1.Introduction

We show a parametric solution for { A+x^2=u^2, A-x^2=v^2 }

2.Theorem
 
There is a parametric solution for simultaneous equation { A+x^2=u^2, A-x^2=v^2 },

      A = (5k^2-4k+1)(1+k^2)
      
      x = 2k(k-1)

      u = 3k^2-2k+1

      v = k^2+2k-1
    
      k: arbitrary


     
Proof.

a+x^2=u^2......................................................(1)

a-x^2=v^2......................................................(2)

From above equation, we obtain u^2-v^2-2x^2=0..................(3).
Since equation (3) has a rational solution (u,v,x)=(3,1,2), we can obtain a parametric solution below.

(x,u,v)=( 2k(k-1)/(1+k^2), (1+3k^2-2k)/(1+k^2), -(-1+k^2+2k)/(1+k^2) )
k is arbitrary.

Then a=u^2-x^2=(5k^2-4k+1)/(1+k^2).

Finally we obtain a parametric solution.

(A,x,u,v)=( (5k^2-4k+1)(1+k^2), 2k(k-1), 3k^2-2k+1, k^2+2k-1 )
 
Q.E.D.