EMAT 6600

Problem Solving


 

 


An algebraic Exploration

 

Problem: Prove that x2 y2 = A3 always has integer solutions (x,y) whenever A is a positive integer.

 

Solution

x2 y2 = A3

(x + y)(x y) = A3 = A2 * A

now if we consider either of the following,

x + y = A2 and x y = A

we then have

x = A2 y and x = A + y

A2 y = A + y

A(A 1) = 2y

hence, y = A(A 1)/2.

or

x + y = A and x y = A2

similarly, we have

x = A y and x = A2 + y

A y = A2 + y

A A2 = 2y

A(1 A)/2 = y

Now, equate the values of y

A(1 A) = A(A 1)

1 A = A 1

2 = 2A 1 = A

thus, A is a positive integer.

 


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