1.Introduction

A. Bremner, AJAI CHOUDHRY and M. Ulas([1]) gave a solution of x^6 + y^6 - 36z^6 + 2w^3 =0 as a special case of PX^6+ QY^6+ Sm^2Z^6+ RW^3= 0.

We show six solutions of ax^6+ by^6+ cz^6+ dw^3= 0 by their method.



2.Method
        

We consider the identity a^2T^6 + b^2 + 2abT^3 = (aT^3+b)^2..........................(1)

Let aT^3 + b = mz^3..................................................................(2)

If we can find the rational solution (T,z) of (2) for some {a,b,m}, we obtain the solution of (3).

a^2X^6 + b^2Y^6 - m^2Z^6 + 2abW^3=0..................................................(3)

Hence if equation (2) has infinitely many rational solutions, equation (3) has infinitely many integer solutions.

     
3.Results


Case 1: 4X^6 + Y^6 - 9Z^6 + 4W^3 = 0
       
By using (a,b,m)=(2,1,3), equation (2) has rational solution (T,z)=(1,1).

We obtain quartic elliptic curve and Weierstrass form,

V^2 = -27k^4+72k^3-108k^2+72k-12.

Y^2 = X^3 - 243.

This rank is 1 and generator: P=(7, 10).

Thus, we can obtain infinitely many integer solutions.

P:    (x,y,z,w)=(4, 5, 1, -20)
2P:   (x,y,z,w)=(488, 655, 253, -319640), (3449, 26309, 18269, 90739741)


Case 2: 4X^6 + 4Y^6 - 25Z^6 + 8W^3 = 0
       
By using (a,b,m)=(2,2,5), equation (2) has rational solution (T,z)=(19,14).

We obtain quartic elliptic curve and Weierstrass form,

V^2 = -14700k^4+43320k^3-47880k^2+23520k-4332.

Y^2 = X^3 -2700.

This rank is 1 and generator: P=(21, 81).

Thus, we can obtain infinitely many integer solutions.

P:    (x,y,z,w)=(43453, 4573, 32004, -198710569)
2P:   (x,y,z,w)=(35838502109, 35885554849, 4172009702, -1286084533138521676541)


Case 3: 9X^6 + Y^6 - 4Z^6 + 6W^3 = 0
       
By using (a,b,m)=(3,1,2), equation (2) has rational solution (T,z)=(-1,-1).

We obtain quartic elliptic curve and Weierstrass form,

V^2 = -12k^4+72k^3-108k^2+72k-27.

Y^2 = X^3 - 243.

This rank is 1 and generator: P=(7, 10).

Thus, we can obtain infinitely many integer solutions.

P:    (x,y,z,w)=(1, 5, 4, 5)
2P:   (x,y,z,w)=(322184141, 56829019, 369033521, 18309408670387679),
                (129900299507, 120418942015, 160841972528, 15642456634064566086605) 


Case 4: 9X^6 + Y^6 - 16Z^6 + 6W^3 = 0
       
By using (a,b,m)=(3,1,4), equation (2) has rational solution (T,z)=(1,1).

We obtain quartic elliptic curve and Weierstrass form,

V^2 = -48k^4+144k^3-216k^2+144k-27.

Y^2 = X^3 - 972.

This rank is 1 and generator: P=(13, 35).

Thus, we can obtain infinitely many integer solutions.

P:    (x,y,z,w)=(5, 7, 2, -35)
2P:   (x,y,z,w)=(1555, 2849, 1436, -4430195),(42901, 265379, 166469, -11385024479)


Case 5: 9X^6 + 4Y^6 - Z^6 + 12W^3 = 0
       
By using (a,b,m)=(3,2,1), equation (2) has rational solution (T,z)=(-1,-1).

We obtain quartic elliptic curve and Weierstrass form,

V^2 = -3k^4+36k^3-54k^2+36k-27.

Y^2 = X^3 - 243.

This rank is 1 and generator: P=(7, -10).

Thus, we can obtain infinitely many integer solutions.

P:    (x,y,z,w)=(253, 488, 655, 123464), (18269, 3449, 26309, -63009781)
2P:   (x,y,z,w)=(322184141, 369033521, 56829019, -118896747963590461), (129900299507, 160841972528, 120418942015, -20893420404683865943696)


Case 6: 16X^6 + Y^6 - 9Z^6 + 8W^3 = 0
       
By using (a,b,m)=(4,1,3), equation (2) has rational solution (T,z)=(-1,-1).

We obtain quartic elliptic curve and Weierstrass form,

V^2 = -27k^4+144k^3-216k^2+144k-48.

Y^2 = X^3 - 972.

This rank is 1 and generator: P=(13, -35).

Thus, we can obtain infinitely many integer solutions.

P:    (x,y,z,w)=(2, 7, 5, 14)
2P:   (x,y,z,w)=(1436, 2849, 1555, -4091164), (166469, 265379, 42901, -44177376751)





4.Reference


[1]. A. Bremner, AJAI CHOUDHRY and M. Ulas, CONSTRUCTIONS OF DIAGONAL QUARTIC AND SEXTIC
     SURFACES WITH INFINITELY MANY RATIONAL POINTS, arxiv.org/pdf/1402.4583, 2014