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=k^{2}g (k≤1, g does not have square factors) and divide both side of the equation by k^{2}, 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 = m^{2}-n^{2}
b = 2mn
c = m^{2}+n^{2}
(m, n ∈ Z)
The area s of this triangle is
s=mn(m^{2}-n^{2})
Let the square factors of s be k, then
s=k^{2}g=mn(m^{2}-n^{2}).
For example, when m=2, n=1, then
a=m^{2}-n^{2}=2^{2}-1^{2}=3
b=2mn=2*2*1=4
and
s=mn(m^{2}-n^{2})=2*1*(2^{2}-1^{2})=6
so 6 is congruent.
When the following simultaneous equations
x^{2}+gy^{2}=z^{2}
x^{2}-gy^{2}=±w^{2}
has a solution, then g is congruent.
When an elliptic curve y^{2}=x^{3}-g^{2}x
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.
Chapter 9 Amicable Numbers |
"Mathematician's Secret Room" | Chapter 11 Number Theoretic Algorithms |
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Chapter 9 (Japanese) | index (Japanese) | Chapter 11 (Japanese) |