## Positive integers (x,y) such that (x+1)^3 – x^3 = y^2

x   and   y   are positive integers that satisfy the equation

$(x+1)^3 \; - \; x^3 \; = \; y^2$

Here are the first few solutions:

x =   7,   104,   1455,   20272,   282359,   3932760,   54776287,   762935264,   10626317415

$(x+1)^3 \; - \; x^3 \; = \; y^2$ ……………………………………….   $y \; = \; a^2 \; + \; (a+1)^2$

$8^3 - 7^3 = 13^2$ ……………………………………………………. $13 = 2^2 + 3^2$
$105^3 - 104^3 = 181^2$ ……………………………………………. $181 = 9^2 + 10^2$
$1456^3 - 1455^3 = 2521^2$ ………………………………………. $2521 = 35^2 + 36^2$
$20273^3 - 20272^3 = 35113^2$ …………………………………. $35113 = 132^2 + 133^2$
$282360^3 - 282359^3 = 489061^2$ ……………………………. $489061 = 494^2 + 495^2$
$3932761^3 - 3932760^3 = 6811741^2$ ………………………. $6811741 = 1845^2 + 1846^2$
$54776288^3 - 54776287^3 = 94875313^2$ ………………….. $94875313 = 6887^2 + 6888^2$
$762935265^3 - 762935264^3 = 1321442641^2$ …………… $1321442641 = 25704^2 + 25705^2$
$10626317416^3 - 10626317415^3 = 18405321661^2$ ….. $18405321661 = 95930^2 + 95931^2$

Prove that y is always expressible as the sum of squares of two consecutive positive integers.