Đề tài bất phương trình diophante tuyến tính

✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆

❚❘❺◆ ❚❘×❮◆● ❙■◆❍

❇❻❚ P❍×❒◆● ❚❘➐◆❍
❉■❖P❍❆◆❚❊ ❚❯❨➌◆ ❚➑◆❍
❈❤✉②➯♥ ♥❣➔♥❤✿

P❍×❒◆● P❍⑩P ❚❖⑩◆ ❙❒ ❈❻P

▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✶✸

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❑❍❖❆ ❍➴❈

◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈
●❙✳❚❙❑❍ ◆●❯❨➍◆ ❱❿◆ ▼❾❯

❍⑨ ◆❐■ ✲ ✷✵✶✺


▼ö❝ ❧ö❝
▼ð ✤➛✉

✷

✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✹

✶✳✶

×î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t✳ ❚❤✉➟t t♦→♥ ❊✉❝❧✐❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷

▲✐➯♥ ♣❤➙♥ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✸

P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✶✳✸✳✶

❚➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❞ü❛ ✈➔♦ ❣✐↔♥ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽

✶✳✸✳✷

❚➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❞ü❛ ✈➔♦ t❤✉➟t t♦→♥ ❊✉❝❧✐❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

◆❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳

✷✹

✶✳✹

✷ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤

✷✻

✷✳✶

❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✻

✷✳✷

❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✧❜à ❝❤➦♥✧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✵

✷✳✸

◆❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✳ ✳ ✳

✸✹

✷✳✸✳✶

▼ët sè ✈➼ ❞ö ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✺

✷✳✸✳✷

❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ ❞↕♥❣ ❧✐➯♥ ♣❤➙♥ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✶

✸ ▼ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥
✸✳✶

✹✸

◆❣❤✐➺♠ ♥❣✉②➯♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✱
❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❧÷ñ♥❣ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✳✷
✸✳✸

✹✸

P❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤
❧÷ñ♥❣ ❣✐→❝ ❝â ✤✐➲✉ ❦✐➺♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✼

❳→❝ ✤à♥❤ ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tr÷î❝ ✳ ✳ ✳ ✳ ✳ ✳

✺✻

❑➳t ❧✉➟♥

✻✹

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✻✺
✶



Pữỡ tr ỏ ồ ữỡ tr t
ởt tr ỳ t ớ t ừ ồ ổ q
ữỡ tr t t ồ t r ữủ ỳ t t
s s ừ số số ỳ t số số ữỡ tr t
ữ sỹ r ớ ừ số ỵ tt ữớ t ỵ tt
t t ữ ữỡ số ồ r
t ữỡ tr t t t tỹ t ữỡ tr
t t t õ ự t số õ t õ ởt t
ợ ữ ờ tr ý t ồ s ọ ờ tổ
r t ổ õ t ồ qt t
t ữỡ tr t t t ừ s ự
t ữỡ tr ợ ố ồ
s ởt t ờ t ổ ồ s tr
q tr ổ t ồ s ọ
ữủ ữỡ
ữỡ ởt số tự
ữỡ t ữỡ tr t t t
ữỡ ởt số t q

t tọ sỹ trồ ỏ t ỡ s s tợ
tớ ữợ ụ
ữ t ừ ồ trỏ tr sốt q tr ồ t
ự ú ù t t
ụ ỷ ớ ỡ t t tợ
Pỏ t ồ ỡ ồ t ổ
t t ủ t õ t t ử ừ



ỡ ổ q t ở ờ
ụ t tốt t t tr sốt tớ t ồ
t t trữớ ồ ồ ỹ ồ ố ở
ũ õ ố ữ tớ tr ở ỏ
õ tr ọ ỳ t sõt t rt
ữủ sỹ õ ỵ ừ t ổ ụ ữ ỗ
ữủ t ỡ
t ỡ
ở t
ồ tỹ

r rữớ




❈❤÷ì♥❣ ✶
▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶ ×î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t✳ ❚❤✉➟t t♦→♥ ❊✉❝❧✐❞
✣à♥❤ ♥❣❤➽❛ ✶✳✶ ✭①❡♠ ❬✶❪✮✳ ❙è ♥❣✉②➯♥ ❝ ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ÷î❝ sè ❝❤✉♥❣ ❝õ❛ ❤❛✐ sè

♥❣✉②➯♥ ❛ ✈➔ ❜ ✭❦❤æ♥❣ ✤ç♥❣ t❤í✐ ❜➡♥❣ ❦❤æ♥❣✮ ♥➳✉ ❝ ❝❤✐❛ ❤➳t ❛ ✈➔ ❝ ❝❤✐❛ ❤➳t ❜✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷ ✭①❡♠ ❬✶❪✮✳ ▼ët ÷î❝ sè ❝❤✉♥❣ ❞ ❝õ❛ ❤❛✐ sè ♥❣✉②➯♥ ❛ ✈➔ ❜ ✭❦❤æ♥❣ ✤ç♥❣
t❤í✐ ❜➡♥❣ ❦❤æ♥❣✮ ✤÷ñ❝ ❣å✐ ❧➔ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ ❛ ✈➔ ❜ ♥➳✉ ♠å✐ ÷î❝ sè ❝❤✉♥❣
❝ ❝õ❛ ❛ ✈➔ ❜ ✤➲✉ ❧➔ ÷î❝ ❝õ❛ ❞✳

❈❤ó þ ✶✳✶✳ ◆➳✉ ❞ ❧➔ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ ❛ ✈➔ ❜ t❤➻ −d ❝ô♥❣ ❧➔ ÷î❝ sè ❝❤✉♥❣ ❧î♥
♥❤➜t ❝õ❛ ❛ ✈➔ ❜✳ ❱➟② t❛ q✉② ÷î❝ r➡♥❣ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ ❛ ✈➔ ❜ ❧➔ sè ♥❣✉②➯♥
❞÷ì♥❣✳
×î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ ❤❛✐ sè ❛ ✈➔ ❜ ✤÷ñ❝ ❦þ ❤✐➺✉ ❧➔ ✭❛✱❜✮ ❤❛② ❣❝❞✭❛✱❜✮ ✭❣r❡❛t❡st
❝♦♠♠♦♥ ❞✐✈✐s♦r✮✳ ◆❤÷ ✈➟② ❞ ❂ ✭❛✱❜✮ ❤❛② ❞ ❂ ❣❝❞✭❛✱❜✮✳

❱➼ ❞ö ✶✳✶✳ ✭✷✺✱✸✵✮ ❂ ✺✱ ✭✷✺✱✲✼✷✮ ❂ ✶✳
✣à♥❤ ♥❣❤➽❛ ✶✳✸ ✭①❡♠ ❬✶❪✮✳ ▼ët sè ♥❣✉②➯♥ ❝ ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ÷î❝ sè ❝❤✉♥❣ ❝õ❛ ♥ sè
♥❣✉②➯♥ a1 , a2 , a3 , . . . , an ✭❦❤æ♥❣ ✤ç♥❣ t❤í✐ ❜➡♥❣ ❦❤æ♥❣✮ ♥➳✉ ❝ ❧➔ ÷î❝ ❝õ❛ ♠é✐ sè ✤â✳

