Nửa nhóm số sinh bởi 3 phần tử

TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI 2
KHOA TOÁN
*************

NGUYỄN THỊ HÒA

NỬA NHÓM SỐ SINH BỞI BA PHẦN TỬ

KHÓA LUẬN TỐT NGHIỆP ĐẠI HỌC
Chuyên ngành: Đại số

HÀ NỘI – 2018


TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI 2
KHOA TOÁN
*************

NGUYỄN THỊ HÒA

NỬA NHÓM SỐ SINH BỞI BA PHẦN TỬ

KHÓA LUẬN TỐT NGHIỆP ĐẠI HỌC
Chuyên ngành: Đại số

Người hướng dẫn khoa học
ThS. ĐỖ VĂN KIÊN

HÀ NỘI – 2018

ớ ỡ
t õ tổ ữủ tọ ỏ t ỡ s
s

ộ

ữớ trỹ t t t ữợ

ữợ tổ tr sốt q tr tổ õ
ừ ỗ tớ tổ ụ t ỡ t ổ tr
tờ số t ổ tr rữớ ồ ữ
ở ừ t tổ t
tốt õ õ t q ữ ổ
ũ õ rt ố s tớ
t ỏ õ ổ t tr ọ ỳ t
sõt rt ữủ sỹ õ õ ỵ ừ t ổ s
ồ
ổ t ỡ

ở t
õ

ỏ


▲í✐ ❝❛♠ ✤♦❛♥
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ❑❤â❛ ❧✉➟♥ ♥➔② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣
tæ✐ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ t❤➛②

❚❤❙✳ ✣é ❱➠♥ ❑✐➯♥

✳ ❚r♦♥❣ ❦❤✐ ♥❣❤✐➯♥

❝ù✉✱ ❤♦➔♥ t❤➔♥❤ ❜↔♥ ❦❤â❛ ❧✉➟♥ ♥➔② tæ✐ ✤➣ t❤❛♠ ❦❤↔♦ ♠ët sè t➔✐ ❧✐➺✉ ✤➣
❣❤✐ tr♦♥❣ ♣❤➛♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳
❚æ✐ ①✐♥ ❦❤➥♥❣ ✤à♥❤ ❦➳t q✉↔ ❝õ❛ ✤➲ t➔✐✿ ✏

❜❛ ♣❤➛♥ tû

◆û❛ ♥❤â♠ sè s✐♥❤ ❜ð✐

✑ ❧➔ ❦➳t q✉↔ ❝õ❛ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♥é ❧ü❝ ❤å❝ t➟♣ ❝õ❛ ❜↔♥

t❤➙♥✱ ❦❤æ♥❣ trò♥❣ ❧➦♣ ✈î✐ ❦➳t q✉↔ ❝õ❛ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ◆➳✉ s❛✐ tæ✐ ①✐♥
❝❤à✉ ❤♦➔♥ t♦➔♥ tr→❝❤ ♥❤✐➺♠✳

❍➔ ◆ë✐✱ ♥❣➔② ✷ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽
❚→❝ ❣✐↔ ❦❤â❛ ❧✉➟♥

◆❣✉②➵♥ ❚❤à ❍á❛


▼ö❝ ❧ö❝
▼ð ✤➛✉
✶ ◆û❛ ♥❤â♠ sè

✶
✸

✶✳✶

◆û❛ ♥❤â♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✷

◆û❛ ♥❤â♠ ❝♦♥ s✐♥❤ ❜ð✐ ♠ët t➟♣ ❤ñ♣

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✸

◆û❛ ♥❤â♠ sè

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✹

▼ët sè ❜➜t ❜✐➳♥ ❝õ❛ ♥û❛ ♥❤â♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✺

P❤➙♥ ❧♦↕✐ ♥û❛ ♥❤â♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✶✳✺✳✶

◆û❛ ♥❤â♠ sè ✤è✐ ①ù♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✶✳✺✳✷

◆û❛ ♥❤â♠ sè ❣✐↔ ✤è✐ ①ù♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✷ ◆û❛ ♥❤â♠ sè s✐♥❤ ❜ð✐ ❜❛ ♣❤➛♥ tû
✷✳✶

✷✶

■✤➯❛♥ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ✈➔♥❤ ♥û❛ ♥❤â♠ sè s✐♥❤ ❜ð✐ ❜❛ ♣❤➛♥
tû

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✶

✷✳✷

●✐è♥❣ ❝õ❛ ♥û❛ ♥❤â♠ sè s✐♥❤ ❜ð✐ ❜❛ ♣❤➛♥ tû

✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✾

✷✳✸

❱➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✹

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✻
✸✻




ỵ ồ t
ỵ tt ỷ õ ởt tữỡ ố tr ừ ồ ữ
ởt ữợ t t ừ số ợ ử t r ừ õ
ró t ữỡ ự ừ tt õ ữủ
t ự ỵ tt ỷ
õ õ s ú t t ữủ tổ t tt t
t ừ ỷ õ ỵ tt ỷ õ õ trỏ q
trồ tr ự ởt số ồ ỡ ữ
ồ t ự ỷ õ số tữỡ ữỡ ợ ữỡ
tr ổ ừ ữỡ t t t ổ t

t t ợ số ữỡ ờ ữủ
ự tr t
ợ ố t s ỹ ữợ õ ở ởt s
sữ ồ tr ừ ởt õ tốt ũ
sỹ ú ù t t ừ t ộ ỹ ồ
t

ỷ õ số s tỷ

ử

ử ừ t ởt số tự ỡ s ỷ õ số
trữ ừ ỷ õ số s
tỷ ố ừ ỷ õ số s tỷ

ử ự

ữợ q ợ ổ t ự ồ ỗ tớ ố
s t tỏ ự ỷ õ số ỷ õ số s
tỷ




✸✳ ✣è✐ t÷ñ♥❣ ♥❣❤✐➯♥ ❝ù✉
◆❣❤✐➯♥ ❝ù✉ ✈➲ ♥û❛ ♥❤â♠ sè✱ ♥û❛ ♥❤â♠ sè s✐♥❤ ❜ð✐ ❜❛ ♣❤➛♥ tû✳

✹✳ ❈➜✉ tró❝ ❦❤â❛ ❧✉➟♥

◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥✱ ❞❛♥❤ ♠ö❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❦❤â❛

