## 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.

• Q1 : m and n for the congruent number 13
• Q2 : m and n for the congruent number 23
• Q3 : m and n for the congruent number 37

### Contents

Results of computation Results of computation 1. Search with definition 1 1.1  Simple loop by m and n 1.2  In case that two of m, n, m+n and m-n are perfect square 2. Search with definition 2 2.1  Simple loop by x and y 2.2 Solution by auxiliary equation 3. Birational transformation between each definitions 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

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