Heights on elliptic curves and the diophantine equation x4 + y4 = cz4

In this paper we give sharp explicit estimates for the difference of the Weil height and the Néron - Tate height on the elliptic curve $v^2 = u^3 - cu$. We then apply this in the proof of the fact that if c > 2 is a fourth power free integer and the rank of $v^2 = u^3 - cu$ is 1 then the equation $x^4 + y^4 = cz^4$ has no nonzero solutions in integer