❧✉➟♥ ❣ç♠ ✷ ❝❤÷ì♥❣✿

•

❈❤÷ì♥❣ ✶✿ ◆û❛ ♥❤â♠ sè✳
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❦❤â❛ ❧✉➟♥ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ♥❤÷
❦❤→✐ ♥✐➺♠ ♥û❛ ♥❤â♠✱ ♥û❛ ♥❤â♠ ❝♦♥ s✐♥❤ ❜ð✐ ♠ët t➟♣ ❤ñ♣✱ ♠ët sè
❜➜t ❜✐➳♥ ❝õ❛ ♥û❛ ♥❤â♠ sè✱ ♥û❛ ♥❤â♠ sè ✤è✐ ①ù♥❣ ✈➔ ♥û❛ ♥❤â♠ sè
❣✐↔ ✤è✐ ①ù♥❣✳

•

❈❤÷ì♥❣ ✷✿ ◆û❛ ♥❤â♠ sè s✐♥❤ ❜ð✐ ❜❛ ♣❤➛♥ tû✳
◆ë✐ ❞✉♥❣ ❝❤õ ②➳✉ ❝õ❛ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ✈➲ ✐✤➯❛♥ ✤à♥❤ ♥❣❤➽❛
❝↔✉ ✈➔♥❤ ♥û❛ ♥❤â♠ sè s✐♥❤ ❜ð✐ ❜❛ ♣❤➛♥ tû ✈➔ ❣✐è♥❣ ❝õ❛ ♥û❛ ♥❤â♠
sè s✐♥❤ ❜ð✐ ❜❛ ♣❤➛♥ tû✳

✷


ữỡ
ỷ õ số
r ữỡ tổ tr ỷ õ ỷ õ số
ởt số t ừ ỷ õ số ỷ õ số

ỷ õ


õ

X



ởt t ủ

õ t ổ

tr X

õ X

ởt

tọ ợ ồ

ỷ

x, y, z X

t

(x y) z = x (y z)
ỡ ỳ tỗ t

eX

s

a e = e a = a, a X

ữủ ồ ởt õ

e



ởt ỷ õ ợ t

X =

õ

t tr

X

A





X

ồ tỷ ỡ ừ ỷ õ

ỷ õ

tự ợ ồ

t

a, b A

ừ

t

X

X

X

A X

õ ờ ợ

a b A

ừ ởt ồ rộ ỷ õ ừ ởt

ỷ õ X ởt ỷ õ ừ X

ử

số tỹ

N

ợ t ởt õ




❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❍á❛

✶✳✷ ◆û❛ ♥❤â♠ ❝♦♥ s✐♥❤ ❜ð✐ ♠ët t➟♣ ❤ñ♣

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳

X

❈❤♦

t➜t ❝↔ ❝→❝ ✈à ♥❤â♠ ❝♦♥ ❝õ❛

A✱

❧➔ ♠ët ✈à ♥❤â♠✱

X

❝❤ù❛

✈à ♥❤â♠ ❝♦♥ ♥➔② ✤÷ñ❝ ❣å✐ ❧➔

A

❑➼ ❤✐➺✉ ❧➔

A

❧➔ ♠ët ✈à ♥❤â♠ ❝♦♥ ❝õ❛

✈à ♥❤â♠ ❝♦♥ s✐♥❤ ❜ð✐ A

X

❝❤ù❛

✳

❚ø ✤à♥❤ ♥❣❤➽❛ t❛ ❝â ♥❤➟♥ ①➨t

❧➔ ✈à ♥❤â♠ ❝♦♥ ♥❤ä ♥❤➜t ❝õ❛

✭✐✐✮ ◆➳✉

❑❤✐ ✤â ❣✐❛♦ ❝õ❛

✳

◆❤➟♥ ①➨t ✶✳✷✳✷✳
✭✐✮

A

A ⊆ X✳

A=∅

X

❝❤ù❛

A✳

t❤➻

A = {λ1 x1 + λ2 x2 + . . . + λn xn | n ∈ N\ {0} , ai ∈ A, λi ∈ N ∀i} .

✶✳✸ ◆û❛ ♥❤â♠ sè

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳

❈❤♦

H ⊆ N✳

❚❛ ♥â✐

H

❧➔ ♠ët ♥û❛ ♥❤â♠ sè ♥➳✉ ♥â

t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉

✭✐✮

0 ∈ H❀

✭✐✐✮

H + H ⊆ H❀

✭✐✐✐✮

|N\H| < ∞✳

◆➳✉

{a1 , a2 , . . . , an }

❧➔ ♠ët

❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ H ✱ tù❝

ai ∈
/ a1 , a2 , . . . , ai−1 , ai+1 , . . . , an , 1 ≤ ∀i ≤ n
t❤➻ t❛ ✈✐➳t

H = a1 , a2 , . . . , an

✳

✹

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❍á❛

❚❤❡♦ ♥❤➟♥ ①➨t ✶✳✷✳✷✭✐✐✮ t❛ ❝â

H = {c1 a1 + c2 a2 + . . . + cn an | c1 , c2 , . . . , cn ∈ N} .

▼➺♥❤ ✤➲ ✶✳✸✳✷✳ ❈❤♦ n ∈ N∗✱ n ≥ 2 ✈➔ H =

tr♦♥❣ ✤â
a1 , a2 , . . . , an ∈ N∗ ✳ ❑❤✐ ✤â H ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
gcd (a1 , a2 , . . . , an ) = 1✳
a1 , a2 , . . . , an ,

❈❤ù♥❣ ♠✐♥❤✳
✣✐➲✉ ❦✐➺♥ ❝➛♥✳ ●✐↔ sû H ❧➔ ♠ët ♥û❛ ♥❤â♠ sè✱ ❦❤✐ ✤â ❜ð✐ ✤✐➲✉ ❦✐➺♥ ✭✐✐✐✮
tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ✶✳✸✳✶ t❛ ❝â
❚❛ ❝❤ù♥❣ ♠✐♥❤
✣➦t

|N\H| < ∞✳

gcd (a1 , a2 , . . . , an ) = 1.

d = gcd (a1 , a2 , . . . , an )✳

H

▼å✐ sè t❤✉ë❝

d>1

t❤➻ t➜t ❝↔ ❝→❝ sè tü ♥❤✐➯♥ ❝â ❞↕♥❣

t❤✉ë❝

H✳

❱➟②

❉♦ ✤â t➟♣

d=1

❤❛②

N\H

✤➲✉ ❝❤✐❛ ❤➳t ❝❤♦

nd + 1

✈î✐

n∈N

d

♥➯♥ ♥➳✉

✤➲✉ ❦❤æ♥❣

❧➔ ✈æ ❤↕♥✱ ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳

gcd (a1 , a2 , . . . , an ) = 1.

