直角三角形與同餘數 (congruent numbers) 台師大數學系 紀文鎮 2007.10.2
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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