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

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 c a ， b ， c 是有理數 a 2 +b 2 =c 2

有理直角三角形 area Congruent numbers 6,5,… 3 4 5

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 x 2 ±n are both squares. i.e. x 2 -n, x 2, x 2 +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 } 1－11－1 對應 3-term arithmetic progression of squares with common difference n 1－11－1 對應 Pf:

根據上述定理， n is a congruent number 存在一個有理平方 s 2 使得 s 2 -n 和 s 2 +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±v4w2=u4±v4

Naïve algorithm: (1) 基礎數論： ( 尋找 integral right triangles) Primitive Pythagorean triples: (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: 1－11－1 對應 n is congruent has a rational solution (x,y) with y≠0.

方程式 定義了一條橢圓曲線 (elliptic curve)

E n : y 2 =x(x+n)(x-n), n :squarefree positive integer. 定理： E n (Q) tors = {(0,0), (n,0), (-n,0), ∞ } 定理： n is congruent if and only if there is (x,y) in E n (Q) with y ≠0. if and only if rank(E n (Q)) ≧ 1. In other words, E n (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(E n, 1) = 0. 反之，若 B-SD conjecture 成立，則 L(E n,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 conjectureFLT 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