Possible sum

Number Theory Level pending

x 2 + y 2 = 1997 ( x − y ) x^2 + y^2 = 1997(x-y)

What is the sum of all integer solutions to the equation above?

Note: If there are n n integer-pair solutions ( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) , ⋯ ( x n , y n ) (x_1, y_1), (x_2, y_2), (x_3, y_3), \cdots (x_n, y_n) , give your answer as ∑ k = 1 n ( x k + y k ) \displaystyle \sum_{k=1}^n (x_k+y_k) .


The answer is 0.

Sahar Bano by Sahar Bano , 38, India

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1 solution

Sahar Bano
Mar 18, 2020

Let (a,b) an integer solution of the equation

Then (-b,-a) will be another integer solution for this equation

Therefore if we sum all solutions we will get 0

Hence the answer is 0

0 Helpful 0 Interesting 1 Brilliant 0 Confused

Not necessarily another solution. If ( a , b ) = ( 0 , 0 ) (a,b) = (0,0) or ( 1997 , − 1997 ) (1997,-1997) then ( − b , − a ) (-b,-a) is the same solution. The solutions consist of ( 0 , 0 ) (0,0) , ( 1997 , − 1997 ) (1997,-1997) and a collection of ( a , b ) , ( − b , − a ) (a,b),(-b,-a) pairs.

The two roots ( 0 , 0 ) (0,0) and ( 1997 , − 1997 ) (1997,-1997) have x + y x+y values equal to 0 0 , and the ( a , b ) , ( − b − a ) (a,b),(-b-a) pairs have combined x + y x+y sums of 0 0 , so the result is still true.

Mark Hennings - 1 year, 2 months ago

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