## Equation: x^2 + (x+1)(x+2) = y^2

$x^2 \; + \; (x+1) \,(x+2) \; = \; y^2$

$x, y$   are positive integers.

Here are the first few solutions:

$2^2 \; + \; (3\times 4) \; = \; 4^2$

$7^2 \; + \; (8\times 9) \; = \; 11^2$

$94^2 \; + \; (95\times 96) \; = \; 134^2$

$263^2 \; + \; (264\times 265) \; = \; 373^2$

$3218^2 \; + \; (3219\times 3218) \; = \; 4552^2$

$8959^2 \; + \; (8960\times 8961) \; = \; 12671^2$

$109342^2 \; + \; (109343\times 109344) \; = \; 154634^2$

$304367^2 \; + \; (304368\times 304369) \; = \; 430441^2$

$3714434^2 \; + \; (3714435\times 3714436) \; = \; 5253004^2$

$10339543^2 \; + \; (10339544\times 10339545) \; = \; 14622323^2$

$126181438^2 \; + \; (126181439\times 126181440) \; = \; 178447502^2$

$351240119^2 \; + \; (351240120\times 351240121) \; = \; 496728541^2$

$4286454482^2 \; + \; (4286454483\times 4286454484)$
$= \; 6061962064^2$

$11931824527^2 \; + \; (11931824528\times 11931824529)$
$= \; 16874148071^2$

$145613270974^2 \; + \; (145613270975\times 145613270976)$
$= \; 205928262674^2$

$405330793823^2 \; + \; (405330793824\times 405330793825)$
$= \; 573224305873^2$

$4946564758658^2 \; + \; (4946564758659\times 4946564758660)$
$= \; 6995498968852^2$

$13769315165479^2 \; + \; (13769315165480\times 13769315165481)$
$= \; 19472752251611^2$

$168037588523422^2 \; + \; (168037588523423\times 168037588523424)$
$= \; 237641036678294^2$

$467751384832487^2 \; + \; (467751384832488\times 467751384832489)$
$= \; 661500352248901^2$

$5708331445037714^2 \; + \; (5708331445037715\times 5708331445037716)$
$= \; 8072799748093144^2$