45x^6 + 20y^6 - 65z^6 = w^2


We show diophantine equation 45x^6 + 20y^6 - 65z^6 = w^2 has infinitely many integer solutions.

45x^6 + 20y^6 - 65z^6 = w^2..................................................................(1)

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

Let (a,b,c)=(45, 20, -65),(p,q)=(2, -5/4) the we obtain

u^2 = 185042812500t^4+504562500000t^3+675675000000t^2+327600000000t+140400000000..............(2)

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

Y^2 = X^3-1121040375X -7663918681250.

Rank is 2 and generator is (X,Y)=[-14639394277474599/506360328100 , -274721628795845290941851/360320945872679000 ],
                                 [-3700063973450217177650/428405830667806761 , -328687698500640935282045195188750/280403260670962906730070741 ].

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

Numerical example:

[X,Y,Z,W] = [198827, 32638, 127607, 49995044203554450].