Congruent number

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In mathematics, a congruent number is a positive rational number that is the area of a right-triangle with three rational number sides.

For example, 5 is a congruent number because it is the area of a 41/6, 3/2, 20/3 triangle. Similarly, 6 is a congruent number because it is the area of a 3,4,5 triangle. 3 is not a congruent number.

It is worth noting that if q is a congruent number then s²q is also a congruent number for any rational number s (just by multiplying each side of the triangle by s). This leads to the observation that whether a nonzero rational number q is a congruent number depends only on its residue in the group

$\mathbb{Q}^{*}/\mathbb{Q}^{*2}$ .

Every residue class in this group contains exactly one square free positive integer, and it is common, therefore, only to consider square free positive integers, when speaking about congruent numbers.

1 The congruent number problem
2 Relation to elliptic curves
3 Current progress
4 References

[edit] The 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 (as of 2005) 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.

[edit] Relation to elliptic curves

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. This is described in detail in chapter 1 of Koblitz's book Introduction to Elliptic Curves and Modular Forms (ISBN 0-387-97966-2). An alternate 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:

$\begin{matrix} a^2 + b^2 &=& c^2\\ \frac{ab}{2} &=& n \end{matrix}$

Then set x = n(a+c)/b and y = 2n²(a+c)/b². A calculation shows

$y^2 = x^3 -n^2x \,\!$

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 - x = 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.

[edit] Current progress

Much work has been done classifying congruent numbers.

For example, it is know that if p is a prime number then

if p = 3 mod 8, then p is not a congruent number, but 2p is a congruent number.
if p = 5 mod 8, then p is a congruent number.
if p = 7 mod 8, then p and 2p are congruent numbers.

Proofs are given in Paul Monsky's article Mock Heegner Points and Congruent Numbers.

[edit] References

A short discussion of the current state of the problem with many good references can be found in Alice Silverberg's Open Questions in Arithmetic Algebraic Geometry (Postscript).
Richard Guy's book, Unsolved Problems in Number Theory (ISBN 0-387-20860-7), has many good references as well.
For a history of the problem see Leonard Eugene Dickson's book History of the Theory of Numbers Volume II (ISBN 0-8218-1935-6) Chapter XVI.
A nice overview of the problem can be found in Ronald Alter's article The Congruent Number Problem (article available at JSTOR).
An overview of the problem can be found in the article The Congruent Number Problem [1] in the journal Resonance.
Tunnell's paper:Tunnell, Jerrold B. (1983). "A classical Diophantine problem and modular forms of weight 3/2". Inventiones Mathematicae 72 (2): 323–334. DOI:10.1007/BF01389327.