1.Introduction

F. IZADI AND K. NABARDI showed the integer solutions of X^4 + Y^4 = 2(U^4 + V^4).

We show the condition for the integer solution of px^4 + py^4 = 2pu^4 + qv^4.

2. Theorem

There is no integer solution of px^4 + py^4 = 2pu^4 + qv^4 if the elliptic curve E(Q) has rank 0.

p,q are arbitrary.

Proof.

px^4 + py^4 = 2pu^4 + qv^4......................................(1)

Let x=u+t, y=u-t then we obtain the quartic curve below.

U=v/t, V=6pu.
V^2 = 3pqU^4-6p^2...............................................(2)

Quartic curve (2) is transformed to the elliptic curve E(Q).

X=3pqU^2, Y=3pqUV.
E(Q): Y^2 = X^3-18p^3qX.........................................(3)

If rank of elliptic curve (3) is 0, then equation (1) has no integer solution.

Q.E.D.


3.Examples

1. (p,q)=(1,2): x^4 + y^4 = 2u^4 + 2v^4 ( IZADI's case )
   rank 1:
   [x,y,u,v]=[21, 19, 20, 7]

2. (p,q)=(1,3): x^4 + y^4 = 2u^4 + 3v^4
   rank 1:
   [x,y,u,v]=[2209, 1391, 409, 1740]

3. (p,q)=(11,15): 11x^4 + 11y^4 = 22u^4 + 15v^4
   rank 1:
   [x,y,u,v]=[8409293, 890729, 4650011, 7389690]

4. (p,q)=(21,31): 21x^4 + 21y^4 = 42u^4 + 31v^4
   rank 1:
   [x,y,u,v]=[251574629, 110897563, 70338533, 229677840]

5. (p,q)=(31,27): 31x^4 + 31y^4 = 62u^4 + 27v^4
   rank 1:
   [x,y,u,v]=[7069099, 3601889, 1733605, 7425182]



4.Reference

[1] F. IZADI AND K. NABARDI, DIOPHANTINE EQUATION X^4 + Y^4 = 2(U^4 + V^4 ), https://arxiv.org/abs/1501.05803