直角三角形與同餘數 (Congruent Numbers) 台師大數學系 紀文鎮 2007.10.2.

直角三角形與同餘數 (Congruent Numbers)

台師大數學系紀文鎮

2007.10.2.

三邊長都是有理數的直角三角形稱為「有理直角三角形」 (rational right triangle)

b

a

ca ， b ， c 是有理數

a2+b2=c2

有理直角三角形area Congruent numbers

6,5,…

3

4

5

6

41

3

20

2

3

Congruent Number Problem

Find all congruent numbers among squarefree positive integers.

在第十世紀，這個問題就備受數學家的重視。

為什麼稱之 congruent number 呢 ?

Fibonacci 1225 “Liber Quadratorum” (The Book of squares).

定義： An integer n is called a “congruum” if there is an integer x such that x2±n are both squares. i.e. x2-n , x2 , x2+n is a 3-term arithmetic progression of squares with common difference n.

Congruum

Congruent

拉丁文 “ Congruere”

“to meet together”.

定理： 設 n>0

{ right triangles with area n }1 －1對應

3-term arithmetic progression of squares with common difference n

nabcba

cbacba

2

1,

0),,( 222

nst

nrs

tsr

tsr22

22

0

),,(1 －1對應

Pf:

),,()2,,(

)2

,2

,2

(),,(

tsrsrtrt

abcabcba

根據上述定理，

n is a congruent number 存在一個有理平方 s2 使得 s2-n 和 s2+n 都是平方

尋找 Congruent numbers ： Arab (10th Century) ： 5, 6 Fibonacci (13th Century) ： 7

Is 1 a congruent number ?

Fibonacci said “no”

But the first acceptable proof due to Fermat.

定理 (Fermat): 1 and 2 are not congruent numbers.

w2=u4±v4

Naïve algorithm:

(1) 基礎數論： ( 尋找 integral right triangles)

Primitive Pythagorean triples:

)2(mod,1),(,0),,2,( 2222 lklklklkkllk

(2)Find an integral right triangle, then the square free part n of its area is a congruent numbers.

背景定理： For n>0, there is a 1-1 correspondence between the following two sets:

nabcba

cbacba

2

1,

0),,( 222

1 －1對應 0,),( 232 yxnxyyx

n is congruent xnxy 232 has a rational solution (x,y) with y≠0.

),(),2

,(

)2

,(),,(

2222

2

yxy

nx

y

nx

y

nx

ac

n

ac

nbcba

方程式 0),)((232 nnxnxxxnxy

定義了一條橢圓曲線 (elliptic curve)

En: y2=x(x+n)(x-n), n :squarefree positive integer.

定理： En(Q)tors = {(0,0), (n,0), (-n,0), ∞}

定理： n is congruent if and only if there is (x,y) in En(Q) with y≠0. if and only if rank(En(Q)) 1.≧ In other words, En(Q) is infinite.

Corollary : If there is one rational right triangle with area n, then there are infinitely many.

Corollary: If there is a right triangle with rational sides and area n, then L(En, 1) = 0.

反之，若 B-SD conjecture 成立，則 L(En,1)=0 implies n is congruent.

定理 (1983)

猜測： If n is positive, squarefree, and n≡ 5, 6, or 7 (mod 8), then there is a rational right triangle with area n.

This has been verified for n <1,000,000

Serre’s Conjecture

T-W conjecture FLT

Serre

Ribet

A. Wiles proved T-W conjecture, hence proved FLT.

Summer School on Serre's Modularity Conjecture Luminy, July 9-20, 2007

今年 7 月在法國的學術會議證實：印度人 Chandrashekhar khare, 及法國人 Jean –Pierre Wintenberger兩人已證明了 Serre’s conjecture.

Clay Mathematics Institute Millennium Problems

Birch and Swinnerton-Dyer Conjecture Hodge Conjecture Navier-Stokes Equations P vs NP Poincaré Conjecture Riemann Hypothesis Yang-Mills Theory