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. 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 Every residue class in this group contains exactly onesquare-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 , has solutions. Given the solutions, one can obtain the such that , and from ,,
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. 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
Smallest solutions
The following is a list of the rational solution to and with congruent number n and the smallest numerator for c..
n
a
b
c
5
6
3
4
5
7
13
14
15
4
20
3
21
12
22
23
24
6
8
10
28
29
30
5
12
13
31
34
24
37
38
39
41
45
20
46
47
52
53
54
9
12
15
55
56
21
60
8
15
17
61
...
...
...
...
101
...
...
...
...
157
Current progress
Much work has been done classifying congruent numbers. 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.