✣à♥❤ ♥❣❤➽❛ ✶✳✹ ✭①❡♠ ❬✶❪✮✳ ▼ët ÷î❝ sè ❝❤✉♥❣ ❞ ❝õ❛ ♥ sè ♥❣✉②➯♥ a1, a2, a3, . . . , an
✭❦❤æ♥❣ ✤ç♥❣ t❤í✐ ❜➡♥❣ ❦❤æ♥❣✮ ✤÷ñ❝ ❣å✐ ❧➔ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ a1 , a2 , a3 , . . . , an
♥➳✉ ♠å✐ ÷î❝ sè ❝❤✉♥❣ ❝ ❝õ❛ a1 , a2 , a3 , . . . , an ✤➲✉ ❧➔ ÷î❝ ❝õ❛ ❞✳
❚÷ì♥❣ tü✱ t❛ ❝ô♥❣ q✉② ÷î❝ r➡♥❣ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ ♥ sè ♥❣✉②➯♥ a1 , a2 , a3 , . . . , an
❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ×î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ a1 , a2 , a3 , . . . , an ❦þ ❤✐➺✉ ❧➔ ✭a1 , a2 , a3 , . . . , an ✮
❤❛② ❣❝❞✭a1 , a2 , a3 , . . . , an ✮✳ ◆❤÷ ✈➟② ❞ ❂ (a1 , a2 , a3 , . . . , an ) ❤❛② ❞ ❂ ❣❝❞✭a1 , a2 , a3 , . . . , an ✮✳
✹


✣à♥❤ ❧➼ ✶✳✶✳ ✭✈➲ sü tç♥ t↕✐ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ ♥❤✐➲✉ sè✱ ①❡♠ ❬✶❪✮ ❈❤♦ ❝→❝ sè
♥❣✉②➯♥ a1 , a2 , a3 , . . . , an ❦❤æ♥❣ ✤ç♥❣ t❤í✐ ❜➡♥❣ ❦❤æ♥❣✳ ❑❤✐ ✤â tç♥ t↕✐ ÷î❝ sè ❝❤✉♥❣ ❧î♥
♥❤➜t ❝õ❛ a1 , a2 , a3 , . . . , an ✳

❚➼♥❤ ❝❤➜t ✶✳✶ ✭①❡♠ ❬✶❪✮✳ ❈❤♦ ❛✱ ❜✱ q✱ r ❧➔ ❝→❝ sè ♥❣✉②➯♥ ✭a2 + b2 = 0✮✳ ◆➳✉ a = bq + r

✈➔ 0 ≤ r < |b| t❤➻ ✭❛✱❜✮ ❂ ✭❜✱r✮✳

❚❤✉➟t t♦→♥ ❊✉❝❧✐❞ ✭t❤✉➟t t♦→♥ t➻♠ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ ❤❛✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✮✳

●✐↔ sû r0 = a, r1 = b ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ❚❛ →♣ ❞ö♥❣ ❧✐➯♥ t✐➳♣ t❤✉➟t t♦→♥ ❝❤✐❛

ri = ri+1 qi+1 + ri+2 ,
tr♦♥❣ ✤â 0 ≤ ri+2 < ri+1 , ∀i = 0, 1, 2, . . . ✈➔ ♥❤➟♥ ✤÷ñ❝ ❝→❝ ♣❤➛♥ ❞÷ r1 , r2 , . . . ✈î✐

r1 > r2 > . . . ✤➳♥ ❦❤✐ ❧➛♥ ✤➛✉ t✐➯♥ ♥❤➟♥ ✤÷ñ❝ ♣❤➛♥ ❞÷ rn = 0 ✭n ≥ 2, 0 < ri+2 <
ri+1 , ∀i = 0, 1, . . . , n − 3✮✳ ❑❤✐ ✤â
(a, b) = (r0 , r1 ) = (r1 , r2 ) = . . . = (rn−2 , rn−1 ) = (rn−1 .qn−1 , rn−1 ) = rn−1 .
❱➟②

(a, b) = rn−1 .

❱➼ ❞ö ✶✳✷✳ ❉ò♥❣ t❤✉➟t t♦→♥ ❊✉❝❧✐❞ t➻♠ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ ✸✹✽✹ ✈➔ ✸✷✼✻✳
▲í✐ ❣✐↔✐✳

❚❛ ❝â

3484 = 3276.1 + 208
3276 = 208.15 + 156
208 = 156.1 + 52
156 = 52.3 + 0.
❱➟②

gcd(3484, 3276) = 52.

❱➼ ❞ö ✶✳✸✳ ❚➻♠ ♠ët ❝➦♣ sè ♥❣✉②➯♥ ①✱ ② ✤➸

3484x + 3276y = 52.
▲í✐ ❣✐↔✐✳

❚❤❡♦ ✈➼ ✈ö tr➯♥ t❛ ❝â

✺


52 = 208 156.1
52 = 208 (3276 208.15) .1 = 16.208 3276
156 = 3276 208.15
52 = 3276 + 16.208
52 = 3276 + 16. (3484 3276.1) = 16.3484 17.3276
208 = 3484 3276.1
õ

3484.16 + 3276.(17) = 52.


(x; y) = (16; 17).

số
số ỳ số ỳ õ ở
n N tự õ

1

a0 +

1

a1 +

1

a2 + . . . +

an1 +

1
an

tr õ a0 số ai số ữỡ i = 1, 2, . . . , n an > 1 ợ

n > 0 số tr ữủ ỵ [a0 ; a1 , a2 , . . . , an ]

số ổ a0, a1, a2, . . . ổ
số ai > 0 ợ i 1 ợ ộ t Ck = [a0 ; a1 , a2 , . . . , ak ] õ tỗ t
ợ

lim Ck = .

k+

) ữợ

ỹ tỗ t s ữủ õ ró tr t t (

ú t ồ tr ừ số ổ [a0 ; a1 , a2 , . . .] ỵ

= [a0 ; a1 , a2 , . . .] .

t ộ số ỳ t ởt số ỳ
ự

a
, b > 0, a, b Z t r0 = a, r1 = b t õ
b
r0 = r1 q1 + r2
(0 < r2 < r1 )
r1 = r2 q2 + r3
(0 < r3 < r2 )
...
rn2 = rn1 qn1 + rn
(0 < rn < rn1 )
rn1 = rn qn
+0

sỷ x =

❙✉② r❛

x=

a
r0
r2
1
1
1
=
= q1 +
= q 1 + r1 = q 1 +
= . . . = q1 +
r
3
1
b
r1
r1
q2 +
q2 + . . . +
r2
r2
1
qn−1 +
qn

⇒ x = [q1 ; q2 , . . . , qn ] .

243 62
❱➼ ❞ö ✶✳✹✳ ❍➣② ❜✐➸✉ ❞✐➵♥ ❝→❝ sè ❤ú✉ t✛ 327 ✱ 243
✱−
✱
t❤➔♥❤ ❧✐➯♥ ♣❤➙♥ sè✳
37
37 23
▲í✐ ❣✐↔✐✳

❚❛ ❝â

32 = 4.7 + 4
7 = 1.4 + 3
4 = 1.3 + 1
3 = 3.1
⇒

32
= [4; 1, 1, 3] = 4 +
7

1

❧➔ ❧✐➯♥ ♣❤➙♥ sè ❝â ✤ë ❞➔✐ ✸✳ ❚÷ì♥❣ tü t❛ ❝ô♥❣ ❝â

1

1+

1

3
243
243
62
= [6; 1, 1, 3, 5]✱ −
= [−7; 2, 3, 5]✱
= [2; 1, 2, 3, 2]✳
37
37
23
1+

❚➼♥❤ ❝❤➜t ✶✳✸✳ ✭❱➲ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥✱ ①❡♠ ❬✸❪✮ ❙ü ❜✐➸✉ ❞✐➵♥
♠ët sè ❤ú✉ t➾ q ❞÷î✐ ❞↕♥❣ ❧✐➯♥ ♣❤➙♥ sè [a0 ; a1 , a2 , . . . , an ] ❧➔ ❞✉② ♥❤➜t✳

❚➼♥❤ ❝❤➜t ✶✳✹✳ ✭❈æ♥❣ t❤ù❝ t➼♥❤ ❣✐↔♥ ♣❤➙♥✱ ①❡♠ ❬✸❪✮ ❈❤♦ ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥
[a0 ; a1 , a2 , . . . , an ]✳ ❳➨t ❤❛✐ ❞➣② (pk )nk=0 ✈➔ (qk )nk=0 ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉
p 0 = a0
p 1 = a1 a0 + 1
pk = ak pk−1 + pk−2

,

q0 = 1
q 1 = a1
qk = ak qk−1 + qk−2

, ∀k = 2, 3, . . .

pk

❑❤✐ ✤â ❣✐↔♥ ♣❤➙♥ t❤ù ❦ ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè [a0 ; a1 , a2 , . . . , an ] ❧➔ Ck = [a0 ; a1 , . . . , ak ] = ✳
qk
pk
❈❤ù♥❣ ♠✐♥❤✳
❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ Ck = [a0 ; a1 , . . . , ak ] =
❜➡♥❣ q✉② ♥↕♣ t❤❡♦ ❦✱
qk
1
✈î✐ ❧÷✉ þ ❧➔ Ck ✭❦ ❃ ✵✮ ✤÷ñ❝ s✉② r❛ tø Ck−1 ❜➡♥❣ ❝→❝❤ t❤❛② ak−1 ❜ð✐ ak−1 + ✳
ak
❚❤➟t ✈➟②
a0
p0
C0 = [a0 ] = a0 =
=
,
1
q0
1
a1 a0 + 1
p1
C1 = [a0 ; a1 ] = a0 +
=
=
,
a1
a1
q1
✼

a1 +
C2 = [a0 ; a1 , a2 ] =

1
a2

a1 +
=

a0 + 1
1
a2

a2 (a1 a0 + 1) + a0
a2 p 1 + p 0
p2
=
= .
a2 a1 + 1
a2 q 1 + q 0
q2

●✐↔ sû

Ck = [a0 ; a1 , . . . , ak ] =

ak pk−1 + pk−2
pk
= , k ≥ 2.

ak qk−1 + qk−2
qk

❑❤✐ ✤â

ak +
Ck+1 =
ak +
=

1
ak+1
1
ak+1

pk−1 + pk−2
qk−1 + qk−2

ak+1 pk + pk−1
ak+1 (ak pk−1 + pk−2 ) + pk−1
=
.
ak+1 (ak qk−1 + qk−2 ) + qk−1
ak+1 qk + qk−1

❉♦ ✤â

Ck+1 =

pk+1

.
qk+1

❱➟② t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝

Ck = [a0 ; a1 , . . . , ak ] =

pk
.
qk

❱➼ ❞ö ✶✳✺✳ ❚➻♠ ❝→❝ ❣✐↔♥ ♣❤➙♥ ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè [6; 1, 1, 3, 5]✳
▲í✐ ❣✐↔✐✳

❚❛ ❝â ❜↔♥❣ s❛✉
❦
ak
pk
qk

✵
✻
✻
✶

✶
✶
✼
✶

✷
✶
✶✸
✷

✸
✸
✹✻
✼

✹
✺
✷✹✸
✸✼

❱➟②

C0 = 6, C1 = 7, C2 =

❚➼♥❤ ❝❤➜t ✶✳✺

13
46
243
, C3 = , C4 =
.
2
7
37

✳

✭①❡♠ ❬✸❪✮ ❈❤♦ Ck ❧➔ ❣✐↔♥ ♣❤➙♥ t❤ù ❦ ❝õ❛ [a0 ; a1 , a2 , . . . , an ]✱ ✈î✐

✶✳✹)✳ ❑❤✐ ✤â

1 ≤ k ≤ n ✈➔ pk , qk ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ t➼♥❤ ❝❤➜t (

pk qk−1 − pk−1 qk = (−1)k−1 .

❚➼♥❤ ❝❤➜t ✶✳✻ ✭①❡♠ ❬✸❪✮✳ ●✐↔ sû {Ck } ❧➔ ❞➣② ❣✐↔♥ ♣❤➙♥ ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥
[a0 ; a1 , a2 , . . . , an ]✳ ❑❤✐ ✤â t❛ ❝â ❝→❝ ♠è✐ ❧✐➯♥ ❤➺ s❛✉
✽


✐✮ Ck − Ck−1

(−1)k−1
=
✱ ✈î✐ 1 ≤ k ≤ n.
qk qk−1

✐✐✮ Ck − Ck−2 =

ak (−1)k
✱ ✈î✐ 2 ≤ k ≤ n.
qk qk−2

❚➼♥❤ ❝❤➜t ✶✳✼ ✭①❡♠ ❬✸❪✮✳ ❱î✐ ❝→❝ ❣✐↔♥ ♣❤➙♥ Ck ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥ [a0; a1, a2, . . . , an]
t❛ ❝â ❝→❝ ❞➣② ❜➜t ✤➥♥❣ t❤ù❝ s❛✉

✐✮ C1 > C3 > C5 > . . .
✐✐✮ C0 < C2 < C4 < . . .
✐✐✐✮ ♠é✐ ❣✐↔♥ ♣❤➙♥ ❧➫ C2j−1 ✤➲✉ ❧î♥ ❤ì♥ ♠é✐ ❣✐↔♥ ♣❤➙♥ ❝❤➤♥ C2i ✳

❚➼♥❤ ❝❤➜t ✶✳✽ ✭①❡♠ ❬✸❪✮✳ ❱î✐ ♠å✐ k = 0, 1, . . . , n t❤➻ (pk , qk ) = 1 ✭tù❝ ❧➔ pk , qk
♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉✮✳

❚➼♥❤ ❝❤➜t ✶✳✾ ✭①❡♠ ❬✸❪✮✳ ❈❤♦ a0, a1, a2, . . . ❧➔ ❞➣② ✈æ ❤↕♥ ❝→❝ sè ♥❣✉②➯♥✱ ai > 0 ✈î✐
∀i ≥ 1✳ ❱î✐ ♠é✐ ❦✱ ✤➦t Ck = [a0 ; a1 , a2 , . . . , ak ]✳ ❑❤✐ ✤â tç♥ t↕✐ ❣✐î✐ ❤↕♥
lim Ck .

k→+∞

❚❤❡♦ t➼♥❤ ❝❤➜t (

❈❤ù♥❣ ♠✐♥❤✳

✶✳✼) t❛ ❝â

C1 > C3 > C5 > . . . > C2n−1 > C2n+1 > . . .
C0 < C2 < C4 < . . . < C2n−2 < C2n < . . .
C2j−1 > C2i ,
✈î✐ ♠å✐ ✐✱ ❥✳ ❚ø ✤â s✉② r❛ ❞➣② {C2k+1 }✱ k = 0, 1, . . . ❧➔ ❞➣② ❣✐↔♠ ✈➔ ❜à ❝❤➦♥ ❞÷î✐ ❜ð✐