✣✐➲✉ ❦✐➺♥ ✤õ✳
●✐↔ sû

gcd (a1 , a2 , . . . , an ) = 1✱

H

t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤

❧➔ ♥û❛ ♥❤â♠ sè✳

❇➡♥❣ ❝→❝❤ ❦✐➸♠ tr❛ ✸ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ♥û❛ ♥❤â♠ sè ✿
✭✐✮ ❉➵ t❤➜② r➡♥❣

0 ∈ H✳
n

✭✐✐✮

H + H ⊆ H✳

❚❤➟t ✈➟②✱ ✈î✐ ♠å✐

x, y ∈ H, x =

n

ci ai , y =
i=1

✈î✐ ❝→❝

ci , di ∈ N, ∀i = 1, n

t❛ ❝â

n

x+y =

n

ci ai +
i=1

✭✐✐✐✮ ❚❛ ❝❤ù♥❣ ♠✐♥❤

n

(ci + di )ai ∈ H.

di ai =
i=1

|N\H| < ∞

i=1

❜➡♥❣ q✉② ♥↕♣ t❤❡♦

✺

n✳

di ai ,
i=1


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❱î✐

n=2

✈➔ ❣✐↔ sû

◆❣✉②➵♥ ❚❤à ❍á❛

H = a, b ✱ gcd (a, b) = 1✳

❑❤✐ ✤â

H

❝â ❞↕♥❣

H = {c1 a + c2 b | c1 , c2 ∈ N} .
◆➳✉

a=1

◆➳✉

a, b = 1✱

❤♦➦❝

b=1

t❤➻

H≡N

s✉② r❛

N\H = ∅

❦❤æ♥❣ ♠➜t tê♥❣ q✉→t t❛ ❣✐↔ sû

❚❛ t❤➜② ✈î✐ ♠å✐

m ∈ Z✱

❤❛②

|N\H| < ∞✳

1 < a < b✳

tç♥ t↕✐ ❞✉② ♥❤➜t ❝➦♣

x, y ∈ Z

s❛♦ ❝❤♦

m = ax + by, 0 ≤ y < a.
❚❤➟t ✈➟②✱ ✈➻

gcd (a, b) = 1

♥➯♥ tç♥ t↕✐

u, v ∈ Z

s❛♦ ❝❤♦

au + bv = 1✳

❙✉② r❛

m = amu + bmv = amu + b (aq + y)

✈î✐

0≤y
= a (mu + bq) + by = ax + by.
●✐↔ sû

m = ax + by = ax + by

❙✉② r❛

ax + by = ax + by ✳

❱➻

gcd (a, b) = 1

y=y

♥➯♥

♥➯♥

✈î✐

❉♦ ✤â
✳

|y − y | ✳✳ a

0 ≤ y, y < a✳

a (x − x ) = b (y − y )✳

♠➔

|y − y | < a

❞♦ ✤â

|y − y | = 0

❤❛②

x=x✳

❱➟② ✈î✐ ♠å✐

m∈Z

tç♥ t↕✐ ❞✉② ♥❤➜t

x, y ∈ Z

s❛♦ ❝❤♦

m = ax + by, 0 ≤ y < a.
❚ø ✤â

m∈H

❦❤✐ ✈➔ ❝❤➾ ❦❤✐

x ≥ 0✳

❉♦ ✤â sè ❧î♥ ♥❤➜t ❦❤æ♥❣ t❤✉ë❝

c = a (−1) + b (a − 1) = ab − a − b.
❉♦ ✤â ✈î✐ ♠å✐

m>c

t❤➻

m ∈ H✱

✈➟② ♥➯♥

✻

|N \ H| ≤ c✳

H

❧➔


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
●✐↔ sû
❳➨t

n>2

✈➔

(iii)

◆❣✉②➵♥ ❚❤à ❍á❛

✤ó♥❣ ✈î✐

H = a1 , a2 , . . . an

n − 1✳

✳

an−1
a1
gcd
,...,
= 1✳
d
d
a1
an−1
❚❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣ t❤➻ N \
,...,
< ∞ tù❝
d

d
an−1
a1
,...,
✳
m1 ∈ N s❛♦ ❝❤♦ ✈î✐ ♠å✐ m ≥ m1 t❤➻ m ∈
d
d
❙✉② r❛ ✈î✐ ♠å✐ m ≥ m1 t❤➻ md ∈ a1 , . . . , an−1 ✳
✣➦t

✣➦t

d = gcd (a1 , a2 , . . . , an−1 )

s✉② r❛

❧➔ tç♥ t↕✐

c = dm1 + (d − 1) an + 1 ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✈î✐ ♠å✐ m ≥ c t❤➻ m ∈ H ✳

❚❤➟t ✈➟②✱ ✈➻

gcd (d, an ) = 1

♥➯♥

m

❝â ❜✐➸✉ ❞✐➵♥ ❞✉② ♥❤➜t ❧➔

m = dx + an y

✈î✐

0 ≤ y < d.

❉♦ ✤â

dx = m − an y ≥ (d − 1) an + dm1 + 1 − an y
= (d − 1 − y) an + m1 d + 1 ≥ dm1 .
❉♦ ✤â
❱➟②

x ≥ m1

♥➯♥

dx ∈ a1 , . . . , an−1

✈➔ s✉② r❛

m = dx + an y ∈ H ✳

|N\H| < c < ∞.

✶✳✹ ▼ët sè ❜➜t ❜✐➳♥ ❝õ❛ ♥û❛ ♥❤â♠ sè

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✶✳
❝❤♦

a1 , a2 , . . . , an

❈❤♦

H = a1 , a2 , . . . , an

❧➔ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛

H✳

❧➔ ♠ët ♥û❛ ♥❤â♠ sè s❛♦

❑❤✐ ✤â

• m (H) := minH\ {0} = min {a1 , a2 , . . . , an }
• g (H) := |N\H|
• emb (H) := n

❣✐è♥❣ H
❝❤✐➲✉ ♥❤ó♥❣

✤÷ñ❝ ❣å✐ ❧➔

✤÷ñ❝ ❣å✐ ❧➔

❝õ❛

✳

❝õ❛

✼

❣å✐ ❧➔

H✳

❜ë✐

❝õ❛

H✳


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
• F (H) := max (N\H)
•

❱➔♥❤

◆❣✉②➵♥ ❚❤à ❍á❛
✤÷ñ❝ ❣♦✐ ❧➔

sè ❋r♦❜❡♥✐✉s

k [H] := k th |h ∈ H = k [ta1 , . . . , tan ]

