## Triangular & Second pentagonal numbers

Suppose the positive integers   $x, \; y$   satisfy

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

Or

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

$x \,(2 \,x + 1)$   is a Triangular number  [   $T_{2 \,n} = n \,(2 \,n + 1)$   ]

numbers of the form   $n \, (3 \, n + 1)/2$   are Second pentagonal numbers.

Equation   (1)   gives us the Pell equation:

$48 \, (x + 1/4)^2 \; - \; 72 \, (y + 1/6)^2 \; = \; 1$

The first few solutions are:

(x, y)   =   (0,0),   (22,18),   (2180,1780),   (213642,174438),   (20934760,17093160),   (2051392862,1674955258)

Note that

$x - y \; = \; 0, \; 2^2, \; 20^2, \; 198^2, \; 1960^2, \; 19402^2$

$2x + 2y + 1 \; = \; 1, \; 9^2, \; 89^2, \; 881^2, \; 8721^2, \; 86329^2$

$3x + 3y + 1 \; = \; 1, \; 11^2, \; 109^2, \; 1079^2, \; 10681^2, \; 105731^2$

Show that for all   $x, \; y$   that satisfy   $2 \, x^2 \; + \; x \; = \; 3 \, y^2 \; + \; y$

$x - y, \; 2x + 2y + 1, \; 3x + 3y + 1$   are all perfect squares.

Summarizing results: