I would like to find some source books or articles which discuss the Diophantine equation $$ x^2+y^2=z^2+t^2,\qquad |y-z|=1 $$ for which $x,z$ are odd positive and $y,t$ are even positive integers. Any brief explanation is also welcome and appreciated.
Thanks!
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What to do is just substitute your conditions on your equations. Since $x,z$ are odd and $y,t$ are even and $|y-z|=1$, then,
$$(2m-1)^2+(2u)^2 = (2u-1)^2+(2n)^2\tag1$$
Expanding, you can see that $2$nd powers of $u$ will cancel, so you'll only have a linear equation in $u$. Letting Walpha do the work, we get the condition,
$$u = -m^2+m+n^2$$
which makes $(1)$ true for any $m,n$.
Is not so hard. $x^2-t^2=z^2-y^2 $, so $(x-t)*(x+t)=(z-y)*(z+y)$. Now suppose that$ z=y+1$.So $ (x-t)*(x+t)=y+z=2y+1$. Now it's so easy, you must to keep account by initial conditions.