Nagell–Lutz theorem
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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 "y2" divides "D".
The result is named for its two independent discoverers, the Norwegian Trygve Nagell (1895–1988) who published it in 1935, and Elizabeth Lutz (1937).
