Congruent number


In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.
The sequence of congruent numbers starts with
12345678
————CCC—
910111213141516
————CCC—
1718192021222324
———SCCCS
2526272829303132
———SCCC—
3334353637383940
—C——CCC—
4142434445464748
C———SCC—
4950515253545556
———SCSCS
5758596061626364
———SCCS—
6566676869707172
C———CCC—
7374757677787980
————CCCS
8182838485868788
———SCCCS
8990919293949596
———SCCCS
979899100101102103104
————CCC—
105106107108109110111112
————CCCS
113114115116117118119120
———SSCCS

For example, 5 is a congruent number because it is the area of a triangle. Similarly, 6 is a congruent number because it is the area of a triangle. 3 and 4 are not congruent numbers. The triangle sides demonstrating a number is congruent can have very large numerators and denominators, for example 263 is the area of a triangle whose two shortest sides are 16277526249841969031325182370950195/2303229894605810399672144140263708 and 4606459789211620799344288280527416/61891734790273646506939856923765.
If is a congruent number then is also a congruent number for any natural number, and vice versa. This leads to the observation that whether a nonzero rational number is a congruent number depends only on its residue in the group
where is the set of nonzero rational numbers.
Every residue class in this group contains exactly one square-free integer, and it is common, therefore, only to consider square-free positive integers when speaking about congruent numbers.

Congruent number problem

The question of determining whether a given rational number is a congruent number is called the congruent number problem., this problem has not been brought to a successful resolution. Tunnell's theorem provides an easily testable criterion for determining whether a number is congruent; but his result relies on the Birch and Swinnerton-Dyer conjecture, which is still unproven.
Fermat's [right triangle theorem], named after Pierre de Fermat, states that no square number can be a congruent number. However, in the form that every congruum is non-square, it was already known to Fibonacci. Every congruum is a congruent number, and every congruent number is a product of a congruum and the square of a rational number. However, determining whether a number is a congruum is much easier than determining whether it is congruent, because there is a parameterized formula for congrua for which only finitely many parameter values need to be tested.

Solutions

n is a congruent number if and only if the system
has a solution where, and are integers.
Given a solution, the three numbers,, and will be in an arithmetic progression with common difference.
Furthermore, if there is one solution, then there are infinitely many: given any solution,
another solution can be computed from
For example, with, the equations are:
One solution is . Another solution is
With this new and, the new right-hand sides are still both squares:
Using as above gives
Given, and, one can obtain, and such that
from
Then and are the legs and hypotenuse of a right triangle with area.
The above values produce. The values give. Both of these right triangles have area.

Relation to elliptic curves

The question of whether a given number is congruent turns out to be equivalent to the condition that a certain elliptic curve has positive rank. An alternative approach to the idea is presented below.
Fix nonzero. Suppose,, are numbers which satisfy the following two equations:
Then set and
A calculation shows
and is not 0.
Conversely, if and are numbers which satisfy the above equation and is not 0, set
, and. A calculation shows these three numbers
satisfy the two equations for,, and above.
These two correspondences between and are inverses of each other, so we have a one-to-one correspondence between any solution of the two equations in
,, and and any solution of the equation in and with nonzero. In particular, from the formulas in the two correspondences, for rational we see that,, and are rational if and only if the corresponding and are rational, and vice versa.
Thus a positive rational number is congruent if and only if the equation
has a rational point with not equal to 0.
It can be shown
that the only torsion points on this elliptic curve are those with equal to 0, hence the
existence of a rational point with nonzero is equivalent to saying the elliptic curve has positive rank.
Another approach to solving is to start with integer value of n denoted as N and solve
where

Current progress

For example, it is known that for a prime number, the following holds:
  • if, then is not a congruent number, but 2 is a congruent number.
  • if, then is a congruent number.
  • if, then and 2 are congruent numbers.
It is also known that in each of the congruence classes, for any given there are infinitely many square-free congruent numbers with prime factors.