# Integer solutions of Quaterni0n norm

I would like to know all integer solutions of the Diophantine equation $x^2-ay^2-bz^2+abw^2=1$ where $a,b$ are fixed positive integers. If you know the answer, I appreciate your reply. (To be more specific, $a,b$ are such that $x^2-ay^2-bz^2+abw^2=0$ has no solution)

• Decisions can be recorded through the solution of the equivalent equation Pell. The formula is somewhat cumbersome. You look like you need? – individ Mar 27 '17 at 4:20

$$x^2-ay^2-bz^2+abw^2=1$$
To make the change. $y=kS$ ; $z=tS$ ; $w=pS$
$$x^2-(ak^2+bt^2-abp^2)S^2=1$$