Congruent Numbers

area

53

a =	1472112483 202332130
b =	21447205780 1472112483
c =	4850493897329785961 297855654284978790
area =	53

A positive integer is called congruent if it is the area of a right triangle whose side lengths are rational numbers.

For example, the right triangle with legs 20/7 and 357/5 has area 102, and the hypotenuse 2501/35 is rational, so 102 is a congruent number. This Demonstration shows representative triangles for all known congruent numbers under 1000. Finding a triangle for area

d

is equivalent to solving the elliptic curve

2

y

=

3

x

-

2

d

x

. Completely solving this problem is similar to solving the Birch and Swinnerton-Dyer conjecture, which is an unsolved problem with a million-dollar prize.

Details

Data for these triangles is from[1].

References

[1] M. Fiorentini. "Numeri Congruenti Minori di 1000." bitman.name/math/table/29.

[2] Wikipedia. "Congruent Number." (Mar 14, 2013) en.wikipedia.org/wiki/Congruent_number.

External Links

Congruent Number (Wolfram MathWorld)

Swinnerton-Dyer Conjecture (Wolfram MathWorld)

Permanent Citation

Ed Pegg Jr

"Congruent Numbers"
http://demonstrations.wolfram.com/CongruentNumbers/
Wolfram Demonstrations Project
Published: November 8, 2013