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Hi everybody. I'd like to know if the diophantine equation

(1) $$X^2 - Y^2 - Z^2 = \pm 1$$

has been studied and if the set of its solutions $(X,Y,Z)$ is known. I appreciate any reference. Thank you very much.

P.S. If instead we look at the diophantine equation

$$X^2 - Y^2 - Z^4 = \pm 1$$

surely we can solve it imposing conditions on the solution of (1) so that Z be a square. However, is there a quicker method?

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    $\begingroup$ This is an old question, but in case someone else comes across it, we have the identity $$(289p^4+14p^2q^2-239q^4)^2-(17p^2-12pq-13q^2)^4 -(17p^2+12pq-13q^2)^4 = -1$$ where $p,q$ solve the Pell equation, $$q^2-17p^2=-1$$ So we can actually make both the $Y,Z$ of your first eqn as squares. $\endgroup$ Commented Jan 3, 2015 at 6:37

2 Answers 2

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In general, if $Q(x,y,z)$ is a nonsingular ternary quadratic form with integral coefficients, then the integral solutions of $Q(x,y,z)=n$ fall into finitely many orbits of the integral automorphism group of $Q$, and these orbits can be effectively determined. For your case there is a shortcut as follows.

Write $a=x+y$, $c=x-y$, $b=z$, then your equation becomes $b^2-ac=\mp 1$. In the language of binary quadratic forms, this means that the form $au^2+2buv+cv^2$ has discriminant $\mp 4$. From classical theory it follows that after an invertible linear change of variables $u'=pu+qv$, $v'=ru+sv$ the form becomes $u'^2\pm v'^2$, i.e. there are $p,q,r,s\in\mathbb{Z}$ such that $ps-qr=1$ and $$ au^2+2buv+cv^2=(pu+qv)^2\pm(ru+sv)^2. $$ Equating coefficients, $$ a=p^2\pm r^2,\quad c=q^2\pm s^2,\quad b=pq\pm rs, $$ i.e. $$ x=(p^2\pm r^2+q^2\pm s^2)/2,\quad y=(p^2\pm r^2-q^2\mp s^2)/2,\quad z=pq\pm rs. $$ To summarize, all integer solutions of $x^2-y^2-z^2=\pm 1$ are of this form, with integers $p,q,r,s$ satisfying $ps-qr=1$. Using congruences one can easily distinguish integer solutions from half-integer ones.

As references, I suggest Rose: A course in number theory, and Cassels: Rational quadratic forms.

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  • $\begingroup$ @GH Thank you! Your answer is very helpful. I have added a P.S. to my answer. $\endgroup$ Commented Nov 28, 2012 at 9:02
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It is also true that the automorphism group of the quadratic form is known. See Is there a topograph for Pythagorean triples? and the three matrices. If you have any particular $x^2 + y^2 - z^2 = n,$ write $(x,y,z)$ as a column vector. Multiply by any of the three square matrices or its inverse and you get another solution for $n.$ Multiply again you get another, and so on for any combination of group elements.

I see, for your ordering you need to switch first and last elements to use these three matrices.

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