## Equation : x^2 + y^2 – x = 4*x*y

The first few solutions of the equation   $x^2 \; + \; y^2 \; - \; x \; = \; 4 \, x \, y$   are:

x =   $1$   …………………… y = 0   or   y = 4
x = 16 =   $4^2$   …………….. y = 4   or   y = 60
x = 225 =   $15^2$   …………… y = 60   or   y = 840
x = 3136 =   $56^2$   ………….. y = 840   or   y = 11704
x = 43681 =   $209^2$   ………… y = 11704   or   y = 163020
x = 608400 =   $780^2$   ……….. y = 163020   or   y = 2270580
x = 8473921 =   $2911^2$   ……… y = 2270580   or   y = 31625104
x = 118026496 =   $10864^2$   …… y = 31625104   or   y = 440480880
x = 1643897025 =   $40545^2$   ….. y = 440480880   or   y = 6135107220

$3(1^2) \; + \; 1 \; = \; 2^2 \; = \; ( \,2(1) - 0 \,)^2$
$3(4^2) \; + \; 1 \; = \; 7^2 \; = \; ( \,2(4) - 1 \,)^2$
$3(15^2) \; + \; 1 \; = \; 26^2 \; = \; ( \,2(15) - 4 \,)^2$
$3(56^2) \; + \; 1 \; = \; 97^2 \; = \; ( \,2(56) - 15 \,)^2$
$3(209^2) \; + \; 1 \; = \; 362^2 \; = \; ( \,2(209) - 56 \,)^2$
$3(780^2) \; + \; 1 \; = \; 1351^2 \; = \; ( \,2(780) - 209 \,)^2$
$3(2911^2) \; + \; 1 \; = \; 5042^2 \; = \; ( \,2(2911) - 780 \,)^2$
$3(10864^2) \; + \; 1 \; = \; 18817^2 \; = \; ( \,2(10864) - 2911 \,)^2$
$3(40545^2) \; + \; 1 \; = \; 70226^2 \; = \; ( \,2(40545) - 10864 \,)^2$