C0 ✱ ❝á♥ ❞➣② {C2k }✱ k = 0, 1, . . . ❧➔ ❞➣② t➠♥❣ ✈➔ ❜à ❝❤➦♥ tr➯♥ ❜ð✐ C1 ✳ ❚❤❡♦ ❧þ t❤✉②➳t ✈➲
❣✐î✐ ❤↕♥ ❝õ❛ ❞➣② sè t❤➻ tç♥ t↕✐ ❝→❝ ❣✐î✐ ❤↕♥

lim C2k+1 = α ,

k→+∞

lim C2k = β.

k→+∞

✶✳✻) t❛ ❝â

❚❤❡♦ t➼♥❤ ❝❤➜t (

C2k+1 − C2k

1
(−1)2k
=
> 0.
=
q2k+1 q2k
q2k+1 q2k
✾

✭❛✮


t t õ


qk k.
s ự q ữ s
ợ t q0 = 1 q0 > 0
ợ t q1 = a1 q1 1 a1 ữỡ

sỷ ú tự qk k.
ợ t õ

qk+1 = ak+1 qk + qk1 .
tt q t qk k, qk1 k 1 ỏ ak 1 t õ

qk+1 k + k 1 k + 1 ợ k 2
ú ợ ợ ồ k = 0, 1, . . . t qk k.
ữ t s r

0<

1



q2k+1 q2k

1
, k 1.
2k (2k + 1)

k + t t ữủ

lim

k+

1
q2k+1 q2k

= 0.



lim (C2k+1 C2k ) = 0.

k+

ứ õ t õ

= .



tự ự tọ tỗ t ợ

lim Ck = .

k+

õ ự

t ứ t t () s r r ự ợ a0, a1, a2, . . . t tỗ t
lim [a0 ; a1 , a2 , . . . , ak ] = lim Ck = õ ỵ s t ồ

k+

k+

số ổ [a0 ; a1 , a2 , . . . , ak , . . .]




❚➼♥❤ ❝❤➜t ✶✳✶✵ ✭①❡♠ ❬✸❪✮✳ ❈❤♦ a0, a1, a2, . . . ❧➔ ❞➣② ✈æ ❤↕♥ ❝→❝ sè ♥❣✉②➯♥✱ ai > 0 ✈î✐
∀i ≥ 1✳ ❳➨t ❧✐➯♥ ♣❤➙♥ sè ✈æ ❤↕♥ α = [a0 ; a1 , a2 , . . .]✳ ❑❤✐ ✤â α ❧➔ sè ✈æ t➾✳
❈❤ù♥❣ ♠✐♥❤✳

●✐↔ sû α ❧➔ sè ❤ú✉ t➾✱ tù❝ ❧➔

a
α = , a, b ∈ Z, b > 0, (a, b) = 1.
b

✶✳✾) t❛ ❝â

❚❤❡♦ ♣❤➛♥ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❝❤➜t (

lim C2k = α

k→+∞

♠➔ ❞➣② {C2k }✱ k = 0, 1, . . . ❧➔ ❞➣② t➠♥❣ ♥➯♥ C2k ≤ α✱ ✈î✐ ♠å✐ k = 0, 1, . . .

lim C2k+1 = α

k→+∞

♠➔ ❞➣② {C2k+1 }✱ k = 0, 1, . . . ❧➔ ❞➣② ❣✐↔♠ ♥➯♥ α ≤ C2k+1 ✱ ✈î✐ ♠å✐ k = 0, 1, . . .

❱➟② t❛ ❝â

C2n ≤ α ≤ C2n+1 , ∀n = 0, 1, . . .
❉♦ α = [a0 ; a1 , a2 , . . .] ♥➯♥ α = Ck ✱ ✈î✐ ♠å✐ k = 0, 1, . . .
❚ø ✤â t❛ ❝â

C2n < α < C2n+1 , ∀n = 0, 1, . . .
1

⇒ 0 < α − C2n < C2n+1 − C2n =
▼➦t ❦❤→❝ ❞♦ C2n =

, ∀n = 0, 1, . . .

p2n
♥➯♥ t❤❡♦ tr➯♥ t❛ s✉② r❛
q2n
0 < αq2n − p2n <

❚❤❛② α =

q2n+1 q2n

1
q2n+1

, ∀n = 0, 1, . . .

a
t❛ ✤✐ ✤➳♥

b
0 < aq2n − bp2n <

b
q2n+1

≤

b
, ∀n = 0, 1, . . .
2n + 1

❉♦ aq2n − bp2n ❧➔ sè ♥❣✉②➯♥ ♥➯♥ t❤❡♦ tr➯♥ t❛ s✉② r❛

1≤

b
, ∀n = 0, 1, . . .
2n + 1

❈❤♦ n → +∞ t❛ t❤✉ ✤÷ñ❝

1≤0
✤✐➲✉ ♥➔② ❧➔ ✈æ ❧þ✳ ❱➟② α ❧➔ sè ✈æ t➾✳
✶✶


❚➼♥❤ ❝❤➜t ✶✳✶✶ ✭①❡♠ ❬✸❪✮✳ ▼å✐ sè ✈æ t➾ α ✤➲✉ ❜✐➸✉ ❞✐➵♥ ✤÷ñ❝ ♠ët ❝→❝❤ ❞✉② ♥❤➜t ❞÷î✐
❞↕♥❣ ♠ët ❧✐➯♥ ♣❤➙♥ sè ✈æ ❤↕♥✳
❈❤ù♥❣ ♠✐♥❤✳

✐✮ ●✐↔ sû α = α0 ❧➔ sè ✈æ t➾✳ ❚❛ ①➙② ❞ü♥❣ ❞➣② a0, a1, a2, . . . ♠ët ❝→❝❤ tr✉② ❤ç✐ ♥❤÷ s❛✉
1
,
α 0 − a0
1
α2 =
,
α 1 − a1
1
α3 =
,
α 2 − a2
...
1
αk+1 =
.
αk − ak

a0 = [α0 ] ,

α1 =

a1 = [α1 ] ,
a2 = [α2 ] ,
...
ak = [αk ] ,

❉♦ α0 ❧➔ sè ✈æ t➾ ♥➯♥ 0 < α0 − a0 < 1✱ ❞♦ ✤â α1 tç♥ t↕✐✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ αk ❧➔ sè ✈æ
t➾✱ ✈î✐ ♠å✐ k = 0, 1, . . . ❜➡♥❣ q✉② ♥↕♣ ♥❤÷ s❛✉

✲ ✈î✐ ❦ ❂ ✵ t❤➻ rã r➔♥❣ α0 ❧➔ sè ✈æ t➾✳
✲ ❣✐↔ sû αk ❧➔ sè ✈æ t➾ ✭k ≥ 0 ✮✱ ❦❤✐ ✤â 0 < αk − ak < 1 ✈➔ ak ❧➔ sè ♥❣✉②➯♥ ♥➯♥ αk − ak
1
tç♥ t↕✐ ✈➔ ❧➔ sè ✈æ t➾✱ ✤ç♥❣ t❤í✐ αk+1 > 1✳ ❑❤✐ ✤â
❧➔ sè ✈æ t➾✱ ❞♦ ✤â αk+1 =
α k − ak

ak+1 = [αk+1 ] ≥ 1.
❚â♠ ❧↕✐ t❛ ❝â a0 , a1 , a2 , . . . ❧➔ ❝→❝ sè ♥❣✉②➯♥✱ ai > 0 ✈î✐ ∀i ≥ 1✳

