The Nagell–Lutz theorem states that a nonzero rational torsion point on an integral short Weierstrass curve has integral coordinates; if its -coordinate is nonzero, then its square divides the discriminant.
The Nagell–Lutz theorem is a result in the theory of Diophantine equations, specifically concerning the representation of integers as sums of powers of natural numbers. It states that if a prime \( p \) can be expressed as a sum of two square numbers, i.e.
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