## Equation : x^2 + y^2 – x = 5*x*y

Consider the equation      $x^2 \; + \; y^2 \; - x \; = \; 5 \, x \, y$

we get 3 families of solutions.
Here are the first few solutions:

(1)

x = 0,   y = 0

x = 576 =   $24^2$
y = 120,   y = 2760

x = 6969600 =   $2640^2$
y = 1454640,   y = 33393360

x = 84318221376 =   $290376^2$
y = 17598237480,   y = 403992869400

x = 1020081835238400 =   $31938720^2$
y = 212903475581280,   y = 4887505700610720

x = 12340949958395942976 =   $3512968824^2$
y = 2575706229984090840,   y = 59129043561995624040

(2)

x = 1
y = 0,   y = 5

x = 13225 =   $115^2$
y = 2760,   y = 63365

x = 159997201 =   $12649^2$
y = 33393360,   y = 766592645

x = 1935646125625 =   $1391275^2$
y = 403992869400,   y = 9274237758725

x = 23417446667815201 =   $153027601^2$
y = 4887505700610720,   y = 112199727638465285

x = 283304267851582177225 =   $16831644835^2$
y = 59129043561995624040,   y = 1357392295695915262085

(3)

x = 25 =   $5^2$
y = 5,   y = 120

x = 303601 =   $551^2$
y = 63365,   y = 1454640

x = 3672966025 =   $60605^2$
y = 766592645,   y = 17598237480

x = 44435542668001 =   $6665999^2$
y = 9274237758725,   y = 212903475581280

x = 537581191524511225 =   $733199285^2$
y = 112199727638465285,   y = 2575706229984090840

x = 6503657210627994133201 =   $80645255351^2$
y = 1357392295695915262085,   y = 31160893757444055403920