non linear diphantine equation.

[afif mansib ch]

proof that the system of simultaneous equations
\[x^2+y^2=z^2+1\]
and
\[x^2-y^2=W^2+1\]
has infinitely many sols in positive integers x,y,z,w.

[afif mansib ch]

i thnk i've slvd it bt i'm confused.plz check if it's correct.
substracting the 2nd eq from the 1st one we get,
\[2y^2=z^2-w^2\]
so for cnstnt u,v
\[y=2uv\]
\[z=2u^2+v^2\]
\[w=v^2-2u^2\]
now frm eq 1
\[x^2=4u^4+v^4+1\]
now substituting the magnitudes on l.h.s. of eq 2 we get,
\[(v^2-2u^2)^2+1\]
which's equal 2 the r.h.s. of eq 2
....so is the sol o.k.?