## Square triangular numbers | X^3 + Y^2 = Z^4

Explain the following result:

Each of the first few consecutive square triangular number yields a solution of

$X^3 \; + \; Y^2 \; = \; Z^4$

$( \,T_n \,)^2 \; - \; ( \,T_{n-1} \,)^2$
$= \; ( \,n \,(n+1)/2 \,)^2 \; - \; ( \,n \,(n-1)/2 \,)^2 \; = \; n^3$

$6^4 = 8^3 + 28^2$
$35^4 = 49^3 + 1176^2$
$204^4 = 288^3 + 41328^2$
$1189^4 = 1681^3 + 1412040^2$
$6930^4 = 9800^3 + 48015100^2$
$40391^4 = 57121^3 + 1631375760^2$
$235416^4 = 332928^3 + 55420360128^2$
$1372105^4 = 1940449^3 + 1882670190576^2$
$7997214^4 = 11309768^3 + 63955420452028^2$
$46611179^4 = 65918161^3 + 2172601941851880^2$
$271669860^4 = 384199200^3 + 73804512448220400^2$
$1583407981^4 = 2239277041^3 + 2507180832055219320^2$
$9228778026^4 = 13051463048^3 + 85170343840128993628^2$
$53789260175^4 = 76069501249^3 + 2893284510097771529376^2$