1.Introduction

I obtained a parametric solution of x^4 + ax^2y^2 + y^4 = u^4 + au^2v^2 + v^4.

However, Choudhry already gave a parametric solution of px^4 + qx^2y^2 + py^4 = pu^4 + qu^2v^2 + pv^4.(Choudhry[1])

Both side, dividing by p and set a=q/p, leads to x^4 + ax^2y^2 + y^4 = u^4 + au^2v^2 + v^4.

Anyway, I show a parametric solution of x^4 + ax^2y^2 + y^4 = u^4 + au^2v^2 + v^4.


2.Theorem
        
     

    x^4 + ax^2y^2 + y^4 = u^4 + au^2v^2 + v^4 has a follwing parametric solution.


    x =8m^7+2a^2nm^6+(8n^2+a^3n^2+2n^2a^2)m^5+(16n^3a-a^3n^3)m^4+(16an^4+2a^2n^4-16n^4)m^3+24n^5m^2+(2a^2n^6+8n^6-4n^6a)m+4n^7a
    
    y =-4am^7+(8n-4an+2a^2n)m^6-24n^2m^5+(-16n^3+16n^3a+2a^2n^3)m^4+(a^3n^4-16an^4)m^3+(2a^2n^5+8n^5+a^3n^5)m^2-2a^2n^6m+8n^7
    
    u =4am^7+(8n-4an+2a^2n)m^6+24n^2m^5+(-16n^3+16n^3a+2a^2n^3)m^4+(-a^3n^4+16an^4)m^3+(2a^2n^5+8n^5+a^3n^5)m^2+2a^2n^6m+8n^7
    
    v =-8m^7+2a^2nm^6+(-8n^2-a^3n^2-2n^2a^2)m^5+(16n^3a-a^3n^3)m^4+(-2a^2n^4-16an^4+16n^4)m^3+24n^5m^2+(-8n^6+4n^6a-2a^2n^6)m+4n^7a
    


 
Proof.

x^4 + ax^2y^2 + y^4 = u^4 + au^2v^2 + v^4..................................(1)

x=pt+m, y=qt+n, u=pt+n, v=qt-m.............................................(2)

Substitute (2) to (1), and simplifying (1), we obtain

(-4np^3-2anpq^2+2ap^2mq+4mp^3+2ampq^2+2ap^2nq+4nq^3+4mq^3)t^2
+(-6n^2p^2-an^2q^2+8ampnq-ap^2m^2-6m^2q^2+6m^2p^2+am^2q^2+ap^2n^2+6n^2q^2)t
+(4m^3p-4n^3p+2an^2mq-2anpm^2+4m^3q+2am^2nq+2ampn^2+4n^3q)

Decide {p,q} to {4m^3p-4n^3p+2an^2mq-2anpm^2+4m^3q+2am^2nq+2ampn^2+4n^3q=0}, then

We obtain p=2m^3+am^2n+an^2m+2n^3 and  q=-2m^3+2n^3-an^2m+am^2n, then

t = -(m^2n^2a^2-2am^4-2an^4-12m^2n^2)
  /(a^3m^4n^2+a^3n^4m^2+2a^2m^4n^2+2a^2n^4m^2-4am^6-4n^6a+8m^6-16m^4n^2-16n^4m^2+8n^6)
  
Substitute p, q and t to (2), and obtain a parametric solution.            


   
Q.E.D.　
 
　　                  
       
3.Example



Case a=1: x^4 + x^2y^2 + y^4 = u^4 + u^2v^2 + v^4

x=  8m^7+2nm^6+11n^2m^5+15n^3m^4+2n^4m^3+24n^5m^2+6n^6m+4n^7

y= -4m^7+6nm^6-24n^2m^5+2n^3m^4-15n^4m^3+11n^5m^2-2n^6m+8n^7

u=  4m^7+6nm^6+24n^2m^5+2n^3m^4+15n^4m^3+11n^5m^2+2n^6m+8n^7

v= -8m^7+2nm^6-11n^2m^5+15n^3m^4-2n^4m^3+24n^5m^2-6n^6m+4n^7

Especially, (m,n)=(3,1): 11567^4 + 11567^2*5174^2 + 5174^4 = 9817^4 + 9817^2*8674^2 + 8674^4


Case a=3: x^4 + 3x^2y^2 + y^4 = u^4 + 3u^2v^2 + v^4

x=  8m^7+18nm^6+53n^2m^5+21n^3m^4+50n^4m^3+24n^5m^2+14n^6m+12n^7

y= -12m^7+14nm^6-24n^2m^5+50n^3m^4-21n^4m^3+53n^5m^2-18n^6m+8n^7

u=  12m^7+14nm^6+24n^2m^5+50n^3m^4+21n^4m^3+53n^5m^2+18n^6m+8n^7

v= -8m^7+18nm^6-53n^2m^5+21n^3m^4-50n^4m^3+24n^5m^2-14n^6m+12n^7

 
4.References


[1].Ajai Choudhry: Parametric Solutions of Quartic Diophantine Equation f(x,y)=f(u,v), Rocky Mountain Journal of Mathematics, vol.35, 2005.