Congruent number

In number theory congruent numbers are whole numbers that can be represented as area of ​​a right triangle with rational side lengths.

The sequence of congruent numbers ( sequence A003273 in OEIS ) starts with

For example, the integer 6 is a congruent number because the right-angled triangle with short sides and has the area and according to the Pythagorean theorem, the hypotenuse. So the integer 6 is a congruent number as area of ​​a right triangle with rational side lengths.

For every positive integer is an integer if and only a Kongruenzzahl when a Kongruenzzahl is. Therefore, one can solve the problem Kongruenzzahl limited to square-free numbers.

More generally, all rational numbers, which occur as the area of a right triangle with side length rational called congruent figures.

Matching numbers in the range 1 to 20

The following integers in the range 1 to 20 are congruent, since they can be represented as area of ​​a right triangle with rational and catheti and rational hypotenuse:

Fermat's theorem

The French mathematician Pierre de Fermat proved that the area of ​​a right triangle with integer side lengths can not be a square number. This is equivalent to the fact that not every other square number 1 is still a matching number. His result he shared with 1659 in a letter to Pierre de Carcavi, the evidence he noted in a comment that was published posthumously in 1670. Fermat based on the known since ancient representation of a primitive Pythagorean triple from as x2 y2, x2 - y2, 2xy and uses the method introduced by him of infinite descent, a variant of mathematical induction. His proof also shows that the equation a4 b4 = c4 no solution with positive integers a, b, c has ( a special case of Fermat's theorem ).

Set of Tunnell

The set of Tunnell, named after Jerrold B. Tunnell, are necessary conditions to ensure that a number is congruent. Assuming the correctness of the conjecture of Birch and Swinnerton - Dyer these conditions are also sufficient.

For a square-free integer defining

If an odd Kongruenzzahl, then must be, if a straight Kongruenzzahl is, then his needs.

If the conjecture of Birch and Swinnerton - Dyer is true for elliptic curves of the form, then these conditions are also sufficient.