algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

, a Mordell curve is an

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

of the form ''y''² = ''x''³ + ''n'', where ''n'' is a fixed non-zero

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

. These curves were closely studied by

Louis Mordell Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory. He was born in Philadelphia, United States, in a Jewish family of Lithuanian extraction. Educati ...

, from the point of view of determining their integer points. He showed that every Mordell curve contains only finitely many integer points (''x'', ''y''). In other words, the differences of

perfect squares In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usu ...

and

perfect cube In arithmetic and algebra, the cube of a number is its third power, that is, the result of multiplying three instances of together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example or . ...

s tend to infinity. The question of how fast was dealt with in principle by

Baker's method In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by , subsumed many earlier results in transcendent ...

. Hypothetically this issue is dealt with by

Marshall Hall's conjecture In mathematics, Hall's conjecture is an open question, , on the differences between perfect squares and perfect cubes. It asserts that a perfect square ''y''2 and a perfect cube ''x''3 that are not equal must lie a substantial distance apart. This ...

Properties

If (''x'', ''y'') is an integer point on a Mordell curve, then so is (''x'', ''-y''). There are certain values of ''n'' for which the corresponding Mordell curve has no integer solutions; these values are: : 6, 7, 11, 13, 14, 20, 21, 23, 29, 32, 34, 39, 42, ... . : −3, −5, −6, −9, −10, −12, −14, −16, −17, −21, −22, ... . The specific case where ''n'' = −2 is also known as Fermat's Sandwich Theorem.

List of solutions

The following is a list of solutions to the Mordell curve ''y''² = ''x''³ + ''n'' for , ''n'', ≤ 25. Only solutions with ''y'' ≥ 0 are shown. In 1998, J. Gebel, A. Pethö, H. G. Zimmer found all integers points for 0 < , ''n'', ≤ 10⁴. In 2015, M. A. Bennett and A. Ghadermarzi computed integer points for 0 < , ''n'', ≤ 10⁷.

References

{{Reflist, colwidth=30em

External links

* J. Gebel
Data on Mordell's curves for –10000 ≤ ''n'' ≤ 10000
* M. Bennett

Algebraic curves Diophantine equations Elliptic curves