10x^6 + 6y^6 - 16z^6  = w^2


We show diophantine equation 10x^6 + 6y^6 - 16z^6  = w^2 has infinitely many integer solutions.

10x^6 + 6y^6 - 16z^6  = w^2.................................................(1)

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

Let (a,b,c)=(10, 6, -16),(p,q)=(2, -2/3) the we obtain

u^2 = 29137920t^4+84879360t^3+101606400t^2+58060800t+18662400...............(2)

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

Y^2 = -8103375X -4942653750.

Rank is 1 and generator is (X,Y)=[-2450,14300].

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

Numerical example:

[X,Y,Z,W] = [4915, 9907, 6787, 2061562950720].