1.Introduction

H. W. Richmond[1] showed equation ax^4+ by^4+ cz^4+ dw^4= 0 has infinitely many integer solutions if abcd is a square number.
Bremner, Choudhry, and Ulas[2] showed quation ax^4+ by^4+ cz^4+ dw^4= 0 has infinitely many integer solutions even if abcd is not a square number.
We show similar result even if abcd is not a square number.
We treat equation a^2X^8 + 2ab^4Y^4 + Z^8 = (a+b^4)^2W^4 as an example.
                 
                 
2.Theorem
     
Diophantine equation a^2X^8 + 2ab^4Y^4 + Z^8 = (a+b^4)^2W^4 has infinitely many integer solutions.

a,b are arbitrary.
 
Proof.

a^2X^8 + 2ab^4Y^4 + Z^8 = (a+b^4)^2W^4..................................(1)

We consider an identity: a^2T^8 + 2aT^4b^4 + b^8 = (aT^4+b^4)^2.........(2)

Let aT^4+b^4=mu^2.......................................................(3)

Let m=a+b^4 and v=mu, the equation (3) becomes to below equation.

v^2 = (a+b^4)aT^4+(a+b^4)b^4............................................(4)

Since quartic equation (4) has a rational point Q(T,v)=(1,a+b^4), then this quartic equation is birationally equivalent to an elliptic curve below.

Y^2+4aYX+(8a^3+16a^2b^4+8ab^8)Y = X^3+(2a^2+6ab^4)X^2+(-4a^4-12a^3b^4-12a^2b^8-4ab^12)X-8a^6-48a^5b^4-96a^4b^8-80a^3b^12-24a^2b^16

Transformation is given, 
U = (4a^3+16a^2b^4+2aX+12ab^8+Y+2b^4X)/Y
V = (4b^16Xa+8b^12Ya+24b^20a^2+104b^16a^3+176b^12a^4+56b^4a^6+144b^8a^5+8a^7+X^3b^4+72a^4b^4X+6a^3X^2+12Xa^5+aX^3-8a^3Yb^4+144a^3b^8X+88a^2b^12X+24a^2X^2b^4+18aX^2b^8)/(Y^2)
X = (2aV-2a^2+2b^4V+2b^8+4a^2*U+4ab^4*U)/(U^2-2*U+1)
Y = (4a^3*U^2-16a^2b^4*U+12a^2b^4+16a^2b^4*U^2+4a^2V-16b^8a*U+8ab^4V+12b^8a*U^2+16ab^8+4b^12+4b^8V)/(U^3-3*U^2+3*U-1).

The point corresponding to point Q is P(X,Y)=(-2a^2-6ab^4, 8a^2b^4-8ab^8).
Hence we get 2Q(T,v)=(-(a^2-6ab^4-3b^8)/(3a^2+6ab^4-b^8), (a+b^4)(a^4+28a^3b^4+6a^2b^8+28ab^12+b^16)/((3a^2+6ab^4-b^8)^2)).

This point P is of infinite order, and the multiples mP, m = 2, 3, ...give infinitely many points.

Hence we can obtain infinitely many integer solutions for equation (1).

Case : m=2

X = -a^2+6ab^4+3b^8
Y = (a^2-6ab^4-3b^8)(3a^2+6ab^4-b^8)
Z = b(3a^2+6ab^4-b^8)
W = b^16+28ab^12+6a^2b^8+28a^3b^4+a^4

a,b are arbitrary.

Case : m=3

X = (a^2+2ab^4+5b^8)(a^4-52a^3b^4-26a^2b^8+12ab^12+b^16)
Y = (a^2+2ab^4+5b^8)(a^4-52a^3b^4-26a^2b^8+12ab^12+b^16)(5a^2+2ab^4+b^8)(a^4+12a^3b^4-26a^2b^8-52ab^12+b^16)
Z = b*(5a^2+2ab^4+b^8)(a^4+12a^3b^4-26a^2b^8-52ab^12+b^16)
W = (-4a^3b^4+70a^2b^8-4ab^12+a^4+b^16)*(a^8+216b^4a^7+860b^8a^6+744b^12a^5+454b^16a^4+744b^20a^3+860b^24a^2+216b^28a+b^32)
       
Q.E.D.


3.References

[1]. H. W. Richmond, On the Diophantine equation F ≡ ax^4+ by^4+ cz^4+ dw^4= 0, the product
abcd being a square number, J. London Math. Soc. 19, (1944)
[2]. ANDREW BREMNER, AJAI CHOUDHRY, AND MACIEJ ULAS, Constructions of diagonal quartic and sextic surfaces 
with infinitely many rational points, arXiv:1402.4583, (2014)