Mordell Curve

موضوع: Mordell Curve الإثنين يوليو 21, 2008 1:40 pm

An elliptic curve of the form

for

an integer. This equation has a finite number of solutions in integers for all nonzero

. If

is a solution, it therefore follows that

is as well.

Uspensky and Heaslet (1939) give elementary solutions for

,

, and 2, and then give

,

, and 1 as exercises. Euler found that the only integer solutions to the particular case

(a special case of Catalan's conjecture) are

,

, and

. This can be proved using Skolem's method, using the Thue equation

, using 2-descent to show that the elliptic curve has rank 0, and so on. It is given as exercise 6b in Uspensky Heaslet (1939, p. 413), and proofs published by Wakulicz (1957), Mordell (1969, p. 126), Sierpiński (1988, pp. 75-80), and Metsaenkylae (2003).

Solutions of the Mordell curve with

موضوع: رد: Mordell Curve الإثنين يوليو 21, 2008 6:53 pm

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