December 26, 2004

 


»º º

December 26, 2004

 


1

1

1.1

. . . . 1

1.2

. . . . 2

1



 
 








 

 


 

 
 





 
  






    

  

  





 

 

  

??

  









 



 







1.1



a, b

(a, b)

a

b

(a, b)



 
  



 
 

 
 





1.1 (

)

m

n

n

m



q

r

m = nq + r, 0 ≤ r < n, (m, n) = (n, r)



d = (m, n), h = (n, r)

d | m, d | n ⇒ d | m − nq ⇒ d | r, d | n ⇒ d | h, h | n, h | r ⇒ h | nq + r ⇒ h | m, h | n ⇒ h | d,



d = h



1.1

(851, 299)

x, y

851x + 299y = (851, 299).

  

851 = 299 · 2 + 253 299 = 253 · 1 + 46 253 = 46 · 5 + 23

46 = 23 · 2 + 0,



(851, 299) = (299, 253) = (253, 46) = (46, 23) = (23, 0) = 23

!"



23 = 253 − 46 · 5 = 253 − (299 − 253) · 5 = 253 · 6 − 299 · 5

= (851 − 299 · 2) · 6 − 299 · 5 = 851 · 6 − 299 · 17.



x = 6, y = −17

1.2

#




1.2 (

)

m, n

x 0 , y 0

mx ₀ + ny ₀ = 1.




 

mx + ny = 1

x = x ₀ + nt,

y = y 0 − mt,



t




x ₀ , y ₀

 
&

x, y

mx + ny = 1 mx + ny = 1 = mx ₀ + ny ₀ .



m(x − x ₀ ) = n(y ₀ − y)

m, n

x = x ₀ + nt, y = y ₀ − mt,



t



1.2

23x + 4y = 1

x

y

  

23 = 4 · 5 + 3, 4 = 3 · 1 + 1, 3 = 1 · 3 + 0,

 !"

1 = 4 − 3 · 1 = 4 − (23 − 4 · 5) · 1 = 23 · (−1) + 4 · 6.

#



23x + 4y = 1

x = −1 + 4t,

y = 6 − 23t,



t



1.1

(299, 481) =

299x + 481y = (299, 481)







1.2

n

233 · n

377

1

n



1.3

n

F _n = 2 ^{2 n} + 1

m

n

F _m



F n



1.4

m, n

(2 ^m − 1, 2 ⁿ − 1) =

 '



1.5

a

n

p

a ⁿ ≡ 1 (mod p).



d

a ^d ≡ 1 (mod p)

d|n.



1.6

a, b

ax + by = n

(1)

n

ab − a − b

x, y

(2)

n = ab − a − b

x, y

¹

(3)

m

n

m

n

 ' 
('

)


%
 
*+


* , *
*  


*

% * 
(

 -.
   
/!
 


'
* , * 
((

0

1 n ∈ {0, 1, 2, · · · , ab − a − b}

ax + by = n

n

Þ  

⇒




" 1
 
#Ë


)



a

b

a > b

ö¡ 

2

13

14

100

!
 


(



2$

350


  4
 4+4  

 




a

b

 

(1)

5 × min{a

, b

},

 

min {x, y}

x

y

(2)

1970

²

2.078 × {1 + log(max{a, b})} ,

 

max {x, y}

x

y

2



Journal of Number Theory 2 (1970), pp. 414-422.