Time of Birth

We are looking for solutions to the equation $x^2 = y^2 + z^2$ where the woman was born in the year $y$ (between $0$ A.D. and $1997$), she lived $z$ years (at least $1$ year and not more than $120$ years) and she died in the year $x$ where $x < 1997$ . There are twenty possible solutions. If she was born in $0$ A.D. there are ten possible solutions. The remaining solutions are:

Born	Lived	Died
$y^2$	$z^2$	$x^2$
$9$	$16$	$25$
$16$	$9$	$25$
$36$	$64$	$100$
$64$	$36$	$100$
$144$	$25$	$169$
$144$	$81$	$225$
$225$	$64$	$289$
$576$	$49$	$625$
$576$	$100$	$676$
$1600$	$81$	$1681$

This can never happen in the future (taking $x^2 > 1997$). Even if the woman had a longer life there are still no solutions in the past for a lifespan of $121$ years, but there are four solutions for a lifespan of $144$ years. In the next $2000$ years (assuming a lifespan of no more than $400$ years) the only solutions are:

Born	Lived	Died
$y^2$	$z^2$	$x^2$
$2304$	$196$	$2500$
$2304$	$400$	$2704$
$3600$	$121$	$3721$
$3969$	$256$	$4225$

The sets of numbers $x$, $y$ and $z$ are Pythagorean triples.