❧➔ ❜✐➳♥ sè✱ ✤÷ñ❝ ❣å✐ ❧➔

❱➼ ❞ö ✶✳✹✳✷✳

❈❤♦

✈➔♥❤ ♥û❛ ♥❤â♠ sè

H = 3, 5, 7

❝õ❛

✈î✐

k

❝õ❛

H✳

H✳

❧➔ ♠ët tr÷í♥❣✱

t

❧➔ ♠ët ♥û❛ ♥❤â♠ sè✳ ❚❛ ❝â

• m (H) = 3
• g (H) = 3
• emb (H) = 3
• F (H) = 4

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✸✳
H\ {0}✳
H

❚❛ ❣å✐ t➟♣

t÷ì♥❣ ù♥❣ ✈î✐

❈❤♦ ♠ët ♥û❛ ♥❤â♠ sè

H = a1 , a2 , . . . , an

Ap (H, a) = {h ∈ H|h − a ∈
/ H}

❧➔

✈➔

t➟♣ ❆♣➨r②

a∈
❝õ❛

a✳

▼➺♥❤ ✤➲ ✶✳✹✳✹✳ ❈❤♦ H ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ✱ a ∈ H\ {0}✳ ❑❤✐ ✤â
Ap (H, a) = {0 = ω (0) , ω (1) , . . . , ω (a − 1)} ,

tr♦♥❣ ✤â ω (i) ❧➔ ♣❤➛♥ tû ❜➨ ♥❤➜t t❤✉ë❝ H s❛♦ ❝❤♦ ω (i) ≡ i (mod a) ✈î✐
♠å✐ i = 0, 1, 2, . . . , a − 1✳
❈❤ù♥❣ ♠✐♥❤✳
❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤

{h ∈ H|h − a ∈
/ H} = {0 = ω (0) , ω (1) , . . . , ω (a − 1)} .
✣➛✉ t✐➯♥ t❛ s➩ ❝❤➾ r❛

{h ∈ H| h − a ∈
/ H} ⊆ {0 = ω (0) , ω (1) , . . . , ω (a − 1)} ,
✽


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
tù❝ ✈î✐ ♠å✐

◆❣✉②➵♥ ❚❤à ❍á❛

h ∈ Ap (H, a)

i ∈ {0, . . . , a − 1}

tç♥ t↕✐

s❛♦ ❝❤♦

h = ω (i)✳

❚❛ ❝â t❤➸ ✈✐➳t

h = aq + i, 0 ≤ i ≤ a − 1.
❱➻

h ∈ H, h ≡ i (mod a)

❚❛ ❝â
❤❛②

s✉② r❛
◆➳✉

h, ω (i) ≡ i (mod a)

h − ω (i) = ap✳

❉♦

ω (i) ≤ h✳

♥➯♥

h ≡ ω (i) (mod a)

♥➯♥

h ≥ ω (i)

♥➯♥

p≥0

❞♦ ✤â

s✉② r❛

✳

(h − ω (i)) ✳✳ a

h − ω (i) − ap = 0,

h − a − ω (i) − a (p − 1) = 0, ❤❛② h − a = ω (i) + a (p − 1) .

p>0

s✉② r❛

ω (i) + a (p − 1) ∈ H

h − a ∈ H✱

❞♦ ✤â

✤✐➲✉ ♥➔② ❧➔

♠➙✉ t❤✉➝♥✳
❱➟②

p=0

✈➔

h = ω (i)✳

◆❣÷ñ❝ ❧↕✐ ✈î✐ ♠å✐
●✐↔ sû

ω(i)

ω (i)

ω (i) − a ∈ H

t❛ s✉② r❛

ω (i) ∈ H ✳

ω (i) − a ≡ i (mod a)✱

♥➯♥

ω (i) ≤ ω (i) − a✱

♥➔② ❧➔ ♠➙✉ t❤✉➝♥ ❞♦
❱➟②

t❛ ❝â

t❤❡♦ t➼♥❤ ♥❤ä ♥❤➜t ❝õ❛

❜➜t ✤➥♥❣ t❤ù❝ ♥➔② ❝❤ù♥❣ tä

a ∈ H \ {0}

❝❤♦ ♥➯♥

a ≤ 0✱

✤✐➲✉

ω (i) − a ∈
/ H✳

ω (i) ∈ Ap (H, a)✳

❱➼ ❞ö ✶✳✹✳✺✳

❈❤♦

H = 5, 9, 13

✳ ❑❤✐ ✤â

Ap(H, 5) = ω(i) | i = 0, 4
= {0, 9, 13, 22, 26} .

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✻✳

❈❤♦

H

❧➔ ♠ët ♥û❛ ♥❤â♠ sè✳

✭✐✮ ❚❛ ❣å✐ sè ♥❣✉②➯♥ ❧î♥ ♥❤➜t ❦❤æ♥❣ t❤✉ë❝
❦➼ ❤✐➺✉ ❧➔

F (H)✱

tù❝

H

F (H) = max(Z\H)✳

✾

❧➔

sè ❋r♦❜❡♥✐✉s

❝õ❛

H✱


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❍á❛

✭✐✐✮ ❚❛ ❣å✐ t➟♣ ❤ñ♣

P F (H) = {x ∈
/ H| x + h ∈ H, ∀h ∈ H\ {0}} ,

t➟♣ ❝→❝ sè ❣✐↔ ❋r♦❜❡♥✐✉s
❣✐↔ ❋r♦❜❡♥✐✉s
❧➔

sè

✱ ♠é✐ ♣❤➛♥ tû ❝õ❛ ♥â ✤÷ñ❝ ❣å✐ ❧➔

✳ ❙è ♣❤➛♥ tû ❝õ❛ t➟♣

H✱

❦➼ ❤✐➺✉ ❧➔

P F (H)

✤÷ñ❝ ❣å✐ ❧➔

❦✐➸✉

❝õ❛

t (H)✳

◆❤➟♥ ①➨t ✶✳✹✳✼✳
✶✳ ❚❤❡♦ ✤✐➲✉ ❦✐➺♥ ✭✐✐✐✮ ❝õ❛ ✤à♥❤ ♥❣❤➽❛ ✶✳✸✳✶ t❤➻ sè ❋r♦❜❡♥✐✉s ❝õ❛

H

❧➔

tç♥ t↕✐✳

✷✳

F (H) ∈ P F (H)✳
❚❤➟t ✈➟②✱ ❣✐↔ sû ♥❣÷ñ❝ ❧↕✐
s❛♦ ❝❤♦

F (H) + h ∈
/H

F (H) + h > F (H)✳

F (H) ∈
/ P F (H)

✤✐➲✉ ♥➔② ✈æ ❧➼ ✈➻

❉♦ ✤â

❧➔ ❤ú✉ ❤↕♥ ♥➯♥

h ∈ H\ {0}

F (H) = max (Z \ H)