✐✐✮ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ q✉② ♥↕♣

α = [a0 ; a1 , a2 , . . . , ak , αk+1 ] , k = 0, 1, . . .
❚❤➟t ✈➟②

1
= a0 + (α0 − a0 ) = α0 = α✳
α1
1
1
1
= a0 +
= a0 +
= α✳
✲ ✈î✐ ❦ ❂ ✶ t❤➻ [a0 ; a1 , α2 ] = a0 +
1
a1 + (α1 − a1 )
α1
a1 +
α2

✲ ❣✐↔ sû ❝â [a0 ; a1 , a2 , . . . , ak , αk+1 ] = α✳ ❑❤✐ ✤â

✲ ✈î✐ ❦ ❂ ✵ t❤➻ [a0 ; α1 ] = a0 +

[a0 ; a1 , a2 , . . . , ak+1 , αk+2 ]
1

= a0 +

1

a1 +
a2 + . . . +

1

= a0 +
a1 +

1
ak+1 +

1

a2 + . . . +

1
αk+2
✶✷

,

1
ak +

1
αk+1


❞♦ ak+1 +
❉♦ ✤â

1
αk+2

= ak+1 + (αk+1 − ak+1 ) = αk+1 ✳

[a0 ; a1 , a2 , . . . , ak+1 , αk+2 ] = [a0 ; a1 , a2 , . . . , ak , αk+1 ] = α.
❱➟② t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝

α = [a0 ; a1 , a2 , . . . , ak , αk+1 ] , k = 0, 1, . . .

✐✐✐✮ ❚✐➳♣ t❤❡♦ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤
α = lim Ck ,
k→+∞

✈î✐ Ck = [a0 ; a1 , a2 , . . . , ak ]✳
❚❤➟t ✈➟②✱ t❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ ❣✐↔♥ ♣❤➙♥ t❛ ❝â

α − Ck =

tr♦♥❣ ✤â

pk+1 pk
−
,
qk+1
qk

pk+1 = αk+1 pk + pk−1
qk+1 = αk+1 qk + qk−1 .

❱➟② s✉② r❛

α − Ck =

− (pk qk−1 − pk−1 qk )
(−1)k
αk+1 pk + pk−1 pk
−
=
=
.
αk+1 qk + qk−1
qk
qk (αk+1 qk + qk−1 )
qk (αk+1 qk + qk−1 )

❱➻ αk+1 qk + qk−1 > ak+1 qk + qk−1 = qk+1 ♥➯♥

|α − Ck | <

1
1
≤
.
qk qk+1
k (k + 1)

❈❤♦ k → +∞ t❛ t❤✉ ✤÷ñ❝

α = lim Ck .
k→+∞

◆❤÷ ✈➟②

α = [a0 ; a1 , a2 , . . .] .

✐✈✮ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ❜✐➸✉ ❞✐➵♥ ✤â ❧➔ ❞✉② ♥❤➜t✳
●✐↔ sû α ❝â ❤❛✐ ❜✐➸✉ ❞✐➵♥

α = [a0 ; a1 , a2 , . . .] = [b0 ; b1 , b2 , . . .] .
❚❛ ❜✐➳t r➡♥❣

C2n < α < C2n+1 , ∀n = 0, 1, . . .
✶✸


❉♦ ✤â

C0 < α < C1

❤❛②

a0 < α < a0 +

1
.
a1

❉♦ a0 ♥❣✉②➯♥ ✈➔ a1 ≥ 1 ♥➯♥ s✉② r❛

[α] = a0 .
❚÷ì♥❣ tü

[α] = b0 .
◆❤÷ ✈➟② t❛ t❤✉ ✤÷ñ❝

a0 = b0 , [a1 ; a2 , a3 , . . .] = [b1 ; b2 , b3 , . . .]
❚✐➳♣ tö❝ q✉→ tr➻♥❤ ♥❤÷ tr➯♥ t❛ ❝â

ai = bi , [ai+1 ; ai+2 , ai+3 , . . .] = [bi+1 ; bi+2 , bi+3 , . . .] , ∀i = 0, 1, . . .
✣✐➲✉ ✤â ❝â ♥❣❤➽❛ ❧➔

ak = bk , ∀k = 0, 1, . . .
❱➟② ♠å✐ sè ✈æ t➾ α ✤➲✉ ❜✐➸✉ ❞✐➵♥ ✤÷ñ❝ ♠ët ❝→❝❤ ❞✉② ♥❤➜t ❞÷î✐ ❞↕♥❣ ♠ët ❧✐➯♥ ♣❤➙♥ sè
✈æ ❤↕♥✳

❱➼ ❞ö ✶✳✻✳ ❍➣② ❜✐➸✉ ❞✐➵♥ ❝→❝ sè

√

√
3+ 7
5 ✈➔ α =
t❤➔♥❤ ❧✐➯♥ ♣❤➙♥ sè ✈æ ❤↕♥✳
2

▲í✐ ❣✐↔✐✳

✐✮ ❜✐➸✉ ❞✐➵♥ sè √5 t❤➔♥❤ ❧✐➯♥ ♣❤➙♥ sè ✈æ ❤↕♥✳
a0 =
a1 = √
a2 = √

√
5 = 2,

√
1
=
5 + 2 = 4,
5−2

√
1
=
5 + 2 = 4.
5+2−4

❱➟②

a0 = 2, an = 4 , ∀n = 1, 2, . . .
✶✹


❉♦ ✤â

✐✐✮

√
5 = [2; 4, 4, 4, . . .] .

√
3+ 7
❜✐➸✉ ❞✐➵♥ sè α =
t❤➔♥❤ ❧✐➯♥ ♣❤➙♥ sè ✈æ ❤↕♥✳
2
a0 = [α] = 2 ,
√
√
7+1
7+1
1
a1 =
=
,
= 1, β =
α−2
3
3
√

√
1
7 + 2 = 4, γ = 7 + 2,
=
β−1
√
√
1
7+2
7+2
a3 =
=
= 1, θ =
,
λ−4
3
3
√
√
1
7+1
7+1
=
= 1, δ =
,
a4 =
θ−1
2
2
√

1
7+1
=
a5 =
= [β] = 1.
δ−1
3
a2 =

❱➟②

a0 = 2, a4n+1 = a4n+3 = a4n+4 = 1, a4n+2 = 4 , ∀n = 0, 1, . . .
❉♦ ✤â

√
3+ 7
α=
= 2; 1, 4, 1, 1
2

✭❧✐➯♥ ♣❤➙♥ sè ✈æ ❤↕♥ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ✹✮✳

✶✳✸ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
✣à♥❤ ♥❣❤➽❛ ✶✳✼ ✭①❡♠ ❬✸❪✮✳ P❤÷ì♥❣ tr➻♥❤ ❉✐♣♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝â
❞↕♥❣

a1 x 1 + a2 x 2 + . . . + an x n = c
tr♦♥❣ ✤â ❝→❝ ❤➺ sè ai , c ∈ Z,

n

i=1

a2i = 0✱ ❝→❝ ❜✐➳♥ sè xi ∈ Z, ∀i = 1, 2, . . . , n✳

❱➼ ❞ö ✶✳✼✳
32x + 40y = 24
❧➔ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✈î✐ ❤❛✐ ➞♥ ①✱ ②✳

✶✺


✣à♥❤ ❧➼ ✶✳✷ ✭①❡♠ ❬✸❪✮✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
Ax + By = C.

