1.Introduction

Susil Kumar Jena[1] showed that x1^4 + 4X2^4 = X3^8 + 4X4^8 has infinitely many integral solutions.

We show that x1^4 + 2nX2^4 = X3^8 + n^2X4^8 has infinitely many integral solutions. 

Furthermore, we searched the rank of Y^2=X^3-nX with n<100 and give the parametric solution of x1^4 + 2nX2^4 = X3^8 + n^2X4^8. 

                 
2.Theorem
     
Diophantine equation x1^4 + 2nX2^4 = X3^8 + n^2X4^8 has infinitely many integral solutions
if the elliptic curve Y^2=X^3-nX has nonzero rank.

n is integer.
 
Proof.

x1^4 + 2nX2^4 = X3^8 + n^2X4^8....................................(1)

We consider an identity: (c^4-nd^4)^2+2n(cd)^4=c^8+n^2d^8.........(2)

Let v^2=c^4-nd^4..................................................(3)

Let X=(c/d)^2 and Y=cv/d^3, the equation (3) becomes to below elliptic curve.

Y^2=X^3-nX........................................................(4)

When the elliptic curve Y^2=X^3-nX has the rational point P(X,Y), rational point 2P(X) is always perfect square number.

Thus we can obtain the integral solution (c,d) from the solution of X=(c/d)^2.

Hence if the elliptic curve Y^2=X^3-nX has nonzero rank, then equation (1) infinitely many integral solutions. 

       
Q.E.D.


3.Results

There are parametric solution of x1^4 + 2nX2^4 = X3^8 + n^2X4^8 with n=m^2+1.

We show only the solution for the point 2P and 4P.

[x1,x2,x3,x4] = [m^4-4m^2-4, 2m(2+m^2), 2+m^2, 2m]

              = [(256+1024m^2+256m^4-2816m^6-4000m^8-2112m^10-496m^12-80m^14+m^16), 4(m^8+24m^6+40m^4+32m^2+16)m(m^2+2)(m^4-4m^2-4), m^8+24m^6+40m^4+32m^2+16, 4m(m^2+2)(m^4-4m^2-4)]

n  rank
[2, 1]
[5, 1]
[6, 1]
[7, 1]
[10, 1]
[12, 1]
[14, 1]
[15, 1]
[17, 2]
[20, 1]
[21, 1]
[22, 1]
[23, 1]
[25, 1]
[26, 1]
[30, 1]
[31, 1]
[32, 1]
[34, 1]
[36, 1]
[37, 1]
[38, 1]
[39, 1]
[41, 1]
[42, 1]
[45, 1]
[46, 1]
[47, 1]
[49, 1]
[50, 1]
[52, 1]
[53, 1]
[54, 1]
[55, 1]
[56, 2]
[57, 1]
[58, 1]
[60, 1]
[62, 1]
[65, 2]
[66, 1]
[69, 1]
[70, 1]
[71, 1]
[72, 1]
[73, 1]
[74, 1]
[76, 1]
[77, 2]
[78, 1]
[79, 1]
[80, 1]
[82, 3]
[84, 1]
[85, 1]
[86, 1]
[87, 1]
[89, 1]
[90, 2]
[94, 1]
[95, 1]
[96, 1]
[97, 2]
[99, 1]


4.Reference

[1]. Susil Kumar Jena, On X1^4 + 4X2^4 = X3^8 + 4X4^8 and Y1^4 = Y2^4 + Y3^4 + 4Y4^4,
     Communications in Mathematics, Vol. 23 (2015).