♠➔

F (H) ∈ P F (H)✳

✸✳ ❚➟♣ ❝→❝ sè ❣✐↔ ❋r♦❜❡♥✐✉s ❝õ❛

(N \ H)

t❤➻ tç♥ t↕✐

H

❧➔ ❝♦♥ ❝õ❛ t➟♣

(N \ H)✱

♠➔ t➟♣

t(H) < ∞✳

▼➺♥❤ ✤➲ ✶✳✹✳✽✳ ❈❤♦ ≤H ❧➔ ♠ët q✉❛♥ ❤➺ t❤ù tü ①→❝ ✤à♥❤ tr➯♥ Z ❜ð✐

♥➳✉ y − x ∈ H ✳ ❑❤✐ ✤â t➟♣ ❝→❝ sè ❣✐↔ ❋r♦❜❡♥✐✉s ❝õ❛ H ❧➔ ♥❤ú♥❣
♣❤➛♥ tû ❝ü❝ ✤↕✐ ❝õ❛ Z \ H t❤❡♦ q✉❛♥ ❤➺ ≤H ✳

x ≤H y

❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ ♠å✐ x ∈ P F (H) s✉② r❛ x ∈ Z \ H ✳
●✐↔ sû tç♥ t↕✐
◆➳✉

y−x>0

y ∈Z\H
t❤➻ ❞♦

x ≤H y

s❛♦ ❝❤♦

x ∈ P F (H)

♥➯♥

s✉② r❛

y − x ∈ H✳

x + (y − x) ∈ H

♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳
❉♦ ✤â

y−x=0

❤❛②

x = y✳

❱➟②

x ∈ max≤H (Z \ H)✳
✶✵

❤❛②

y ∈ H✱

✤✐➲✉


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
◆❣÷ñ❝ ❧↕✐✱ ✈î✐ ♠å✐

◆❣✉②➵♥ ❚❤à ❍á❛

x ∈ max≤H (Z \ H) ❣✐↔ sû tç♥ t↕✐ h ∈ H \ {0} s❛♦ ❝❤♦

x+h∈
/ H✳
❙✉② r❛

(x + h) − x = h ∈ H ✳
x ≤H x + h✱

❉♦ ✤â

✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈î✐

x ∈ max≤H (Z \ H).

❚➟♣ ❆♣❡r② ❝❤♦ t❛ ♠ët ❝æ♥❣ t❤ù❝ ✤➸ t➻♠ sè ❋r♦❜❡♥✐✉s ✈➔ ❝→❝ sè ❣✐↔
❋r♦❜❡♥✐✉s ♥❤÷ s❛✉

▼➺♥❤ ✤➲ ✶✳✹✳✾✳ ❈❤♦ H =
a ∈ H\ {0}✳

a1 , a2 , . . . , an

❑❤✐ ✤â

❧➔ ♠ët ♥û❛ ♥❤â♠ sè✱

✶✳ F (H) = maxAp (H, a) − a;
✷✳ P F (H) = {ω − a |ω ∈ max≤

H

Ap (H, a)}

❈❤ù♥❣ ♠✐♥❤✳
✶✳ ❈❤ù♥❣ ♠✐♥❤

F (H) = maxAp (H, a) − a✳

maxAp (H, a) ∈ Ap (H, a)

t❛ s✉② r❛

❉♦ ✤â

maxAp (H, a) − a ≤ F (H)

●✐↔ sû

F (H) + a > maxAp (H, a)✳

❚❛ ❝â

a > 0, F (H) + a ∈ H

♠➔

❚❤❡♦ ✤à♥❤ ♥❣❤➽❛

maxAp (H, a) − a ∈
/ H✳

❤❛②

F (H) + a ≥ max Ap (H, a)✳

F (H) + a − a ∈
/H

s✉② r❛

F (H)+a ∈ Ap (H, a)✱ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈î✐ ✤à♥❤ ♥❣❤➽❛ maxAp (H, a)✳
❱➟②

F (H) = maxAp (H, a) − a✳

✷✳ ❈❤ù♥❣ ♠✐♥❤
❱î✐ ♠å✐

x ∈ P F (H) t❛ ❝❤ù♥❣ ♠✐♥❤ x ∈ {ω − a | ω ∈ max≤H Ap (H, a)}✳

❚❤➟t ✈➟②✱ t❛ ❝â
✣➦t
❝❤♦
◆➳✉

P F (H) = {ω − a | ω ∈ max≤H Ap (H, a)}✳

x+a∈H

✈➔

ω = x + a ∈ Ap (H, a)✳
ω ≤H w

s✉② r❛

w−x−a > 0

(x + a) − a ∈
/H

x + a ∈ Ap (H, a)✳

❍ì♥ ♥ú❛ ❣✐↔ sû tç♥ t↕✐

w−ω ∈H
t❤➻

♥➯♥

s✉② r❛

s❛♦

w − x − a ∈ H✳

x + (w − x − a) ∈ H
✶✶

w ∈ Ap (H, a)

❤❛②

w − a ∈ H✱

✤✐➲✉ ♥➔②


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
w ∈ Ap (H, a) .

♠➙✉ t❤✉➝♥ ✈î✐ ❣✐↔ t❤✐➳t
❉♦ ✤â

w =x+a=ω

◆❣÷ñ❝ ❧↕✐✱ ✈î✐ ♠å✐

◆❣✉②➵♥ ❚❤à ❍á❛

❤❛②

ω ∈ max≤H Ap (H, a)

ω ∈ max≤H Ap (H, a)✱

✈➔

x = ω − a✳

t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤

ω − a ∈ P F (H)✳
❉♦

ω ∈ Ap (H, a)

♥➯♥

ω−a∈

/ H✳

●✐↔ sû

ω−a ∈
/ P F (H)

❙✉② r❛

(ω + ai ) − a ∈
/H

♠➔

◆❤÷♥❣

ω ≤H ω + ai

ω = ω + ai ✱

✈➔

t❤➻ tç♥ t↕✐

ai , 1 ≤ i ≤ n

ω + ai ∈ H

♥➯♥

s❛♦ ❝❤♦

ω + ai ∈ Ap (H, a)✳

✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈î✐ ❣✐↔ t❤✐➳t

ω ∈ max≤H Ap (H, a)✳
❉♦ ✤â
❱➟②

ω − a ∈ P F (H)✳

P F (H) = {ω − a |ω ∈ max≤H Ap (H, a)}✳

❱➼ ❞ö ✶✳✹✳✶✵✳

❈❤♦

❚➟♣ ❆♣➨r② ❝õ❛

H

H 4, 7, 9

ù♥❣ ✈î✐

7

✳ ❑❤✐ ✤â t❛ ❝â

❧➔

Ap(H, 7) = {0, 4, 8, 9, 12, 13, 17} ,
✈➔ t➟♣

max≤H Ap(H, 7) = {12, 17} .
❉♦ ✤â

P F (H) = {5, 10}✳

✶✷

ω − a + ai ∈
/ H✳


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❍á❛

◆û❛ ♥❤â♠ sè ✤è✐ ①ù♥❣ ✈➔ ♥û❛ ♥❤â♠ sè ❣✐↔ ✤è✐ ①ù♥❣ ❧➔ ❝→❝ ✤è✐ t÷ñ♥❣
r➜t q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♥û❛ ♥❤â♠ sè✳ ❚r♦♥❣ ♣❤➛♥ t✐➳♣
t❤❡♦ tæ✐ ✤÷❛ r❛ ✤à♥❤ ♥❣❤➽❛ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤❛✐ ❧♦↕✐ ♥û❛ ♥❤â♠ sè
♥➔②✳

