In mathematics, 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 integer congruent numbers starts with
For example, 5 is a congruent number because it is the area of a (20/3, 3/2, 41/6) triangle. Similarly, 6 is a congruent number because it is the area of a (3,4,5) triangle. 3 is not a congruent number.
If q is a congruent number then s2q is also a congruent number for any natural number s (just by multiplying each side of the triangle by s), and vice versa. This leads to the observation that whether a nonzero rational number q is a congruent number depends only on its residue in the group
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.
The question of determining whether a given rational number is a congruent number is called the congruent number problem. This problem has not (as of 2019) 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 (the difference between consecutive elements in an arithmetic progression of three squares) is non-square, it was already known (without proof) 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.
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 (as can essentially also be found in the introduction to Tunnell's paper).
Suppose a, b, c are numbers (not necessarily positive or rational) which satisfy the following two equations:
Then set x = n(a+c)/b and y = 2n2(a+c)/b2. A calculation shows
and y is not 0 (if y = 0 then a = -c, so b = 0, but ()ab = n is nonzero, a contradiction).
Conversely, if x and y are numbers which satisfy the above equation and y is not 0, set a = (x2 - n2)/y, b = 2nx/y, and c = (x2 + n2)/y. A calculation shows these three numbers satisfy the two equations for a, b, and c above.
These two correspondences between (a,b,c) and (x,y) are inverses of each other, so we have a one-to-one correspondence between any solution of the two equations in a, b, and c and any solution of the equation in x and y with y nonzero. In particular, from the formulas in the two correspondences, for rational n we see that a, b, and c are rational if and only if the corresponding x and y are rational, and vice versa. (We also have that a, b, and c are all positive if and only if x and y are all positive; from the equation y2 = x3 - xn2 = x(x2 - n2) that if x and y are positive then x2 - n2 must be positive, so the formula for a above is positive.)
Thus a positive rational number n is congruent if and only if the equation y2 = x3 - n2x has a rational point with y not equal to 0. It can be shown (as a nice application of Dirichlet's theorem on primes in arithmetic progression) that the only torsion points on this elliptic curve are those with y equal to 0, hence the existence of a rational point with y 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
Much work has been done classifying congruent numbers.
For example, it is known that for a prime number p, the following holds:
It is also known that in each of the congruence classes 5, 6, 7 (mod 8), for any given k there are infinitely many square-free congruent numbers with k prime factors.