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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.
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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
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有理直角三角形 area Congruent numbers 6,5,… 3 4 5
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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.
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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:
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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.
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定理 (Fermat): 1 and 2 are not congruent numbers. w2=u4±v4w2=u4±v4
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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.
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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.
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方程式 定義了一條橢圓曲線 (elliptic curve)
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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.
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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.
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定理 (1983)
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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
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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.
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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
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