✶✳✺ P❤➙♥ ❧♦↕✐ ♥û❛ ♥❤â♠ sè

✶✳✺✳✶ ◆û❛ ♥❤â♠ sè ✤è✐ ①ù♥❣
✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✶✳ H
❈❤♦

♥➳✉ ✈î✐ ♠å✐

x∈Z

◆❤➟♥ ①➨t ✶✳✺✳✷✳

t❤➻

❧➔ ♠ët ♥û❛ ♥❤â♠ sè t❛ ♥â✐

x∈H

❤♦➦❝

H

❧➔

✤è✐ ①ù♥❣

F (H) − x ∈ H.

H ❧➔ ✤è✐ ①ù♥❣ t❤➻ F (H) ❧➫✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû
F (H)
F (H)
F (H) ❧➔ sè ❝❤➤♥✱ tù❝ ❧➔
∈ Z✳ ◆➳✉
∈ H t❤➻
2

2
F (H)
F (H)
2.
= F (H) ∈ H ✱ ✈æ ❧➼✳ ❈❤♦ ♥➯♥
∈
/ H t❤➻ ❞♦ H ✤è✐ ①ù♥❣
2
2
F (H)
F (H)
♥➯♥ F (H) −
∈ H ✱ s✉② r❛
∈ H ✱ ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳
2
2
❱➟② F (H) ❧➔ sè ❧➫✳
◆➳✉

◆û❛ ♥❤â♠ sè ✤è✐ ①ù♥❣ ❝á♥ ✤÷ñ❝ ✤➦❝ tr÷♥❣ ❜ð✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✳

▼➺♥❤ ✤➲ ✶✳✺✳✸✳ ❈❤♦ H ❧➔ ♥û❛ ♥❤â♠ sè✱ a ∈ H\ {0}✳ ✣➦t t➟♣ Ap (H, a) =
{0 = w1 < w2 < . . . < wa }✳

❈→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ t÷ì♥❣✳

✭✶✮ H ✤è✐ ①ù♥❣✳
✭✷✮ wi + wa−i+1 = wa ✈î✐ 2 ≤ i ≤ a − 1✳
✭✸✮ t (H) = 1✳
✭✹✮ P F (H) = {F (H)}✳

✶✸


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❍á❛

✭✺✮ 2g (H) = F (H) + 1✳
❈❤ù♥❣ ♠✐♥❤✳
(1) ⇒ (2)✳
❱➻

H

✤è✐ ①ù♥❣ ♥➯♥ ✈î✐ ♠å✐

s✉② r❛
✣➦t

F (H) = maxAp (H, a) − a = wa − a ∈
/ H✳

❚❛ ❝â

wi ∈ Ap (H, a)

t❛ ❝â

F (H) − (wi − a) ∈ H ✱

wa − wi ∈ H ✳

j = wa −wi ∈ H ✱ ♥➳✉ j ∈
/ Ap (H, a) t❤➻ j−a ∈ H

❞➝♥ tî✐

wa −wi −a ∈

H✳
❙✉② r❛
❱➟②

wa − a = (wa − wi − a) + wi ∈ H ✱

j ∈ Ap (H, a)

▼➦t ❦❤→❝

(2) ⇒ (1)✳

tù❝ ❧➔

wa − wi = w k

w1 < w2 < . . . < wa−1

♥➯♥

✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳

✈î✐

wk ∈ {w1 , w2 , . . . , wa }✳

k = a − i + 1✳

F (H) = maxAp (H, a) − a = wa − a✱

❚❛ ❝â

✈➔

P F (H) = {ω − a | ω ∈ max≤H Ap (H, a)} = {wa − a} = F (H)✳
❑❤✐ ✤â ✈î✐ ♠å✐
❝❤♦

x ≤H α✱

x ∈ Z\H

♠➔

t❛ s✉② r❛ tç♥ t↕✐

max≤H (Z \ H) = {F (H)}

α ∈ max≤H (Z \ H)

♥➯♥

x ≤H F (H) .

s❛♦

❉♦ ✤â

F (H) − x ∈ H ✳
❱➟②

H

❧➔ ✤è✐ ①ù♥❣✳

(1) ⇒ (4)✳❱î✐
❱➻

H

◆➳✉
❱➻

♠å✐

✤è✐ ①ù♥❣ ✈➔

x ∈ P F (H)✱
x∈
/H

F (H) − x = 0

x ∈ P F (H)

t❤➻

♥➯♥

♥➯♥

t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤

x = F (H)✳

F (H) − x ∈ H ✳

0 < F (H) − x ∈ H ✳

x + (F (H) − x) ∈ H

❞♦ ✤â

F (H) ∈ H ✱

✤✐➲✉ ♥➔② ❧➔

♠➙✉ t❤✉➝♥✳
❙✉② r❛

F (H) − x = 0

(4) ⇒ (1)✳

❱î✐ ♠å✐

●✐↔ sû tç♥ t↕✐
❱➻

❤❛②

F (H) = x✳

x∈Z\H

✱ t❛ ❝➛♥ ❝❤➾ r❛

y ∈ max≤H (Z \ H)

s❛♦ ❝❤♦

max≤H (Z \ H) = P F (H) = {F (H)}
✶✹

F (H) − x ∈ H ✳

x ≤H y ✳

♥➯♥

y = F (H)✳

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❙✉② r❛

x ≤H F (H)

(1) ⇔ (5)✳ H
t❤➻

❤❛②

◆❣✉②➵♥ ❚❤à ❍á❛

F (H) − x ∈ H ✳
x ∈ N\H

❧➔ ♥û❛ ♥❤â♠ sè ✤è✐ ①ù♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈î✐ ♠å✐

