In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a congruent number is a positive integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

 that is the area of a right triangle

Right triangle

A right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...

 with three rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

 sides. A more general definition includes all positive rational numbers with this property.

The sequence of integer congruent numbers starts with

5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, …

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 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

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

.

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.

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 2009) been brought to a successful resolution. Tunnell's theorem

Tunnell's theorem

In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution. The congruent number problem asks which rational numbers can be the area of a right triangle with all three sides rational...

 provides an easily testable criterion for determining whether a number is congruent; but his result relies on the Birch and Swinnerton-Dyer conjecture

Birch and Swinnerton-Dyer conjecture

In mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. Its status as one of the most challenging mathematical questions has become widely recognized; the conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay...

, which is still unproven.

Fermat's right triangle theorem, named after Pierre de Fermat

Pierre de Fermat

Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...

, states that no square number

Square number

In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...

 can be a congruent number.

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

Elliptic curve

In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

 has positive rank

Rank of an abelian group

In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...

. 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

Dirichlet's theorem on arithmetic progressions

In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ≥ 0. In other words, there are infinitely many primes which are...

 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.

Current progress

Much work has been done classifying congruent numbers.

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

• if p ≡ 3 (mod
  Modular arithmetic
  In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
  
  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.