Mathematics Appreciation
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Mathematics Appreciation. 数学欣赏. 主讲:张文俊. 第六章 数学问题. 几个著名数学问题的历史与现状. 名人语录. 问题是数学的心脏。 —— P.R.HALMOS 意义深刻的数学问题从来都不是一找出答案就完事了。 …… 每一代数学家都重新思考并重新改造他们的前辈所发现的解答,并把这些解答纳入当代流行的概念和符号体系之中。 ——L. BERS.

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Mathematics Appreciation



P.R.HALMOS

L. BERS



;


zwj@szu.edu.cn


In This Section

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=

2=

51882

zwj@szu.edu.cn


1



Zenon, 490---429


  • :

    .


2


1.

5Anaxagoras, 500428



=


=


1882C.L.F. Lindemann18521939


2.

429Delos


?

2=


8



1837P.L. Wantzel, 1814--1848


3.

4


N

P

H

S


N

P

H

S


N

P

H

S

aSHSH

a

?



1837


3


1

2

3

4

5


17


1837P.L. Wantzel, 1814--1848



1

2

3a = cosA, A/3, x = cos(A/3), x


1

1837

2 1882C.L.F. Lindemann18521939


4

=


=

3




90


5


  • 1:

  • .


  • 2:


  • 3:


.


Fermat

zwj@szu.edu.cn


Fermat

16371994


1



  • Pierre de Fermat, 1601---1665,1601820-

    30



  • 17


  • 16701679


1621


8

x2 + y2 = z2

m, n2mn


1637


xn + yn = zn (1)

n > 2

1670


2

n=3,4


1707-1783

Leonhard Euler(1707 - 1783)

  • 18

  • 13173060


n=4

1738

1(a,b,c),

a4 + b4= c4(2)

c (a, b, c)(a1, b1, c1) c1 < cc


n=4

x4 + y4 = z23

2

3


n=3

1753n=3


Carl Friedrich Gauss(1777 - 1855)

  • a + b k1/2i abk


n=3n=3


n = 5

Legendre (1752 - 1833)

  • 1823 n = 5


n = 5

Dirichlet (1805 - 1859)

  • 1828 n = 5

  • 1832 n = 14


n = 7

Gabriel Lam (1795 - 1870)

  • 1839 n = 7

  • 1847


3


Sophie Germain (1776 - 1831)

  • (I)n xyz(II)n xyz


1831

x, y, zn100

n100 x, y, zx, y, znxn+yn=zn


Ernst Edward Kummer (1810 - 1893)

  • 1845 1847

  • n < 100

  • 1857


1847x, y, zn18573000

n100xn+yn=zn


4


100200

  • 1816

  • 1850185320001900 n100206


10

Paul Friedrich Wolfskehl (1856 - 1908)

  • 1883

  • 10100


  • 1941n < 253747887

  • 1977 n < 125000


5


19832810


xn + yn = zn

xn + yn = 1


xn + yn = 1

xn + yn = 1


xn+yn=1


1

2y2 =x3+ax+b, a,bx3+ax+b=0

3 xn + yn = 1n > 2)


1922


xn+yn=1

1983.

xn+yn=1


6



--

1950--:


19271958

1926

--


1985

n >2--

--n >2


--


Andrew Wiles:

1986Andrew Wiles1953---8

Wiles19949

--


Andrew Wiles

  • 1953

  • 10

  • 1975


  • 1993 6 23 --


  • 19939


  • 1994 1 9

  • 9 19


  • 1995 5 100 Annals of Math.


  • 1997 62710


1996

  • Ostrowski

  • Schock

  • Wolf


1997

  • 19988

  • (J.C. Fields)

    20056

  • 2005


7



1. Fermat-Catalan

m, n, k

1/m + 1/n + 1/k < 1

xn + ym = zk

a, b, c.

