38x^6 + 10y^6 - 48z^6 = w^2


We show diophantine equation 38x^6 + 10y^6 - 48z^6 = w^2 has infinitely many integer solutions.

38x^6 + 10y^6 - 48z^6 = w^2.........................................................(1)

For details,ax^6 + by^6 + cz^6 = w^2 Ⅴ

Let (a,b,c)=(38, 10, -48),(p,q)=(2, -14/5) the we obtain

u^2 = 7202903040t^4-3318220800t^3+17619840000t^2+729600000t+2736000000..............(2)

This quartic equation is birationally equivalent to an elliptic curve below.

Y^2 = X^3-281884458375X -28168188878891250.

Rank is 2 and generator is (X,Y)=[-460497650/1089 , 4462111428650/35937 ], [-25928085590006/98942809 , -163814781169223599472/984184121123] .

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

Numerical example:

[X,Y,Z,W] = [281, 65, 236, 102056220].