✭✶✮

✐✮ (1) ❝â ♥❣❤✐➺♠ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ d = (A, B) |C ✳
✐✐✮ ◆➳✉ (x0 , y0 ) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (1) t❤➻ ♠å✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

(1) ✤÷ñ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝


B


 x = x0 + d t


 y = y0 − A t

, t ∈ Z.

d

❈❤ù♥❣ ♠✐♥❤✳

✐✮

⇒) ●✐↔ sû (x0 , y0 ) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (1)✱ tù❝ ❧➔
Ax0 + By0 = C.
❚❛ ❝â d = (A, B) ⇒

d |A
⇒ d |(Ax0 + By0 ) ❤❛② d |C ✳
d |B

⇐) ●✐↔ sû d = (A, B) |C
❑❤æ♥❣ ❣✐↔♠ tê♥❣ q✉→t✱ t❛ ❣✐↔ sû B > 0✳ ❱➻ d = (A, B) |C ♥➯♥ ∃a, b, c ∈ Z s❛♦ ❝❤♦

A = da, B = db, C = dc, (a, b) = 1.
❑❤✐ ✤â (1) t÷ì♥❣ ✤÷ì♥❣ ✈î✐

ax + by = c.

✭✷✮

❱➻ (a, b) = 1 ♥➯♥ t➟♣ ❤ñ♣ {1a, 2a, . . . , ba} ❧➔ ♠ët ❤➺ t❤➦♥❣ ❞÷ ✤➛② ✤õ ♠♦❞✉❧♦ ❜ ♥➯♥

∃x ∈ {1, 2, . . . , b} : ax ≡ c (mod b)
⇒ ∃x ∈ {1, 2, . . . , b} : ax − c ≡ 0 (mod b)

⇒ ∃x ∈ {1, 2, . . . , b} , ∃y ∈ Z : ax − c = by
⇒ ∃x ∈ {1, 2, . . . , b} , ∃y ∈ Z : ax + by = c.
✣✐➲✉ ✤â ❝â ♥❣❤➽❛ ❧➔

∃x, y ∈ Z : ax + by = c.
✶✻


❈❤ù♥❣ tä ♣❤÷ì♥❣ tr➻♥❤ (2) ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥ ✭①✱②✮✱ tù❝ (1) ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥ ✭①✱②✮✳

✐✐✮

⇐) ◆➳✉


B


 x = x0 + d t


 y = y0 − A t

, t ∈ Z t❤➻

d
B
A
Ax + By = A x0 + t + B y0 − t = Ax0 + By0 = C ✳
d

d
❈❤ù♥❣ tä ✭①✱②✮ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (1)✳
⇒) ◆➳✉ (x, y) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (1)✱ tù❝ ♣❤÷ì♥❣ tr➻♥❤ (2) t❤➻
ax + by = c = ax0 + by0 ⇒ a(x − x0 ) = b(y0 − y) ⇒ b |a(x − x0 ) ✳
❱➻ (a, b) = 1 ♥➯♥

b |(x − x0 ) ⇒ ∃t ∈ Z : x − x0 = bt ⇔ ∃t ∈ Z : x = x0 + bt✳
❚ø ✤â

y = y0 − at.
❱➟②

x = x0 + bt
, t∈Z
y = y0 − at
❤❛②


B


 x = x0 + d t


 y = y0 − A t

, t ∈ Z.

d

◆❤➟♥ ①➨t ✶✳✷✳ ❱✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ (1) q✉② ✈➲ ✈✐➺❝ t➻♠
✐✮ d = (A, B).
✐✐✮ ▼ët ♥❣❤✐➺♠ r✐➯♥❣ (x0 , y0 ) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (1) .
❈❤ó♥❣ t❛ ❜✐➳t r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ (1) ❝â ♥❣❤✐➺♠ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ d = (A, B) |C ✳ ❚r♦♥❣
tr÷í♥❣ ❤ñ♣ ♥➔② t❛ ❣✐↔ sû ❆ ❂ ❛❞✱ ❇ ❂ ❜❞✱ ❈ ❂ ❝❞ t❤➻ (a, b) = 1 ✈➔ ❦❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤

(1) t÷ì♥❣ ✤÷ì♥❣ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤
ax + by = c.

✭✷✮

◆➳✉ (x0 , y0 ) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ (2) t❤➻ ♠å✐ ♥❣❤✐➺♠ ❝õ❛ (2) ✤÷ñ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝

x = x0 + bt
, t ∈ Z.
y = y0 − at
✶✼


◆❤÷ ✈➟② ✈✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ (2) q✉② ✈➲ ✈✐➺❝ t➻♠ ♠ët ♥❣❤✐➺♠ (x0 , y0 ) ❝õ❛ ♥â✳
❳➨t ♣❤÷ì♥❣ tr➻♥❤
✭✸✮

ax + by = 1.

◆➳✉ (x1 , y1 ) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ (3) t❤➻ (cx1 , cy1 ) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ (2)✳ ❚❤➔♥❤ t❤û t❛
q✉② ✈➲ ❜➔✐ t♦→♥✿ ❈❤♦ (a, b) = 1✳ ❍➣② t➻♠ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (3)✳
a
❚❛ ❜✐➸✉ ❞✐➵♥ ♣❤➙♥ sè
t❤➔♥❤ ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥

|b|

a
= [a0 ; a1 , a2 , . . . , an ] .
|b|
pn−1
pn
●å✐ Cn−1 =
✈➔ Cn =
❧➔ ❤❛✐ ❣✐↔♥ ♣❤➙♥ ❝✉è✐ ❝ò♥❣ ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè ♥➔②✳ ❚❛ ❝â
qn−1
qn
a
pn
= ✱ (a, b) = 1✱ (pn , qn ) = 1 ♥➯♥ a = pn , |b| = qn ✳ ❚❤❡♦ t➼♥❤ ❝❤➜t ( ) t❛ ❝â
|b|
qn

✶✳✺

pn qn−1 − pn−1 qn = (−1)n−1
⇒ aqn−1 − |b| pn−1 = (−1)n−1
⇒ a(−1)n−1 qn−1 − |b| (−1)n−1 pn−1 = 1.
❱➟② ♥➳✉ b > 0 t❤➻ ♣❤÷ì♥❣ tr➻♥❤ (3) ❝â ♠ët ♥❣❤✐➺♠ ❧➔

x1 = (−1)n−1 .qn−1
y1 = (−1)n .pn−1
◆➳✉ b < 0 t❤➻ ♣❤÷ì♥❣ tr➻♥❤ (3) ❝â ♠ët ♥❣❤✐➺♠ ❧➔

x1 = (−1)n−1 .qn−1

y1 = (−1)n−1 .pn−1
✶✳✸✳✶

❚➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❞ü❛ ✈➔♦ ❣✐↔♥ ♣❤➙♥

❚✐➳♥ ❤➔♥❤ t❤ü❝ ❤✐➺♥ t❤❡♦ ❝→❝ ❜÷î❝ s❛✉

❇÷î❝ ✶✿ ❚➻♠ d = (A, B) ✈➔ ✤÷❛ ♣❤÷ì♥❣ tr➻♥❤ (1) ✈➲ ♣❤÷ì♥❣ tr➻♥❤ (2) ✈î✐ (a, b) = 1✳
❇÷î❝ ✷✿ ❱✐➳t |b|a = [a0; a1, a2, . . . , an]✳
n−1
❇÷î❝ ✸✿ ❚➼♥❤ ❣✐↔♥ ♣❤➙♥ Cn−1 = [a0; a1, . . . , an−1] = pqn−1
✳ ❙✉② r❛ pn−1 ✈➔ qn−1 ✳
❇÷î❝ ✹✿ ❙✉② r❛ ♠ët ♥❣❤✐➺♠ r✐➯♥❣ (x0, y0) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (2)
◆➳✉ b > 0 t❤➻
◆➳✉ b < 0 t❤➻

x0 = (−1)n−1 .c.qn−1
y0 = (−1)n .c.pn−1
x0 = (−1)n−1 .c.qn−1
y0 = (−1)n−1 .c.pn−1

