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

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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