Quadratic Diophantine Equation

Number Theory Level 3

x 2 + 2 y = x y x^2 + 2y = xy , here x , y x,y are the natural numbers.Find the sum of all the solution pairs ( x , y ) (x,y) .For example if ( 2 , 3 ) (2,3) and ( 4 , 5 ) (4,5) are two solution then enter your answer as 2 + 3 + 4 + 5 = 14 2+3+4+5=14 .


The answer is 39.

Akash Hossain by Akash Hossain , 18, Bangladesh

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1 solution

X X
May 31, 2018

y = x 2 x − 2 = x + 2 + 4 x − 2 y=\frac{x^2}{x-2}=x+2+\frac4{x-2} ,so x − 2 = 1 x-2=1 or 2 2 or 4 4 .So the solutions are ( 3 , 9 ) , ( 4 , 8 ) , ( 6 , 9 ) (3,9),(4,8),(6,9)

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