1.Introduction


We show a parametric solution of a4x^4+a3x^3y+a2x^2y^2+a1xy^3+a0y^4 = a4u^4+a3u^3v+a2u^2v^2+a1uv^3+a0v^4.

Oliver Couto[2] showed parametric solutions of similar equations for degree 2,3 and 5.

2.Theorem
        
     

    a4x^4+a3x^3y+a2x^2y^2+a1xy^3+a0y^4 = a4u^4+a3u^3v+a2u^2v^2+a1uv^3+a0v^4 has a follwing parametric solution.


    x = (2a2^3+27a3a1^2+72a2^2a0+27a0a3^2-27a0a1^2-9a2a3a1-72a2a0a3
        -72a2a0a1+72a2a0^2+27a1^3-27a2a1^2)m^9
        +(162a0a3a1+216a0^3+27a0a3^2+9a2^2a1+81a0a1^2-27a2a1^2-216a0^2a3
        -108a2a0a3-216a2a0a1+324a2a0^2-a2^3-324a0^2a1+90a2^2a0+27a1^3)n^4m^5
        +(162a0a3a1+216a0^3+27a0a3^2+9a2^2a1+81a0a1^2-27a2a1^2-216a0^2a3
        -108a2a0a3-216a2a0a1+324a2a0^2-a2^3-324a0^2a1+90a2^2a0+27a1^3)n^8m
      
    y = (27a2a1^2+27a0a1^2-72a2^2a0-27a0a3^2-72a2a0^2+72a2a0a3+9a2a3a1
        -27a1^3-2a2^3+72a2a0a1-27a3a1^2)m^9
        +(-432a0^2a1+9a2^2a3-216a2a0a3+378a0a3a1-324a2a0a1+243a0a1^2
        -324a0^2a3+27a2a1^2+90a2^2a0+324a2a0^2-27a1^3+54a2a3a1-17a2^3+81a0a3^2
        -81a3a1^2-27a3^2a1+216a0^3+18a2^2a1)n^4m^5
        +(-432a0^2a1+9a2^2a3-216a2a0a3+378a0a3a1-324a2a0a1+243a0a1^2
        -324a0^2a3+27a2a1^2+90a2^2a0+324a2a0^2-27a1^3+54a2a3a1-17a2^3+81a0a3^2
        -81a3a1^2-27a3^2a1+216a0^3+18a2^2a1)n^8m
    
    u = (162a0a3a1+216a0^3+27a0a3^2+9a2^2a1+81a0a1^2-27a2a1^2-216a0^2a3
        -108a2a0a3-216a2a0a1+324a2a0^2-a2^3-324a0^2a1+90a2^2a0+27a1^3)m^8n
        +(162a0a3a1+216a0^3+27a0a3^2+9a2^2a1+81a0a1^2-27a2a1^2-216a0^2a3
        -108a2a0a3-216a2a0a1+324a2a0^2-a2^3-324a0^2a1+90a2^2a0+27a1^3)n^5m^4
        +(2a2^3+27a3a1^2+72a2^2a0+27a0a3^2-27a0a1^2-9a2a3a1-72a2a0a3-72a2a0a1
        +72a2a0^2+27a1^3-27a2a1^2)n^9
      
    v = (-432a0^2a1+9a2^2a3-216a2a0a3+378a0a3a1-324a2a0a1+243a0a1^2
        -324a0^2a3+27a2a1^2+90a2^2a0+324a2a0^2-27a1^3+54a2a3a1-17a2^3+81a0a3^2
        -81a3a1^2-27a3^2a1+216a0^3+18a2^2a1)m^8n
        +(-432a0^2a1+9a2^2a3-216a2a0a3+378a0a3a1-324a2a0a1+243a0a1^2
        -324a0^2a3+27a2a1^2+90a2^2a0+324a2a0^2-27a1^3+54a2a3a1-17a2^3+81a0a3^2
        -81a3a1^2-27a3^2a1+216a0^3+18a2^2a1)n^5m^4
        +(27a2a1^2+27a0a1^2-72a2^2a0-27a0a3^2-72a2a0^2+72a2a0a3+9a2a3a1
        -27a1^3-2a2^3+72a2a0a1-27a3a1^2)n^9
    
    condition: a4+a2+a0=a3+a1
               m and n are arbitrary.

 
Proof.

a4x^4 + a3x^3y + a2x^2y^2 + a1xy^3 + a0y^4 = a4u^4 + a3u^3v + a2u^2v^2 + a1uv^3 + a0v^4.................(1)

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

Substitute (2) to (1) and using a4 = a1+a3-a2-a0, and simplifying (1), we obtain

(a3p^4-2a3+a1p^4+a0q^4-2a1+a3p^3q+a1pq^3+a2p^2q^2-a0p^4-a2p^4)t^4
+(4a1p^3m-4a0mq^3+8a0n-2a2p^2mq+4a2n-2a1n+3a3p^2mq-4a0p^3m-6a3n-3a1pmq^2+3a3p^3m-4a2p^3m+2a2pmq^2+a1mq^3)t^3
+(-4a2pm^2q+6a1p^2m^2+3a3p^2m^2+3a1pm^2q-6a3n^2+3a3pm^2q-3a1m^2q^2-6a1n^2+8a2n^2+a2m^2q^2-5a2p^2m^2-6a0p^2m^2+6a0m^2q^2)t^2
+(-2a2pm^3+3a1m^3q-2a2m^3q-4a0pm^3+a3m^3q+4a2n^3-4a0m^3q+a3pm^3-2a3n^3-6a1n^3+3a1pm^3+8a0n^3)t..........(3)

Equating to zero the coefficient of t and t^2 in (3),  then we obtain

p = 1/3(6a1n^4+3a1m^4+3a3m^4-2a2n^4-12a0n^4-4a2m^4)/(m^3n(-2a2-4a0+a3+3a1))

q = -1/3(-12a1n^4+3a1m^4+3a3m^4+10a2n^4+12a0n^4-4a2m^4-6n^4a3)/(m^3n(-2a2-4a0+a3+3a1))

Finally, we obtain t as follows,

t =  -(coefficient of t^3) / (coefficient of t^4).
  
Substitute p, q, and t to (2), and obtain a parametric solution.            


   
Q.E.D.　
 
　　                  
       
3.Examples


Case 1: (a4,a3,a2,a1,a0)=(1,2,1,1,1)
        x^4 + 2x^3y + x^2y^2 + xy^3 + y^4 = u^4 + 2u^3v + u^2v^2 + uv^3 + v^4

x= (-37n^8 - 37m^4n^4 + 47m^8)m
y= -(26n^8 + 26m^4n^4 + 47m^8)m
u= (47n^8 - 37m^4n^4 - 37m^8)n
v= -(47n^8 + 26m^4n^4 + 26m^8)n


Case 2: (a4,a3,a2,a1,a0)=(1,7,7,7,6)
        x^4 + 7x^3y + 7x^2y^2 + 7xy^3 + 6y^4 = u^4 + 7u^3v + 7u^2v^2 + 7uv^3 + 6v^4

x= (1388n^4m^4 + 959m^8 + 1388n^8)m
y= -(803n^4m^4 + 959m^8 + 803n^8)m
u= (1388n^4m^4 + 1388m^8 + 959n^8)n
v= -(803n^4m^4 + 803m^8 + 959n^8)n

 




4.Reference


[1]. Ajai Choudhry: On the Quartic Diophantine Equationf(x, y)=f(u, v), Journal of Number Theory 75,(1999)
[2]. Oliver Couto:http://www.celebrating-mathematics.com/