December 26, 2004

 


»º º

December 26, 2004

 


1

1

1.1

. . . . 1

1.2

. . . . 1

1.3

. . . . 2

1




 








 











  




  






DZ


1.1

1.1 (

)

(1)

n

m

n ≥ m + 1

2

(2)

n

m

n ≥ rm + 1

r + 1




 

n

m

n ≥ rm + 1

(r + 1)

1.2



x

x

1

{x}

x = 2.345 → x = 2 + 0.345 → {x} = 0.345, y = −1.62 → y = −2 + 0.38 → {y} = 0.38, z = √

2 → z = 1 + ( √

2 − 1) → {z} = √ 2 − 1.

1.2

N

α

m, n

0 < m + nα < 1

N .



α

nα

n



(0, 1)

N

0, 1

N

 ,

1 N , 2

N

 , · · · ,

N − 1 N , 1

 .


 

N + 1

{1 · α}, {2 · α}, · · · , {N · α}, {(N + 1) · α} ∈ (0, 1)



2

1 ≤ n ≤ N + 1

{n ₁ α}, {n ₂ α}



⎧ ⎪

⎨

⎪ ⎩

n ₁ α = m ₁ + {n ₁ α}

n ₂ α = m ₂ + {n ₂ α}

|{n 1 α} − {n 2 α}| < _N ¹

⇒ |(m ₁ − m ₂ ) − (n ₁ − n ₂ )α| < 1 N .

  

m = m ₁ − m ₂ , n = −n ₁ + n ₂

m = −(m ₁ − m ₂ ), n = n ₁ − n ₂

0 < m + nα < 1

N .

1.3

DZ

5

n

m

mn

 

6

7

n = 23

m = 29

29 × 23 = 667

1990

^{1 2}








1.3

n

m

mn

1

0

n = 7

m = 143

143 × 7 = 1001



n + 1

n + 1

1, 11, 111, · · · ,

  n

111 · · · 1,

  n+1

111 · · · 1



n

n + 1


 


n

a =

  i

111 · · · 1, b =

  j

111 · · · 1; i < j



n | (b − a) =

  j−i

111 · · · 1

  i

000 · · · 0



1.1

3 × 4






√ 5

1 G. Berzsenyi, The Roseberry conjecture, Quantum (May 1990).

2 G. Berzsenyi, At siexes and sevens, Quantum (November 1990).

1.2

 

  
" 



3

1.3

1





12 π

1.4

 

7

1.5

N, p

q

α

m, n



m + nα − q p

  < 1 N .

1.6

m

17 · m

1

0

m



1.7

m

29 · m

1

0

m



' 

7 × 7

49

)
 
 (


 (* 

 (


(

  (


( ((



 
(

4 × 4

( (

(



3

º



+

2n



4

 # 

DZ

 #  
 #



 #
! 

n

*

n

d n

n = 2

√

2


 

d 2 = √

2

n = 4





d ₄ = 1

d _n

d ₂ = √

2, d ₃ = √

6 − √ 2, d ₄ = 1, d ₅ = ^√ ₂ ² , d ₆ = ^√ ₆ ¹³ , d ₇ = 4 − 2 √

3, d ₈ = ^{√ 6−} ₂ ^{√ 2} , d ₉ = ¹ ₂ , d 14 = ^{√ 6−} ₃ ^{√ 2} , d 16 = ¹ ₃ , d ₂₅ = ¹ ₄ .



d 10

0.421

n

d n



√

2

√ 4

3 · √ n






d 3 = √ 6 − √

2

4