✶✽


❱➼ ❞ö ✶✳✽✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
✭✹✮

342x − 123y = 15.
▲í✐ ❣✐↔✐✳

P❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ t÷ì♥❣ ✤÷ì♥❣ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤
✭✹❛✮

114x − 41y = 5.
❚❛ ❝â

114
= [2; 1, 3, 1, 1, 4]✱ ✈î✐ n = 5✳
41
25
p4
= [2; 1, 3, 1, 1] =
q4
9 ⇒
(p4 , q4 ) = 1

C4 =

p4 = 25
q4 = 9.

❉♦ b = −41 < 0 ♥➯♥ ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ (4a) ❧➔

x0 = (−1)5−1 .5.9 = 45
y0 = (−1)5−1 .5.25 = 125.
❱➟② ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (4a)✱ tù❝ ♣❤÷ì♥❣ tr➻♥❤ (4) ❧➔

x = 45 + 41t
, t ∈ Z.
y = 125 + 114t

✶✳✸✳✷

❚➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❞ü❛ ✈➔♦ t❤✉➟t t♦→♥ ❊✉❝❧✐❞

❚✐➳♥ ❤➔♥❤ t❤ü❝ ❤✐➺♥ t❤❡♦ ❝→❝ ❜÷î❝ s❛✉

❇÷î❝ ✶✿ ❚➻♠ d = (|A| , |B|) t❤❡♦ t❤✉➟t t♦→♥ ❊✉❝❧✐❞ ♠ð rë♥❣✳
❇÷î❝ ✷✿ ❇✐➸✉ t❤à ❞ ♥❤÷ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❆ ✈➔ ❇✱ ❝❤➥♥❣ ❤↕♥
d = nA + mB (n, m ∈ Z) .

❇÷î❝ ✸✿ ◆❤➙♥ ❤❛✐ ✈➳ ✤➥♥❣ t❤ù❝ tr➯♥ ✈î✐ Cd t❛ ✤÷ñ❝
A

Cn
Cm
+B
= C.
d
d

❇÷î❝ ✹✿ ❙✉② r❛ ♠ët ♥❣❤✐➺♠ r✐➯♥❣ (x0, y0) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (1) ❧➔

Cn


 x0 =
d




 y0 = Cm
d

✶✾


❱➼ ❞ö ✶✳✾✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
342x − 123y = 15.
▲í✐ ❣✐↔✐✳

✭✹✮

❱➻ (342, −123) = 3 |15 ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠✳ ❚❛ ❝â

342 = 123.2 + 96
123 = 96.1 + 27
96 = 27.3 + 15
27 = 15.1 + 12
15 = 12.1 + 3
12 = 3.4 + 0.
❙✉② r❛

3 = 15 − 12.1 = 15 − (27 − 15.1).1 = 15.2 − 27.1 = (96 − 27.3).2 − 27.1 =
= 96.2 − 27.7 = 96.2 − (123 − 96.1).7 = 96.9 − 123.7 = (342 − 123.2).9 − 123.7 =
= 342.9 − 123.25.
❙✉② r❛

342.45 − 123.125 = 15.
❚ø ✤â ♣❤÷ì♥❣ tr➻♥❤ (4) ❝â ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❧➔

(x0 ; y0 ) = (45; 125).
❙✉② r❛ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (4) ❧➔


123


 x = 45 + 3 t


 y = 125 + 342 t

,t∈Z

3

❤❛②

x = 45 + 41t
, t ∈ Z.
y = 125 + 114t

✣à♥❤ ❧➼ ✶✳✸ ✭①❡♠ ❬✸❪✮✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
a1 x1 + a2 x2 + . . . + an xn = c.
✷✵

✭✺✮


Pữỡ tr (5) õ d = (a1 , a2 , . . . , an ) |c

ữỡ tr (5) õ t õ s õ ổ số
ự



) sỷ (x1 , x2 , . . . , xn ) ởt ừ ữỡ tr (5) tự
n

ai xi = c.
i=1

õ d = (a1 , a2 , . . . , an ) d

n

ai xi d |c

i=1

) ự q t

)

ợ ú t ỵ (
sỷ ú ợ k 2

ợ n = k + 1 t d = (a1 , a2 , . . . , ak+1 ) |c t h = (a1 , a2 , . . . , ak ) õ t õ

d = (h, ak+1 ) |c
r tỗ t t, xk+1 Z

ht + ak+1 xk+1 = c.
h |ht t tt q s tỗ t x1 , x2 , . . . , xk Z
k

ai xi = ht.
i=1

õ

k+1

ai xi = c.
i=1

ự tọ ữỡ tr a1 x1 + a2 x2 + . . . + ak+1 xk+1 = c õ (x1 , x2 , . . . , xk+1 )

ự q t
ợ ú t ỵ ()

sỷ ú ợ k 2 tự ữỡ tr

k

ai xi = c õ

i=1

t s õ ổ số
ợ n = k + 1 t s ự tọ ữỡ tr

k+1
i=1



ai xi = c õ t s õ ổ số


♥❣❤✐➺♠✳ ❚❤➟t ✈➟②
●å✐ (t1 , t2 , . . . , tk+1 ) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

k+1

ai xi = c✱ tù❝ ❧➔

i=1
k+1

ai ti = c.
i=1
k

⇒

ai ti = c − ak+1 tk+1 .
i=1

❳➨t ♣❤÷ì♥❣ tr➻♥❤
k

ai xi = c − ak+1 tk+1

✭✻✮

i=1

❝â ✈➳ ♣❤↔✐ ❧➔ ❤➡♥❣ sè ✈➔ ❝â ♠ët ♥❣❤✐➺♠ ❧➔ (t1 , t2 , . . . , tk ) ♥➯♥ t❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣
t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❝â ✈æ sè ♥❣❤✐➺♠✳ Ù♥❣ ✈î✐ ♠é✐ ♥❣❤✐➺♠ (x1 , x2 , . . . , xk ) ❝õ❛ (6) t❤➻
♣❤÷ì♥❣ tr➻♥❤

k+1

ai xi = c ❝â ♥❣❤✐➺♠ ❧➔ (x1 , x2 , . . . , xk , tk+1 )✳ ❈❤ù♥❣ tä ♣❤÷ì♥❣ tr➻♥❤

i=1
k+1

ai xi = c ❝â ✈æ sè ♥❣❤✐➺♠✳

i=1

❈â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (5) ❝â ❞↕♥❣


n−1


Li1 ti ,
x

+
x
=

1
1


i=1



n−1

Li2 ti ,

x2 = x2 +

i=1



...



n−1




Lin ti .
 xn = xn +
i=1

tr♦♥❣ ✤â (x1 , x2 , . . . , xn ) ❧➔ ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ (5)✱ ti ∈ Z, ∀i = 1, 2, . . . , n✳ ❚✉②
♥❤✐➯♥ ❝→❝ ❤➺ sè Lij (i, j = 1, 2, . . . , n) ❦❤æ♥❣ ❝â ❝æ♥❣ t❤ù❝ t÷í♥❣ ♠✐♥❤✳

❱➼ ❞ö ✶✳✶✵✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
6x + 15y + 10z = 3.
▲í✐ ❣✐↔✐✳

P❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ t÷ì♥❣ ✤÷ì♥❣ ✈î✐

6(x + z) + 15y + 4z = 3.