F (H) − x ∈ H ✳

❚ù❝ ❧➔ tr♦♥❣ t➟♣

{0, 1, . . . , F (H)}

♥û❛ sè ❦❤æ♥❣ t❤✉ë❝

H✳

❝â ✤ó♥❣ ♠ët ♥û❛ sè t❤✉ë❝

H

✈➔ ♠ët

✣✐➲✉ ♥➔② t÷ì♥❣ ✤÷ì♥❣ ✈î✐

g (H) =

F (H) + 1
,
2

❤♦➦❝ t÷ì♥❣ ✤÷ì♥❣ ✈î✐

2g (H) = F (H) + 1.
(3) ⇔ (4)✳

✣✐➲✉ ♥➔② ❧➔ ❤✐➸♥ ♥❤✐➯♥✳

❱➼ ❞ö ✶✳✺✳✹✳
❚❛ ❝â

❈❤♦

H = 4, 5, 6

P F (H) = {7}✳

❑❤✐ ✤â

H

✳
❧➔ ✤è✐ ①ù♥❣✳

✶✳✺✳✷ ◆û❛ ♥❤â♠ sè ❣✐↔ ✤è✐ ①ù♥❣
✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✺✳ H
F (H)
①ù♥❣ F (H)
x ∈ Z\H, x =
2
❈❤♦

♥➳✉

❧➔ ♠ët ♥û❛ ♥❤â♠ sè✱ t❛ ♥â✐

❝❤➤♥ ✈➔ ✈î✐ ♠å✐

H

t❤➻

❧➔

❣✐↔ ✤è✐

F (H) − x ∈

H✳

▼➺♥❤ ✤➲ ✶✳✺✳✻✳ ❈❤♦ H ❧➔ ♠ët ♥û❛ ♥❤â♠ sè✱ F (H) ❝❤➤♥ ✈➔
a ∈ H \ {0}✳

❑❤✐ ✤â ❝→❝ ♠➺♥❤ ✤➲ s❛✉ t÷ì♥❣ t÷ì♥❣✳

✭✶✮ H ❧➔ ❣✐↔ ✤è✐ ①ù♥❣✳
✭✷✮ ❚➟♣ Ap (H, a) ❝â ❞↕♥❣
Ap (H, a) = {0 = w0 < w1 < . . . < wa−2 = F (H) + a}∪
✶✺

F (H)
+a ,
2


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❍á❛

✈➔ wi + wa−i−2 = wa−2,
✭✸✮ P F (H) =

F (H) ,

0 ≤ i ≤ a − 2✳

F (H)
2

✳

✭✹✮ 2g (H) = F (H) + 2✳
❈❤ù♥❣ ♠✐♥❤✳
(1) ⇒ (2)✳ ●✐↔ sû H ❧➔ ❣✐↔ ✤è✐ ①ù♥❣✳
F (H)
+ a ∈ Ap (H, a)✳
❚❛ ❝❤➾ r❛
2
F (H)
F (H)
F (H)
◆➳✉
+a ∈
/ H t❤➻ ✈➻
+a =
2
2
2

✈➔

H

t❛ ❝â

F (H) −

F (H)

+a
2

∈ H,

❤❛②

F (H)
− a ∈ H.
2
❙✉② r❛

F (H)
=a+
2

F (H)
−a
2

❤❛②

F (H)
∈ H,
2
✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳
❉♦ ✤â

F (H)
+ a ∈ H.

2
F (H)
F (H)
❍ì♥ ♥ú❛
+a −a=
∈
/ H.
2
2
F (H)
❉♦ ✤â
+ a ∈ Ap (H, a)✳
2

✶✻

∈ H,

❧➔ ❣✐↔ ✤è✐ ①ù♥❣ ♥➯♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❍á❛

❱➻ ✈➟② t❛ ❝â t❤➸ ✈✐➳t

Ap (H, a) = {0 = w0 < w1 < . . . < wa−2 = F (H) + a} ∪
❈❤ù♥❣ ♠✐♥❤
❱î✐ ♠å✐

❱➻

F (H)
+a .
2

wi + wa−i−2 = wa−2 , 0 ≤ i ≤ a − 2.

0 ≤ i ≤ a − 2✱

wi − a ∈
/ H

✈➔

wa−2 − wi = F (H) + a − wi .
F (H)
♥➯♥ F (H) − (wi − a) ∈ H
wi − a =
2
t❛ ❝â

❤❛②

wa−2 − wi ∈ H ✳
wa−2 − wi − a ∈
/ H ♥➯♥ wa−2 − wi ∈ Ap (H, a)✳
F (H)
▼➦t ❦❤→❝✱ wa−2 −wi =
♥➯♥ wa−2 −wi ∈ {0 = w0 < w1 < . . . < wa−2 }✳

2
❱➟② tç♥ t↕✐ j : 0 ≤ j ≤ a − 2 t❤ä❛ ♠➣♥ wa−2 − wi = wj ✱ ❧➟♣ ❧✉➟♥ t÷ì♥❣
❍ì♥ ♥ú❛

j = a − i − 2✳
F (H)
x ∈ Z\H, x =
t❛ s➩ ❝❤➾
2

tü tr♦♥❣ ♠➺♥❤ ✤➲ ✶✳✺✳✸ t❛ s✉② r❛

(2) ⇒ (1)✳
❚➟♣

❱î✐ ♠å✐

Ap (H, a)

