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=> Algebraic Curves-Mordell Curve
=> Algebraic Curves-Ochoa Curve
=> Algebraic Integer
=> Algebraic Number
=> Algebraic Number Theory
=> Chebotarev Density Theorem
=> Class Field
=> Cyclotomic Field
=> Dedekind Ring
=> Fractional Ideal
=> Global Field
=> Local Field
=> Number Field Signature
=> Picard Group
=> Pisot Number
=> Weyl Sum
=> Casting Out Nines
=> A-Sequence
=> Anomalous Cancellation
=> Archimedes' Axiom
=> B2-Sequence
=> Calcus
=> Calkin-Wilf Tree
=> Egyptian Fraction
=> Egyptian Number
=> Erdős-Straus Conjecture
=> Erdős-Turán Conjecture
=> Eye of Horus Fraction
=> Farey Sequence
=> Ford Circle
=> Irreducible Fraction
=> Mediant
=> Minkowski's Question Mark Function
=> Pandigital Fraction
=> Reverse Polish Notation
=> Division by Zero
=> Infinite Product
=> Karatsuba Multiplication
=> Lattice Method
=> Pippenger Product
=> Reciprocal
=> Russian Multiplication
=> Solidus
=> Steffi Problem
=> Synthetic Division
=> Binary
=> Euler's Totient Rule
=> Goodstein Sequence
=> Hereditary Representation
=> Least Significant Bit
=> Midy's Theorem
=> Moser-de Bruijn Sequence
=> Negabinary
=> Negadecimal
=> Nialpdrome
=> Nonregular Number
=> Normal Number
=> One-Seventh Ellipse
=> Quaternary
=> Radix
=> Regular Number
=> Repeating Decimal
=> Saunders Graphic
=> Ternary
=> Unique Prime
=> Vigesimal
Ziyaretçi defteri
 

Algebraic Curves-Mordell Curve

An elliptic curve of the form y^2=x^3+n for n an integer. This equation has a finite number of solutions in integers for all nonzero n. If (x,y) is a solution, it therefore follows that (x,-y) is as well.

MordellCurve

Uspensky and Heaslet (1939) give elementary solutions for n=-4, -2, and 2, and then give n=-1, -5, -6, and 1 as exercises. Euler found that the only integer solutions to the particular case n=1 (a special case of Catalan's conjecture) are (x,y)=(-1,0), (0,+/-1), and (2,+/-3). This can be proved using Skolem's method, using the Thue equation x^3-2y^3=+/-1, using 2-descent to show that the elliptic curve has rank 0, and so on. It is given as exercise 6b in Uspensky and Heaslet (1939, p. 413), and proofs published by Wakulicz (1957), Mordell (1969, p. 126), Sierpiński and Schinzel (1988, pp. 75-80), and Metsaenkylae (2003).

Solutions of the Mordell curve with 0<y<10^5 are summarized in the table below for small n.

n solutions
1 (-1, 0), (0, 1), (2, 3)
2 (-1, 1)
3 (1, 2)
4 (0, 2)
5 (-1, 2)
6 none
7 none
8 (-2, 0), (1, 3), (2, 4), (46, 312)
9 (-2, 1), (0, 3), (3, 6), (6, 15), (40, 253)
10 (-1, 3)

Values of n such that the Mordell curve has no integer solutions are given by 6, 7, 11, 13, 14, 20, 21, 23, 29, 32, 34, 39, 42, ... (Sloane's A054504; Apostol 1976, p. 192).

 

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