In
algebra
Algebra () is one of the areas of mathematics, 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 mathem ...
, 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, 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 u ...
and
perfect cubes tend to infinity. The question of how fast was dealt with in principle by
Baker's method. Hypothetically this issue is dealt with by
Marshall Hall's conjecture.
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'', ≤ 10
4.
In 2015, M. A. Bennett and A. Ghadermarzi computed integer points for 0 < , ''n'', ≤ 10
7.
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