If n∈N is the value of area of right-angled triangle and the values of
every sides of triangle are in Q,
then n is called as congruent number or congruum.
Let a, b and c ∈N be the sides of right-angled triangle, and a and b close to a right angle,
then the area of this triangle is,
Let k as s=k2g (k≤1, g does not have square factors) and divide both side of the equation by k2, then,
Therefore the definition of congruent number is redefined as
Natural number g omitting the square factors from the area of a right-angled triangle abc.
The sides a, b, c of right-angled triangle are defined by the following equation.
a = m2-n2
b = 2mn
c = m2+n2
(m, n ∈ Z)
The area s of this triangle is
Let the square factors of s be k, then
For example, when m=2, n=1, then
so 6 is congruent.
When the following simultaneous equations
has a solution, then g is congruent.
When an elliptic curve y2=x3-g2x
has a non-trivial rational point (i.e. except (0,0), (±g,0)), then g is congruent.
There are birational transformations between these three definitions.
And definition 1 and 2 also has a double formula which are similar to the rational point on elliptic curve.
There are 361 congruent numbers under 1000.
In this chapter, we try to find m and n for all above congruent numbers.
(All the solution within the range 1 ≤ g ≤ 999 are here.)
In order to understand the difficulty of this problem, please try to answer the following questions.
|1. Search with definition 1||Results of computation|
|2. Search with definition 2||Results of computation|
2.1 Simple loop by x and y
2.2 Solution by auxiliary equation
|3. Birational transformation between each definitions||Results of computation|
4. Double of each definition
5. Godwin's method for definition 1
6. Criteria for congruent number
1 ≤ g ≤ 999 (All results),
|"Mathematician's Secret Room"||Chapter 11
Number Theoretic Algorithms
|Chapter 9 (Japanese)||index (Japanese)||Chapter 11 (Japanese)|