Nagell–Lutz theorem

In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves. Suppose that C defined by

y² = x³ + ax² + bx + c = f(x)

is a non-singular cubic curve with integer coefficients a, b, c, and let D be the discriminant of the cubic polynomial f,

D = −4a³c + a²b² + 18abc − 4b³ − 27c².

Let P = (x,y) be a rational point of finite order on C, for the group law.

Then x and y are integers; and either y = 0, in which case P has order two, or else y divides D, which immediately implies that "$y^{2}$" divides "D".

The result is named for its two independent discoverers, the Norwegian Trygve Nagell (1895–1988) who published it in 1935, and Elisabeth Lutz (1937).