Problems need HardWork 24

Number Theory Level 3

For prime x x , let ( x 1 , y 1 ) , ( x 2 , y 2 ) , … , ( x n , y n ) (x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n) be all the integer solutions of ( x , y ) (x,y) such that they satisfy the equation x − y 4 = 4 x-y^4=4 . Find the value of ∑ j = 1 n ( x j + y j ) \displaystyle \sum_{j=1}^n (x_j+ y_j) .


The answer is 10.

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Dev Sharma by Dev Sharma , 20, India

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1 solution

x = y 4 + 4 x = y^{4} + 4 = y 4 + 4 y 2 + 4 − 4 y 2 y^{4} + 4y^{2} + 4 - 4y^{2} = ( y 2 + 2 ) 2 − 4 y 2 (y^{2} + 2)^{2} - 4y^{2} = ( y 2 + 2 + 2 y ) × ( y 2 + 2 − 2 y ) (y^{2} + 2 + 2y) \times (y^{2} + 2 - 2y)

x is a prime, so either ( y 2 + 2 + 2 y ) = 1 (y^{2} + 2 + 2y) = 1 or ( y 2 + 2 − 2 y ) = 1 (y^{2} + 2 - 2y) = 1

Solving for both equations gives two solutions (5,1) and (5,-1)

The question wants the sum, which is 5 + 1 + 5 - 1 = 10 \boxed{10}

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