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* = 2*n*^{2}(*a*+*c*)/*b*^{2}. 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* = (*x*^{2} – *n*^{2})/*y*, *b* = 2*nx*/*y*, and *c* = (*x*^{2} + *n*^{2})/*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 *y*^{2} = *x*^{3} – *xn*^{2} = *x*(*x*^{2} – *n*^{2}) that if *x* and *y* are positive then *x*^{2} – *n*^{2} 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 *y*^{2} = *x*^{3} – *n*^{2}*x* 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:

http://www.thehcmr.org/issue2_2/**congruent**_**number**.pdf

http://www.ias.ac.in/resonance/Aug1998/pdf/Aug1998p33-45.pdf

http://www.intlpress.com/JPAMQ/p/2005/14-27.pdf