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 2009) 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.
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 (1/2)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; notice 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.
The following deal with the problem in a more rigorous way with some illustrations included: