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  is an algebraic number of degree , then there is an  depending only on  such that

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

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.