✷✷

✭✼✮


✣➦t u = x + z t❛ t❤✉ ✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤

15y + 4z = 3 − 6u.

✭✼❛✮

❚❛ ♥❤➟♥ t❤➜② ♣❤÷ì♥❣ tr➻♥❤

15y + 4z = 1
❝â ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❧➔ (y1 ; z1 ) = (−1; 4) ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ (7a) ❝â ♠ët ♥❣❤✐➺♠ r✐➯♥❣

❧➔ (y0 ; z0 ) = (−3 + 6u; 12 − 24u)✳
❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (7a) ❧➔

y = −3 + 6u + 4t
, u, t ∈ Z.
z = 12 − 24u − 15t
▼➦t ❦❤→❝ u = x + z s✉② r❛ x = u − z = −12 + 25u + 15t.
❱➟② ♣❤÷ì♥❣ tr➻♥❤ (7) ❝â ♥❣❤✐➺♠ tê♥❣ q✉→t ❧➔

x = −12 + 25u + 15t
y = −3 + 6u + 4t
, u, t ∈ Z.
z = 12 − 24u − 15t

◆❤➟♥ ①➨t ✶✳✸ ✭①❡♠ ❬✸❪✮✳ ❈→❝❤ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ♥❤✐➲✉ ➞♥ tr♦♥❣
✈➼ ❞ö tr➯♥ ❞ü❛ tr➯♥ t÷ t÷ð♥❣ s❛✉ ✤➙②✿ ❚ø ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ♥ ➞♥✱
t❛ ✤÷❛ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ n − 1 ➞♥✱ t✐➳♣ tö❝ ♥❤÷ ✈➟② ❝✉è✐ ❝ò♥❣ t❛
♥❤➟♥ ✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✷ ➞♥✳ ▼é✐ ❧➛♥ ❣✐↔♠ sè ➞♥ ♥❤÷ ✈➟② t❛ ❧↕✐
❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✷ ➞♥ ✭♥❤÷♥❣ ♣❤↔✐ q✉❛ t❤❛♠ sè✮✳ ❱➻ ❧➩ ✤â t❛
✤÷ñ❝ ❤➺ ♥❣❤✐➺♠ ♣❤ö t❤✉ë❝ ✈➔♦ n − 1 t❤❛♠ sè✳
◆â✐ ♠ët ❝→❝❤ rã r➔♥❣ ❤ì♥✱ ①➨t ♣❤÷ì♥❣ tr➻♥❤ ✭✺✮ ✈î✐ n ≥ 3✱ ai = 0, ∀i = 1, 2, . . . , n✳
●å✐ d = (an−1 , an )✳ ❑❤✐ ✤â an−1 = d.bn−1 , an = d.bn , (bn−1 , bn ) = 1✳ ❉♦ ✤â ✭✺✮ trð
t❤➔♥❤

a1 x1 + a2 x2 + . . . + an−2 xn−2 + d (bn−1 xn−1 + bn xn ) = c.

✭✺❛✮

✣÷❛ ✈➔♦ ➞♥ ♠î✐ t ❜➡♥❣ ❤➺ t❤ù❝

bn−1 xn−1 + bn xn = t.

✭✺❜✮

a1 x1 + a2 x2 + . . . + an−2 xn−2 + dt = c.

✭✺❝✮

❑❤✐ ✤â ✭✺❛✮ trð t❤➔♥❤

✷✸


●✐↔ sû x1 , x2 , . . . , xn−2 , t ❧➔ ♠ët ♥❣❤✐➺♠ ♥❣✉②➯♥ ❝õ❛ ✭✺❝✮✳ Ù♥❣ ✈î✐ sè ①→❝ ✤à♥❤ t✱
①➨t ♣❤÷ì♥❣ tr➻♥❤
✭✺❞✮

bn−1 xn−1 + bn xn = t.

❉♦ (bn−1 , bn ) = 1 t ♥➯♥ ✭✺❞✮ ♥❤➜t ✤à♥❤ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥✱ ❝❤➥♥❣ ❤↕♥ (xn−1 , xn )✳ ❑❤✐
✤â rã r➔♥❣ (x1 , x2 , . . . , xn−2 , xn−1 , xn ) ❧➔ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❝õ❛ ✭✺✮✳ ❉➵ t❤➜② ♠å✐ ♥❣❤✐➺♠
♥❣✉②➯♥ ❝õ❛ ✭✺✮ ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✺❝✮ ✈î✐ ✤✐➲✉ ❦✐➺♥ ✭✺❜✮✳

✶✳✹ ◆❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦✲
♣❤❛♥t❡ t✉②➳♥ t➼♥❤
❳➨t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♣♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤

a1 x1 + a2 x2 + . . . + an xn = c

✭✺✮

✈î✐ ❝→❝ ❤➺ sè ai , c ∈ Z+ ✱ ❝→❝ ❜✐➳♥ sè xi ∈ Z+ , ∀i = 1, 2, . . . , n✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤
✭✺✮ ❧✉æ♥ ❝â ❤ú✉ ❤↕♥ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ x = (x1 , x2 , . . . , xn )✳ ❚ø ✤➲ ❜➔✐✱ t❛ ❝â t❤➸
❤↕♥ ❝❤➳ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❝→❝ ❜✐➳♥ sè ❜ð✐

1 ≤ xi ≤

(c + ai ) − (a1 + a2 + . . . + an )
ai

, ∀i = 1, 2, . . . , n.

❑❤✐ ✤â✱ ❝→❝❤ ✤ì♥ ❣✐↔♥ ♥❤➜t ✤➸ t➻♠ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ x = (x1 , x2 , . . . , xn ) ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ✭✺✮ ❧➔ t❛ ❝❤♦ ♠ët ❜✐➳♥ sè xi ♥➔♦ ✤â ❧➛♥ ❧÷ñt ❝❤↕② q✉❛ ❝→❝ ❣✐→ trà ❝â t❤➸
❝â ❝õ❛ ♥â ✈➔ t➻♠ ❝→❝ ❜✐➳♥ sè ❝á♥ ❧↕✐ tø ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✳

❱➼ ❞ö ✶✳✶✶✳ ❚➻♠ ❝→❝ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
6x + 15y + 10z = 200.
▲í✐ ❣✐↔✐✳

❚❤❡♦ ❜➔✐ r❛ t❛ ❝â

1 ≤ x ≤ 29
1 ≤ y ≤ 12
1 ≤ z ≤ 17.

✳
✳
❍ì♥ ♥ú❛✱ tø ✤➲ ❜➔✐ s✉② r❛ x✳✳5 ✈➔ y ✳✳2✳
✲ ✈î✐ ② ❂ ✷ t❛ ❝â 3x + 5z = 85 ♥➯♥

x = 5 → z = 14
x = 10 → z = 11
x = 15 → z = 8
x = 20 → z = 5
x = 25 → z = 2
✷✹

✭✽✮