How would one determine solutions to the following quadratic Diophantine equation in three variables:

$$x^2 + n^2y^2 \pm n^2y = z^2$$

where n is a known integer and $x$, $y$, and $z$ are unknown positive integers to be solved. Ideally there would be a parametric solution for $x$, $y$, and $z$.

[Note that the expression $y^2 + y$ must be an integer from the series {2, 6, 12, 20, 30, 42 ...} and so can be written as either $y^2 + y$ or $y^2 - y$ (e.g., 12 = $3^2 + 3$ and 12 = $4^2 - 4$). So I have written this as +/- in the equation above.]



Considering this as a quadratic in $y$, it is sufficient (and necessary) that

$n^4 - 4n^2(x^2-z^2)$ is a perfect square.


$n^2 - 4(x^2 - z^2)$ is a perfect square, say $q^2$.

Rewriting gives us

$(n-q)(n+q) = 4 (x^2 - z^2)$

If $n=2k$ is even, then $q$ needs to be even too (say $2m$) and

$(k-m)(k+m) = (x-z)(x+z)$

If $n=2k+1$ is odd, then $q$ needs to be odd too (say $2m+1$) and

$(k-m)(k+m+1) = (x-z)(x+z)$

Thus you can pick any $m$, and try to factor the left hand side above (choosing the right one depending on the parity of $n$) into two terms of the same parity.

  • $\begingroup$ Not sure of a parametric solution. It sort of depends on $n$. $\endgroup$
    – Aryabhata
    Mar 7 '12 at 15:08

We will consider the more general equation:


Then, if we use the solutions of Pell's equation: $p^2-(q+1)s^2=1$

Solutions can be written in this ideal:




And more:




$L$ - integer and given us.


For the case when the equation: $X^2+qY^2+qY=Z^2$

factor $q$ - is not a square, then solutions can yrazit using Pell's equation: $p^2-qs^2=1$

Then there is another solution:





This is a good problem as it requires to write the (Diophantine) parametric representations of the variables.

This method can be extended to n variables; one way of stating this result is to say there is an algorithm to construct a n variable quadratic system and if our equation fits this construction then the solutions are read off trivial.

I will explain this later.

  • 1
    $\begingroup$ So now is later... $\endgroup$
    – Thomas
    Sep 15 '12 at 21:34
  • $\begingroup$ Is it "later" yet? :) $\endgroup$
    – Ricket
    Sep 28 '14 at 3:53

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