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px

np

np

p

p

a

p

=

1 x Z x2 a p1 x Z x2 a p0 a 0 p

a

p

(a)p12 p

p

q

p

q

=

q

p

p q 3

4

p

q

=

q

p

a b p a

p = b

p

ab

q

=

a

q

b

q

a+n a+ 2n a n Z gcd(a, n) = 1

n n

p

= 1 p > n

n n= 2kqk11 qk22 qkrr

P > n n

P

= 1

pn

p

=

2

p

kq1

p

k1 q2p

k2

qr

p

kr=

2

p

k

n

ki k

P > n P 3 8 n

P

=

2

P

k= (1)kP

218 = 1.

P > n

8k+ 3

8k+ 3

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pi i

8k+ 3 pn

N=

n

i=1

p2i + 2.

gcd(N, pi) = 1i pi 1 i n 8k+ 3 p 3 8 p | N p | N p 1 5 7 8

ni=1p

2i + 2 0 p p | N

=ni=1

p2i 2 p =n

i=1p2i

p

= 1 =

2p

=2

p

=

1p

2

p

= (1)p12 (1)p

218 = 1

p 1 8 p 8k+ 3

N

8k+ 1

8k + 1 8k + 1 N 3 8

8k + 1 8k + 3

ki k1

P > n P 1 8 P t q1 t q1 P 1 qi 2 i r

n

p

=

2

p

k q1

p

k1 q2p

k2

qr

p

kr=

q1

p

k1

=P

q1k1

= t

q1k1

= (1)k1

= 1. P t

n

t+ q1k > q1 t q1 8 q1 q2 qr P

P > n nP

= 1

n n

p

= 1 p > n

n= 2k

qk11 q

k22 q

krr

P > n n

P

= 1

p

n

p

=

2

p

kq1

p

k1 q2p

k2

qr

p

kr=

2

p

k

n

ki k

P > n P 1 8

nP = 2

Pk

= (

1)k

P218 = 1.

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P > n

8k+ 1

8k+ 1

pi

i

8k+ 1 pn

M =

ni=1

pi ; N=M4 + 1

gcd(N, pi) = 1i N p | N N=

M22

+ 12 p 4k + 1 8k + 5 8k+ 1 pi| N p 8k+ 5

M4 + 1 M4 + 2M2 + 1 2M2 0 p= M

2 12 2M2 0 p

= M2 12 2M2

p

M2 12

p

= 1 =

2M2p

=

1p

2

p

M2

p

= (1)p12 (1)p

218

p

8k+ 5

1 = (1) (1) = 1 p 8k+ 1 p1, p2, , pn

x4

1 p p 1 8

ki k1

P > n P 1 8 P s q1 s q1 P 1 qi 2 i r

n

p

=

2

p

k q1

p

k1 q2p

k2

qr

p

kr=

q1

p

k1

=

P

q1

k1=

s

q1

k1= (1)k1 = 1.

P

s+ q1k > q1 s q1 8 q1 q2 qr P

px

np

n= m2

m Z px

n

p

=px

1 d(n) =(x) d(n)

d(n)

n

(x)

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n px

n

p

n

px

n

p

=o ((x))

x

px

n

p

=px

p

n

; = {1, 1}

p x a + kn a n

px

p

n =

apamodnx

a

na,n(x)

a,n(x) a+ n a+ 2n a+ 3n x

a,n(x) 1(n)

(x)

px

p

n

a

pamodnx

a

n

1(n)

(x).

px

pn

a

pamodnx

an

(x)(n)

< a

pamodnx

an

1(n)

(x).

px

n

p

apamodnx

a

n

(x)(n)

< a

pamodnx

1.(x)

(n)

(n) a

pxn

p apamodnx

a

n(x)

(n) < (n)(x)

(n) =(x)

=px

n

p

N n a2 p = n

a

2

+kp

p

N

x

2

, y

2

N x < y

y2 x2 = (y+x) (y x) = p.1 1 p y x = 1 y+ x = p x y x, y > N a2 +kp

12

px

n

p

=o ((x)) = PR(n) =PNR(n) = 1

2.

px

np

=o ((x)) 12

12

px

np

= o ((x))

PR(n) = 1

2=PNR(n) =

px

n

p

=o ((x))

n n 1 4. px

n

p

= 0

x

px

n

p

=px

pn

.

n

12

limxpx

pn

= 0

+1

1

n 3 4 px

n

p

=

px

p1mod4

pn

px

p3mod4

pn

.

n

px

pn =

pxp1mod4

pn+

pxp3mod4

pn = 0

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=px

p3mod4

pn

=

px

p1mod4

pn

n 3 4 px

n

p

= 2

px

p1mod4

pn

=px

n

p

= 2

px

p3mod4

pn

.

px

np

0

x n

n

p

n

p

n

p

= 1

n

P n

P

= 1

limx

px

n

p

= lim

x

pP

n

p

+

P P

limx

pP

n

p

+

P

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Q

a1,

a2, , al

Q

http://en.wikipedia.org/wiki/Quadratic_reciprocityhttp://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithmetic_progressionshttp://en.wikipedia.org/wiki/Prime_number_theorem