In the following all variables are assumed to be integers.

It is easy to write a Diophantine equation which has solutions only when $N$ is a square. i.e.


It's trivial to write a Diophantine equation which has solutions if and only if $N$ is divisible by 4:

$$N = 4A$$

Also it is fairly easy to write a Diophantine which has solutions if and only if $N$ is not divisible by 4:


But how about a Diophantine equation which has solutions if and only if $N$ is not a square number?

(sum, product and minus only can be used).

  • $\begingroup$ $N=A^2+1$ for $A\ne\pm1$ will do, if I understand you right. $\endgroup$ – Qurultay Apr 4 '18 at 21:50
  • $\begingroup$ @Qurultay $N=6$ is not a perfect square but can't be written as $A^2+1$, either... $\endgroup$ – Misha Lavrov Apr 4 '18 at 21:52
  • $\begingroup$ @MishaLavrov Now I understand you. It desires a +1. $\endgroup$ – Qurultay Apr 4 '18 at 21:53

Here's one approach, though possibly not the simplest.

A number $N$ is not a perfect square if $A^2+1 \le N \le (A+1)^2-1$ for some $A$. But how can we encode $X \le Y$? Over the real numbers, the standard trick would be to write $Y = X + Z^2$, because $Z^2$ is always nonnegative. Over the integers, that doesn't quite work, but we know that $X \le Y$ if and only if there are four integers $P,Q,R,S$ such that $Y = X + P^2 + Q^2 + R^2 + S^2$, by using Lagrange's four-square theorem.

So we get $$ N = A^2+1 + B^2 + C^2 + D^2 + E^2 \text{ and } A^2+2A = N + F^2 + G^2 + H^2 + I^2 $$ but we probably want to write this as a single equation. To do this, just take $$ (N - A^2-1-B^2-C^2-D^2-E^2)^2 + (A^2+2A-N-F^2-G^2-H^2-I^2)^2 = 0. $$ Over the integers (or even the reals), $X^2+Y^2=0$ only if $X=Y=0$, giving us what we wanted.

Actually, the above only characterizes positive nonsquares: if $N$ is negative, there's no value of $A$ we can choose. But we can multiply the equation above by an equation representing $N\le-1$ easily, fixing this issue.

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  • $\begingroup$ Oh that's clever using an inequality to sandwich out the numbers we want! I kind of new both tricks but didn't think to use this method. Well done! $\endgroup$ – zooby Apr 4 '18 at 22:02
  • $\begingroup$ Isn't it sufficient to use $Z^2$ since it's guaranteed positive, as you noted? That is, $N = A^2 + 1 + Z_1^2$ and $A^2 + 2A = N + Z_2^2$. Can you please explain what using the 4-square theorem adds? UPDATE: Oh, I think I get it now: instead of only adding squares $Z^2$, we can add any positive integer using the 4-square theorem, allowing N to range across the whole inequality. +1 $\endgroup$ – Lawrence Apr 5 '18 at 1:53

Pell's equation $$X^2-NY^2=1$$ has a solution in integers other than the trivial $(\pm1, 0)$ if and only if $N$ is not a square.

To express that $X^2 \neq 1$, one can add one of the equations from What is the simplest Diophantine equation equivalent to N is not zero?

that is, one of:

  • $Z=1+A^2+B^2+C^2+D^2$
  • $AZ=(2B+1)(3B+1)$
  • $AZ=7+B^2+BC+C^2$

where $Z=X^2-1$

Edit: as in Misha Lavrov's answer, this only characterizes positive non-squares.

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  • $\begingroup$ Ah that's clever. So for example: $(2+A^2+B^2+C^2+D^2)^2-NY^2=1$ $\endgroup$ – zooby Apr 4 '18 at 23:56

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