### 第四章[证明方法] - 南京大学计算机科学与技术系

```证明方法

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（当前）定理的前提
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xy((xy)  (x2y2)) //论域为正实数

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n2=2(2k2+2k)+1
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n2是奇数
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pq  ¬q¬p

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¬q  ¬p
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//直接证明的设想不奏效。
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n是偶数，存在一个整数k使得n=2k
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3n+2=2(3k+1)
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3n+2是偶数 (¬p)
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q  ¬qF

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¬q  r¬r
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There is no rational number whose square is 2.
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Proof
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Extra hypothesis: (p/q)2=2, and p,q are integers which have
no common factors except for 1.
Then, p2=2q2  p2 is even  p is even  p2 is multiple of 4
 q2 is even  q is even  p,q have 2 as common factor 

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p1 ...  pn  q  ¬q  p1  ...  pn  F

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¬q, p1, ..., pn  矛盾 (比如p1 ¬ p1)

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p1 ...  pn  q  (p1  q)  ...  (pn  q)

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p1  q
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…
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pn  q
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n=1, 2, 3, 4.（穷举）

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n0
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n1
(x+y)r < xr+yr, 这里x, y是正实数, r是0<r<1的实数.
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x < xr， y < yr  x+y < xr+yr  (x+y)r < xr+yr

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[ p1p2…pn] [( p1p2) ( p2p3) …( pnp1)]

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p1 p2
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p2 p3
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…
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pn p1
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x P(x)

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1729=103+93=123+13

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y2=2，x= y y，x y=(y y) y=y2=2
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x (P(x)  y (yxP(y) )
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x P(x)  y z (P(y)  P(z)  y = z)

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x P(x) x P(x)

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3
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7
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a=b

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a2=ab

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a2-b2=ab-b2 两边减去b2
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(a-b) (a+b) = (a-b) b
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(a+b) = b
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2b = b
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2=1

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Pierre de Fermat (1601-1665), France
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Fermat’s Last Theorem (1637) （费马大定理）
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xn+yn=zn (n2, xyz0)没有整数解
Andrew Wiles (1953- ), Oxford, England
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1994/1995完成了费马大定理的证明（约10年时间）
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Goldbach Conjecture（1742年给欧拉的信中）
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“a+b”猜想
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1966年陈景润（1933－1996）证明了“1+2”猜想

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Four Color Theorem
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Proposed by in Francis Guthrie 1852
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Proven in 1976 by Kenneth Ira Appel (1932-, New York)
and Wolfgang Haken (1928-, Berlin)
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Percy John Heawood (1861-1955, Britain) proved the five
color theorem in 1890

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Hilbert’s problems (23), ICM’1900, Paris
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Millennium Prize Problems(7) by the Clay
Mathematics Institute in 2000
1. P versus NP problem
2. Hodge conjecture
3. Poincaré conjecture (solved by Perelman)
4. Riemann hypothesis
5. Yang–Mills existence and mass gap
6. Navier–Stokes existence and smoothness
7. Birch and Swinnerton-Dyer conjecture
Grigori Perelman (1966-, Russian)
In November 2002, Perelman posted the first of a series of
eprints to the arXiv, ...
He declined to accept
Fields Medal award in 2006
Millennium Prize award in 2010

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pp.64-65（对应英文教材 pp.85-86）: 25, 35,
39, 41
pp.75-76（对应英文教材 p.103）：11, 23, 29
（将中文版“整数”改为“正整数”）, 30
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