r❛

F (H) − x ∈ H ✳

❧➔ ♠ët ❤➺ t❤➦♥❣ ❞÷ ✤➛② ✤õ ♠æ ✤✉♥

a✱

❞♦ ✤â tç♥ t↕✐

✳

w ∈ Ap (H, a) s❛♦ ❝❤♦ w − x ✳✳ a✱ s✉② r❛ tç♥ t↕✐ k ∈ Z s❛♦ ❝❤♦ w = x + ka✳
❚❛ ❝â

k=0

❍ì♥ ♥ú❛

✈➻

w = x✳

k>0

✈➻ ♥➳✉

k≤0

t❤➻

x = w + (−ka) ∈ H ✱

t❤✉➝♥✳
❚❛ ①➨t ✷ tr÷í♥❣ ❤ñ♣
❚❍✶✳

w=

F (H)
+ a✳

2

❑❤✐ ✤â

F (H) − x = F (H) − w + ka
= F (H) −
=

❱➻

x=

F (H)
2

♥➯♥

F (H)
− a + ka
2

F (H)
+ (k − 1) a.
2

k > 1✳
✶✼

✤✐➲✉ ♥➔② ❧➔ ♠➙✉

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❉♦ ✤â
❚❍✷✳

F (H)
+ a + (k − 2) a ∈ H.
2

F (H) − x =

w=

◆❣✉②➵♥ ❚❤à ❍á❛

F (H)
+ a✱
2

t❛ s✉② r❛ tç♥ t↕✐

i, 0 ≤ i ≤ a − 2

s❛♦ ❝❤♦

w = wi ✳

❑❤✐ ✤â

F (H) − x = wa−2 − a − wi + ka

= wa−2 − wi + (k − 1) a
= wa−i−2 + (k − 1) a ∈ H.
(1) ⇒ (3)✳

●✐↔ sû

H

❧➔ ❣✐↔ ✤è✐ ①ù♥❣✱ t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤

F (H)
✳
2
F (H)
❚r÷î❝ t✐➯♥ ❝❤ù♥❣ ♠✐♥❤
∈ P F (H)✳
2
F (H)
●✐↔ sû tç♥ t↕✐ h = 0 ∈ H s❛♦ ❝❤♦
+h∈
/ H✳
2
F (H)
+h ∈H
❉♦ H ❧➔ ❣✐↔ ✤è✐ ①ù♥❣ ♥➯♥ F (H) −
2
F (H)
− h ∈ H✳
❤❛②
2

F (H)
F (H)
❙✉② r❛
− h + h ∈ H ❤❛②
∈ H ✱ ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳
2
2
F (H)
❉♦ ✤â ❣✐↔ sû s❛✐✱ tù❝ ❧➔ ✈î✐ ♠å✐ h = 0 ∈ H t❤➻
+ h ∈ H✳
2
F (H)
❱➟②
∈ P F (H)✳
2
F (H)
❚❛ s➩ ❝❤➾ r❛ ✈î✐ ♠å✐ x ∈ P F (H) , x = F (H) t❤➻ x =
✳ ❚❤➟t ✈➟②✱
2
F (H)
❣✐↔ sû x =
✱ ❞♦ x ∈
/ H s✉② r❛ F (H) − x ∈ H ✳
2
❱➻ x = F (H) ♥➯♥ F (H) − x > 0✳
P F (H) =

❉♦

F (H) ,

x ∈ P F (H)

♥➯♥

x + (F (H) − x) ∈ H

♠➙✉ t❤✉➝♥✳

✶✽

❤❛②

F (H) ∈ H ✱

✤✐➲✉ ♥➔② ❧➔


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❱➟②

P F (H) =

(3) ⇒ (1)✳
❱➻

F (H) ,

❱î✐ ♠å✐

x ∈ Z\H

max≤H (Z\H)✳

◆❣✉②➵♥ ❚❤à ❍á❛
F (H)
2

✳

F (H)
t❛ s➩ ❝❤➾ r❛ F (H) − x ∈ H ✳
2
y ∈ max≤H (Z\H) s❛♦ ❝❤♦ x ≤H y ♠➔

x ∈ Z\H, x =

♥➯♥ tç♥ t↕✐
❉♦ ✤â

y = F (H) ,
❤♦➦❝

y=

F (H)
.
2

y = F (H) t❤➻ x ≤H F (H) s✉② r❛ F (H) − x ∈ H ✳

F (H)
F (H)
F (H)
t❤➻ x ≤H
s✉② r❛
− x ∈ H✳
◆➳✉ y =
2
2
2
F (H)
F (H)
F (H)
❱➻ x =
♥➯♥ 0 <
− x ♠➔
∈ P F (H) ♥➯♥
2
2
2
◆➳✉

F (H)
+
2

F (H)
−2
2

∈H

❤❛②

F (H) − x ∈ H.

F (H)
t❤➻ t❛ ❝â F (H) − x ∈ H ✳
2
❳➨t t➟♣ A = {0, 1, . . . , F (H)}✱ A ❝â F (H) + 1 ♣❤➛♥ tû✳
F (H)
F (H)
❱➻ H ❧➔ ❣✐↔ ✤è✐ ①ù♥❣ ♥➯♥ tr♦♥❣ t➟♣ A \
s➩ ❝â
♣❤➛♥ tû
2
2
❦❤æ♥❣ t❤✉ë❝ H ✳
F (H)
❉♦ ✤â g(H) =
+1
2
F (H)
+1
(4) ⇒ (1)✳ ❚❛ ❝â g(H) =
2
F (H)
F (H)
❚❛ t❤➜② ✈î✐ ♠å✐ 1 ≤ x ≤
t❤➻

+ 1 ≤ F (H) − x ≤ F (H) − 1✳
2
2
❍ì♥ ♥ú❛ x ✈➔ F (H) − x ❦❤æ♥❣ ❝ò♥❣ t❤✉ë❝ H ✳ ❉♦ ✤â
(1) ⇒ (4)✳

❱î✐ ♠å✐

x ∈ N\H, x =

g(H) ≥

F (H)
− 1.
2

✶✾


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
▲↕✐ ❝â

F (H)
2

✈➔

◆❣✉②➵♥ ❚❤à ❍á❛

F (H) ∈

/ H✱

s✉② r❛

g(H) ≥

F (H)
+ 1.
2

F (H) − x ❝â ♠ët ♣❤➛♥
F (H)
tû t❤✉ë❝ H ✈➔ ♠ët ♣❤➛♥ tû ❦❤æ♥❣ t❤✉ë❝ H ✱ 1 ≤ ∀x ≤
− 1✳ ✭✯✮
2
F (H)
✱ t❛ s✉② r❛ 1 ≤ x ≤ F (H)✳
❇➙② ❣✐í ✈î✐ ♠å✐ x ∈ N\H, x =
2
F (H)
❚❍✶✿ ◆➳✉ 1 ≤ x ≤
− 1.
2
❘ã r➔♥❣ ❜ð✐ ✭✯✮ ♥➳✉ x ∈
/ H t❤➻ F (H) − x ∈ H.
F (H)
❚❍✷✿ ◆➳✉
+ 1 ≤ x ≤ F (H) − 1✳
2
❇ð✐ ✭✯✮ ♥➳✉ x ∈

/ H t❤➻ F (H) − x ∈ H ✳
❉➜✉ ✧❂✧ ❝❤➾ ①↔② r❛ ❦❤✐ tr♦♥❣ ❤❛✐ ♣❤➛♥ tû

❚❍✸✿

x = F (H)

❱➼ ❞ö ✶✳✺✳✼✳

t❤➻

F (H) − x = 0 ∈ H.

H = 3, 4, 5 .
F (H)
❚❛ ❝â P F (H) = {1; 2} =
; F (H)
2
❑❤✐ ✤â H ❧➔ ❣✐↔ ✤è✐ ①ù♥❣✳
❈❤♦

✷✵

✳

x

✈➔