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 "<math>y^2<math>" 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).fr:Théorème de Nagell-Lutz