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Siegel's Theorem

There are at least two Siegel's theorems. The first states that an elliptic curve can have only a finite number of points with integer coordinates.

The second states that if xi is an algebraic number of degree r, then there is an A(xi) depending only on xi such that

|xi-p/q|>(A(xi))/(q^(2r^(1/2)))

for all integer p and q (Landau 1970, pp. 37-56; Hardy 1999, p. 79).

See also

Elliptic Curve, Roth's Theorem

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References

Davenport, H. "Siegel's Theorem." Ch. 21 in Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, pp. 126-125, 1980.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Landau, E. Vorlesungen über Zahlentheorie, Vol. 3. New York: Chelsea, 1970.

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Siegel's Theorem

Cite this as:

Weisstein, Eric W. "Siegel's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SiegelsTheorem.html

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