$. Now you just need to work out for which values of $x$ $f^{-1}(x^2)$ is an integer, or for which values of $x$, $x^2\equiv4$ mod $7$. By checking all residues mod $7$ we conclude that $x$ has to be congruent to $2$ or $5$ mod $7$.

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