congruent numbers-a short introduction

Congruent numbers are simple to define. An rational number is said to be a congruent number if it is the area of some right triangle the lenghts of whose three sides are rational numbers. In other words, a rational number n is a congruent number if there exists three rational numbers a,b,c such that a^2+b^2=c^2,ab=2n. It is clear that if n is a congruent number if and only if s^2n is for some non-zero rational number s. So, whether n is a congruent number depends only on the class \mathbb{Q}^*/\mathbb{Q}^{*2}.

There are various equivalent formulations for the congruent numbers, for example, the followings:

A rational number n is a congruent number if and only if there are three rational numbers A,B,C such that n=A^2-B^2=B^-C^2.

A ratiobal number n is a congruent number if and only if the corresponding elliptic curve Y^2=X^3-n^2X over \mathbb{Q} has non-torsion points.

There are several bijections between the set C(n)of the solutions to the system of equations a^2+b^2=c^2,ab=2n on variables a,b,c and the set E(n)of non-torsion points of the elliptic curve Y^2=X^3-n^2X(we can show that the only torsion points of this elliptic curve are (0,0),(n,0),(-n,0). The proof to this result uses Dirichlet’s theorem on arithmetic progression. cf Theorem 7.2 of this article.). One of them is:

f:C(n)\rightarrow E(n), (a,b,c)\mapsto (n(a+c)/b,2n^2(a+c)/b^2)

A little remark before we go on. The congruent number is perhaps one of the few objets in number theory that do not have generalization to other number fields(there are very few research on this aspect. Searching the Internet, we only find some results like congruent nulmbers over real quadratic fields).

One of the most important results in this field is perhaps Tunnell’s theorem:

We can suppose a representative of an element in \mathbb{Q}^*/\mathbb{Q}^{*2} to be a square-free integer(positive, of course). We set(the same notation as the article, ‘the congruent number problem’ by Keith Conrad),

f(n)=\#\{(x,y,z)\in\mathbb{Z}^3|x^2+2y^2+8z^2=n\},

$latex g(n)=\#\{(x,y,z)\in\mathbb{Z}^3|x^2+2y^2+32z^2=n\}$,

$latex h(n)=\#\{(x,y,z)\in\mathbb{Z}^3|x^2+4y^2+8z^2=n/2\}$,

$latex k(n)=\#\{(x,y,z)\in\mathbb{Z}^3|x^2+4y^2+32z^2=n/2\}$, then for odd n, if n is a congruent number, then f(n)=2g(n). For even n, if n is congruent number, then h(n)=2k(n). If we assume the weak Birch-Swinnerton-Dyer conjecture, then the converse is true, too.

The interesting part of the theorem is of course the converse part. Note that, all the four sets are not so hard to calculate, so, we can just compare the cardinality of these four sets to decide whether a rational number is a congruent number.

Yet, the default of this converse part is that it assumes the BSD conjecture, which is widely an open question.

The first part of the theorem can also be useful, in showing that an integer is not a congruent number(the counterpart of the first part). For example, for n=1. We have that f(1)=2,g(1)=2, this shows that 1 is not a congruent number. If we do not use this theorem, then to show that n=1 is not a congruent number, we can use the descent method due to Fermat. An interesting corollary of this result is that, \sqrt{2} is not rational. Indeed, if it is, then the right triangle with three sides of lengths, \sqrt{2},\sqrt{2},2 is of area 1, contradicting the result that 1 is a congruent number.

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