December 26, 2004



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December 26, 2004

 


1

1

1.1

. . . . 1

1.2

. . . . 1

1.3 N = 3

. . . . 2

1

1.1



N





N

N = 5, 6, 7

  
  

N

)

(1)

²⁰ ₃ , ³ ₂

⁴¹ ₆

5

N = 5



(2)

3, 4

5

6

N = 6

(3)

²⁴ ₅ , ³⁵ ₁₂

³³⁷ ₆₀

7

N = 7



20 3

3 2 41 6

5

4

3 5

6

24 5

35 12 337 60

7



N = 157

= 6803298487826435051217540 411340519227716149383203



= 411340519227716149383203 21666555693714761309610



= 224403517704336969924557513090674863160948472041 8912332268928859588025535178967163570016480830



D. Zagier

6803298487826435051217540 411340519227716149383203

411340519227716149383203 21666555693714761309610 224403517704336969924557513090674863160948472041

8912332268928859588025535178967163570016480830

157

1.2



1225

 ₄₁

12

 ₂

− 5 =  ₃₁

12

 ₂

 ₄₁ ,

12

 ₂

+ 5 =  ₄₉

12

 ₂ .

 


 


  




 


N

x, y, z

x ² − N = y ² , x ² + N = z ² .












1.1

N

(1) N

(2)

N



(1) ⇒ (2)

p, q, r

p ² + q ² = r ² , pq = 2N

 _p+q

2

 ₂

− N =  _r

2

 ₂

 _p−q ,

2

 ₂

+ N =  _r

2

 ₂ .




N

(1) ⇐ (2)

N

x, y, z

x ² − N = y ² , x ² + N = z ² .


 

N

√ x ² + N + √ x ² − N

√ x ² + N − √ x ² − N 2x

N

1.3 N = 3

1.2 N = 3



N = 3

x q , y

q , z q




3

x, y, z, q



x, y, z

z ² = x ² + y ²

m, n

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

x = m ² − n ² , y = 2mn, z = m ² + n ² ,

mn(m ² − n ² ) = 3q ²

3

,



m, n, m ² −n ²

m, n

mn(m ² −n ² ) = 3q ²

 

(1)

3 | m

mn(m ² − n ² ) = 3q ²

m ² − n ²



m ² − n ² = q ² ₁

n ² + q ₁ ² = m ²

3 | n

3 | q ₁

3 | (m, n)

3 | m, n ² + q ² ₁ = m ²

m, n

(2)

3 | n

mn(m ² −n ² ) = 3q ²

m = a ² , n = 3b ² , m ² −n ² = c ²



a, b, c

a ⁴ − 9b ⁴ = c ²

??

(3)

3 | m ² − n ²

m, n

3

mn(m ² − n ² ) = 3q ²

m = a ² , n = b ² ,

a, b

3
ab,

(a ² + b ² )(a ² − b ² ) = a ⁴ − b ⁴ = 3q ₂ ² , q 2

. (1.1)



a, b

3

(a ² − b ² , a ² + b ² ) = 1,

3 | a ² − b ² . (1.2)



(1.1)

(1.2)

a ² + b ² = q ₃ ² ⇒ a, b

3

.



a, b

3

 

N = 3



1.1

(1)

N = 1

x ⁴ − y ⁴ = z ²

(2)

N = 1



1.2

N = 2



1.3

N = 14



1.4

N = 15



1.5

N = 41



1.6 (5/2, 1/2, 7/2)

N = 6





 

 


 
 

 

d 1 , d 2

 








3a

5a

a


Æ


 Æ
 



N = 7

N = 1, 2, 3, 4


$%

N = 13, 14, 15

"

 
'


  (


P

8

3

P

2P

-

-

N

N ≡ 5, 6, 7 (mod 8)

N




 
 




E N : y ² = x ³ − N ² x

N

E N



(x, y)

+







n

N = 6(1 ² + 2 ² + 3 ² + · · · + n ² )

N = 8n ³ − 2n



 
$

   
 


 

a, b, c



&
 

 

u, x, y

⎧ ⎪

⎨

⎪ ⎩

a = ^{u 2 +x} _x ² , b = ^{u 2 +y} _y ² ,

c = ^{u 2 −x} _x ² + ^{u 2 −y} _y ² .




.