1995H.Damon A. Granville 10


2. Beal

m ,n, k 3

xn + ym = zk

222a, b, c

Andrew Beal555


Goldbach

zwj@szu.edu.cn


Goldbach

(1742 ? !)

n > 22n=p+q,

p,q


1



  • 5=5=4+1=3+2=3+1+1=2+2+1

    =2+1+1+1=1+1+1+1+1


p(n)n

P(1)=1, P(2)=2,

P(3)=3, P(4)=5,

P(5)=7, P(6)=11,

P(7)=15,

P(8)=22,

P(100)=190,569,29219

P(200)=3,972,999,029,388(4)



1770(Lagrange, 17361813)


(Hilbert, 18621943)

kc(k),c(k)k.

c(k)


4(m)(n)


:

(mn)1,

n>22n=p+q, p,q


2

Goldbach


C.Goldbach1690--1764,172935

174267


21

4=1+1+1+1=1+1+2=1+3

5=2+3=1+1+3=1+1+1+2=1+1+1+1+1


630

9


:

6

1+11+1


19160

!


3

Goldbach


1.

  • 1912

    4mm


18 m


  • m,n,

    4mn(m+n)

    2N=p1p2pj+q1q2qk, (jm,kn)

    m,n, m=n=1(2N=p1+q1 )


2.


  • (Eratosthenes)

    22235


  • 2020(G.H.Hardy,1877-1947)(Littlewood, 1885-1977).

    1923


  • 2030

    1937


3.

  • 1930,25

    4m

    m800000.


1935 m 2208 ( )

1936 m 271 ( ,,)

1937 m 267 ( )

1950 m 220 ( ,)

1956 m 218 ( )

1976 m 26 ( )


  • m,n,

    4mn(m+n)

    2N = p1p2pj + q1q2qk,

    (jm,kn)

    m,n, m = n =1


19209+9;

19247+7

19326+6

1937(5+7,(4+9)


193819405+54+4

19572+3

19621+4

19651+3


19661+2

:

2n=p+q, 2n=p+q1 q2

1+1


4

Goldbach


19331996

  • 1933522

  • ,GoldbachGoldbach


1966571+2

2

2001973


10

1+1



zwj@szu.edu.cn


18521976


1




2


1852Francis Guthrie


  • Francis GuthrieFrederick Guthrie

  • Frederick GuthrieA.De Morgan(18061871)

  • A.De Morgan18521023W.R.Hamilton(1805-1865),



De MorganHamilton13


  • 1878613A.Cayley (1821-1895)

  • 1879A.De Morgan

  • Cayley


W.R, (HamiltonWilliam Rowan)1805 8 4 1865 9 2 (Dublin)


A. (CayleyArthur) 1821 8161895 1 26


3


CayleyCayleyCayley


nn-1

n-1n


Cayley1879

11



4


11189029P.J.Heawood, 1861--19551879



:

f e v



1


25


6 fk k


kkfk e v




3


  • f =2,3,4,5

  • f k k f =k+1

  • 255


AA

A

3A


1AAab2Aab3Aab


aAbk-1456


A553A4aAbabA5k+15


5


Minkovski(1864-1909)



(MinkowskiHermann)1864 6 22 ()1909 1 12


6


20

  • 1913

  • 1920Franklin25

  • 192627


  • 1936Franklin31

  • 1940Winn35

  • 1968O.Ore40

  • 197552


7


700


19769IllinoisK.I.AppelW.Haken3IBM36012009Illinois J. Math. V.21


Appel

Four Coulors Suffice



19851


2000



Poincare

zwj@szu.edu.cn


2000524

Clay Mathematics Institute


1. RiemannRiemann

Riemann(Bernhard Riemann,18261866)


Riemann


P(x)= anxn + an-1xn-1 + + a1x + a0

P(x)= an(x-x1) (x-x2) (x-xn)


s

2, -4, -6 ,


Riemann1859

1/2

Riemann


Im

:

-2, -4, -6, .

0

1/2

1

Re


G.F.B., (RiemannGeorg Friedrich Bernhard)1826 9 17 (Breselenz)1866 7 20 (Selasca)


2. Poincare

(Poincare)


3. PNP

,



(StephenCook)1971


P()NPPNP? P NP. 200


4. Hodge

PoincareHodge


5. Yang-Mills

YangMills


Yang-Mills

Yang-Mills

GR4GYang-Mills


6. Navier-Stokes

Navier-StokesNavier-StokesRn (n=2 3) x Rnt 0 u(x,t) = (ui(x,t))1in Rnp(x,t) RnNavier-Stokes


7. BirchSwinnerton-Dyer

x2 + y2 = z2


AbelBirchSwinnerton-DyerZeta (s) s = 1

BirchSwinnerton-Dyer

Abel(1) = 0 (1) 0


Thank You !


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