A congruent number can be defined as an integer that is equal to the area of a rational right triangle (Koblitz 1993).
For a positive integer 
, the associated elliptic curve
is
 |
(1)
|
which is a quadratic twist of 
. The integer 
is a congruent number iff 
has a rational point of
infinite order. Equivalently, 
has elliptic curve rank
at least 1 over 
(Koblitz 1993, Watkins et al. 2014).
Watkins et al. (2014) reported that the smallest known parameter producing elliptic curve rank 7 is 
and listed 27 such elliptic
curves. Their extensive searches found no elliptic
curve in this family with elliptic
curve rank 8. They also described a heuristic suggesting
that 7 might be the maximal elliptic curve rank
in this family, although whether the elliptic
curve ranks in the family are bounded
remains unknown.
Numbers 
such that
 |
(2)
|
are also known as congruent numbers. They are a generalization of the congruum problem, which is the case 
.
For example, 
,
the smallest congruent numbers are
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
