Chapter 10 : Congruent Numbers (D27 Congruent numbers)

(June 07, 1999) [Japanese]

Abstract

[Definition 1]

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,

s=ab/2

Let k as s=k2g (k≤1, g does not have square factors) and divide both side of the equation by k2, then,

g=(a/k)(b/k)/2

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

s=mn(m2-n2)

Let the square factors of s be k, then

s=k2g=mn(m2-n2).

For example, when m=2, n=1, then

a=m2-n2=22-12=3
b=2mn=2*2*1=4

and

s=mn(m2-n2)=2*1*(22-12)=6

so 6 is congruent.


[Definition 2]

When the following simultaneous equations

x2+gy2=z2
x2-gy2=±w2

has a solution, then g is congruent.


[Definition 3]

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.

Contents

1. Search with definition 1Results of computation

1.1  Simple loop by m and n

  1 ≤ g ≤  99, 100 ≤ g ≤ 199
200 ≤ g ≤ 299, 300 ≤ g ≤ 399
400 ≤ g ≤ 499, 500 ≤ g ≤ 599
600 ≤ g ≤ 699, 700 ≤ g ≤ 799
800 ≤ g ≤ 899, 900 ≤ g ≤ 999

1.2  In case that two of m, n, m+n and m-n are perfect square

  1 ≤ g ≤  99, 100 ≤ g ≤ 199
200 ≤ g ≤ 299, 300 ≤ g ≤ 399
400 ≤ g ≤ 499, 500 ≤ g ≤ 599
600 ≤ g ≤ 699, 700 ≤ g ≤ 799
800 ≤ g ≤ 899, 900 ≤ g ≤ 999

1.3  In case that three of m, n, m+n and m-n are perfect square

  1 ≤ g ≤ 499
500 ≤ g ≤ 999

2. Search with definition 2Results of computation

2.1  Simple loop by x and y

(1) Birational equivalence to definition 1
(2) Program
(3) In case of the coefficient of w2 < 0
(4) Program

  1 ≤ g ≤ 499 of (2)
500 ≤ g ≤ 999 of (2)

  1 ≤ g ≤ 999 of (4)

2.2 Solution by auxiliary equation

(1) Auxiliary equation
(2) Program

1 ≤ g ≤ 999

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

7. Conclusion

1 ≤ g ≤ 999 (All results),

Rank of elliptic curve y2=x(x2-g2),
(with Cremona's mwrank)
   1 ≤ g ≤  999
1000 ≤ g ≤ 1999
2000 ≤ g ≤ 2999
3000 ≤ g ≤ 3999
4000 ≤ g ≤ 4999
5000 ≤ g ≤ 5999
6000 ≤ g ≤ 6999
7000 ≤ g ≤ 7999
8000 ≤ g ≤ 8999
9000 ≤ g ≤ 9999


Chapter 9
Amicable Numbers
"Mathematician's Secret Room" Chapter 11
Number Theoretic Algorithms
Chapter 9 (Japanese) index (Japanese) Chapter 11 (Japanese)

E-mail : kc2h-msm@asahi-net.or.jp
Hisanori Mishima