December 28, 2004

 


»º º

December 28, 2004

 


1

Elliptic Equation

1

1

Elliptic Equation

1.1

y ² = x ³ + 23

x

y

 

x

y

⎧ ⎪

⎨

⎪ ⎩



2 ≡ 0, 1 (mod 4),



3 ≡ 0, 1, 3 (mod 4), y ² ≡ x ³ + 23 (mod 4).

⇒

 y ² ≡ 0 (mod 4), x ³ ≡ 1 (mod 4).

⇒

 2|y,

x ≡ 1 (mod 4).

 

4(y/2) ² + 4 = x ³ + 3 ³ = (x + 3)(x ² − 3x + 9).



x ≡ 1 (mod 4),

x ² − 3x + 9

4

3

4



3

p

x ² − 3x + 9

p | (y/2) ² + 1 ⇒ (y/2) ² ≡ −1 (mod p)

⇒ (y/2) ⁴ ≡ 1 (mod p).





4 | (p − 1) ⇒ p ≡ 1 (mod 4).



p



1.1

y ² = x ³ − 5

x

y



1.2

y ² = x ³ + 11

x

y

¹



1.3

y ² = x ³ − 17

x

y



1.4

y ² = x ⁴ + x ³ + x ² + x + 1

x

y

(1)

x < −1

x > 3

(2x ² + x) ² < 4x ⁴ + 4x ³ + 4x ² + 4x + 4 < (2x ² + x + 1) ² .

(2)

(1)

x

y

1 ??



1.5




   

y ² + y = x ³ − x



x

y

(1, −1)

(2, 2)



 

 

2

 

 



























 

 


 

 

- -





y ² = x ³ + 7.

(1)

(x, y)

x

(2)

y ² + 1 ² = (x + 2)((x − 1) ² + 3)

4

 

3

1869



  
  






n



 
 
 
Æ




 



2



(2, 2)

Quantum January/February 1997, pp. 5-10



  

-


" 
 


n



n = 3





n


% #


 !

#
 &

 '
  


 



1 1 2



n

n = 2

n = 4

n = 6

!  

  
! 

n = 4




n = 6

n = 8

n = 10

  '
 


n = 12, 14, 16, 18

n = 20, 24, 28



DZ 

n = 0, 14, 34

n ≡ 4, 8, 20, 24, 28 (mod 34).