Ages that are sums of powers in a 19th-century life

An old man whose life spanned the $1800$s and $1900$s made three remarks, with $x,y,m,n$ positive integers: in the calendar year $x^4+y^4$ his age was $x^2+y^2$; in the year $2m^2$ his age was $2m$; and in the year $3n^4$ his age was $3n$. Find $xymn$.

$4650$. The only sum of two fourth powers in range is $5^4+6^4=1921$, when he was $61$, so he was born in $1860$. Then $1860+2m=2m^2$ gives $m=31$, and $1860+3n=3n^4$ gives $n=5$. So $xymn=5\cdot6\cdot31\cdot5=4650$.