Luận văn thạc sỹ toán về phương trình tuyến tính với các số fibonacci

ĐẠI HỌC THÁI NGUYÊNNGƯỜI HƯỚNG DẪN KHOA HỌC THÁI NGUYÊN - 2019.

ĐẠI HỌC THÁI NGUYÊN

NGƯỜI HƯỚNG DẪN KHOA HỌC

THÁI NGUYÊN - 2019

✈✐➺❝ ❤÷î♥❣ ❞➝♥✱ ✤ë♥❣ ✈✐➯♥ ❦❤✉②➳♥ ❦❤➼❝❤ tæ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ tæ✐ t❤ü❝

❤✐➺♥ ✤➲ t➔✐✳ ❚r♦♥❣ q✉→ tr➻♥❤ t✐➳♣ ❝➟♥ ✤➲ t➔✐ ✤➳♥ q✉→ tr➻♥❤ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥

✈➠♥ ❚❤➛② ❧✉æ♥ t➟♥ t➻♥❤ ❝❤➾ ❜↔♦ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ❝❤♦ tæ✐ ❤♦➔♥t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❈❤♦ ✤➳♥ ❜➙② ❣✐í ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛ tæ✐ ✤➣ ✤÷ñ❝ ❤♦➔♥t❤➔♥❤✱ ①✐♥ ❝↔♠ ì♥ ❚❤➛② ✤➣ ✤æ♥ ✤è❝ ♥❤➢❝ ♥❤ð tæ✐✳

❚æ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥ ✈➔ P❤á♥❣

✣➔♦ t↕♦ ❝õ❛ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚æ✐ ①✐♥ tr➙♥trå♥❣ ❝↔♠ ì♥ ❝→❝ ❚❤➛②✱ ❈æ ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ✤↕t ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ q✉þ

❜→✉ ❝ô♥❣ ♥❤÷ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ♥❤➜t ✤➸ tæ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥

♥➔②✳

❚æ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr÷í♥❣ ❚❍P❚

❍♦❛ ▲÷ ❆ ✲ ◆✐♥❤ ❇➻♥❤ ♥ì✐ tæ✐ ❝æ♥❣ t→❝ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï tæ✐ ❤♦➔♥t❤➔♥❤ ❝æ♥❣ ✈✐➺❝ ❝❤✉②➯♥ ♠æ♥ t↕✐ ♥❤➔ tr÷í♥❣ ✤➸ tæ✐ ❤♦➔♥ t❤➔♥❤ ❝❤÷ì♥❣ tr➻♥❤

❤å❝ t➟♣ ❝❛♦ ❤å❝✳

❈✉è✐ ❝ò♥❣✱ tæ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱

♥❤ú♥❣ ♥❣÷í✐ ❦❤æ♥❣ ♥❣ø♥❣ ✤ë♥❣ ✈✐➯♥✱ ❤é trñ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ❝❤♦tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳

❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✾

❚→❝ ❣✐↔

✣✐♥❤ ❚❤à ❍✉②➲♥

❞➣② ❋✐❜♦♥❛❝❝✐ ♥❤÷ ❞➣② k✲❋✐❜♦♥❛❝❝✐✳✳✳ ❍➛✉ ❤➳t ♥❤ú♥❣ t➼♥❤ ❝❤➜t tèt ❝õ❛

♥❤ú♥❣ ❞➣② ♥➔② ✤➲✉ ①✉➜t ♣❤→t tø ❞➣② ❋✐❜♦♥❛❝❝✐✳ ▼ët ❞➣② tç♥ t↕✐ s♦♥❣ s♦♥❣

✈î✐ ❞➣② ❋✐❜♦♥❛❝❝✐ ❧➔ ❞➣② ▲✉❝❛s✳ ❉➣② ♥➔② ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ ✤➦❝ ❜✐➺t tr♦♥❣t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡✳ ❍❛✐ ❞➣② ♥➔② ❧➔ ❝❤ó♥❣ ❝â

❤❛✐ ❜➔✐ t♦→♥ ✼✼✾ ✈➔ ✽✵✹ ✈➔ ❧í✐ ❣✐↔✐ ❝õ❛ ❤❛✐ ❜➔✐ t♦→♥ ♥➔②✳ ❈→❝ ❦➳t q✉↔ ✤➣

❜✐➳t ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷ñ❝ ✈✐➳t t❤❡♦ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✱ ❬✸❪✳

❈❤÷ì♥❣ ✷ t❛ t➟♣ tr✉♥❣ ✤✐ t➻♠ ❤✐➸✉ ❜➔✐ t♦→♥ tê♥❣ q✉→t✱ ❧í✐ ❣✐↔✐ ❜➔✐ t♦→♥

tr♦♥❣ ❦❤✐ m = 3, 4 tø ✤â ✤÷❛ r❛ ❞ü ✤♦→♥ ❧í✐ ❣✐↔✐ ❝❤♦ ❜➔✐ t♦→♥ tê♥❣ q✉→t✳

❈ö t❤➸ tr♦♥❣ ♣❤➛♥ ✷✳✶ ❣✐î✐ t❤✐➺✉ ❜➔✐ t♦→♥ tê♥❣ q✉→t✳ P❤➛♥ ✷✳✷ tr➻♥❤ ❜➔②

❧í✐ ❣✐↔✐ tr♦♥❣ tr÷í♥❣ ❤ñ♣ m = 3 ❤♦➦❝ 4✳ P❤➛♥ ✷✳✸ tr➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ ❝❤♦tr÷í♥❣ ❤ñ♣ tê♥❣ q✉→t ✤â ❧➔ ✣à♥❤ ❧þ ♥❣➝✉ ♥❤✐➯♥✳ P❤➛♥ ✷✳✹ ✤➳♥ ❤➳t ✷✳✽ ❧➔

❝→❝ ❦➳t q✉↔ ①♦❛② q✉❛♥❤ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ✣à♥❤ ❧þ ♥❣➝✉ ♥❤✐➯♥✳ ❈→❝ ❦➳tq✉↔ ✤➣ ❜✐➳t ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷ñ❝ ✈✐➳t t❤❡♦ t➔✐ ❧✐➺✉ ❬✹❪✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❈→❝ ❤➔♠ Fa,b(n) = aϕn + b(1 − ϕ)n ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠s✐♥❤✳

❚✉② ✈➟② ❦❤→❝ ✈î✐ ❞➣② ❋✐❜♦♥❛❝❝✐✱ ❤❛✐ sè ✤➛✉ t✐➯♥ tr♦♥❣ ❞➣② ▲✉❝❛s ❧➔

L0 = 2 ✈➔ L1 = 1 ✭tr♦♥❣ ❞➣② ❋✐❜♦♥❛❝❝✐ ❧➔ ✵ ✈➔ ✶✮✳ ❈❤➼♥❤ ✈➻ t❤➳ ♠➔ ♠ët sèt➼♥❤ ❝❤➜t ❝õ❛ sè ▲✉❝❛s s➩ ❦❤→❝ ✈î✐ sè ❋✐❜♦♥❛❝❝✐✳

▼➺♥❤ ✤➲ ✶✳✶✳✺✳ ❱î✐ ♠å✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ n, t❛ ❝â

Ln = 1 +

√52

!n

− 1 −

√52

!n

❚ø ▼➺♥❤ ✤➲ ✶✳✶✳✸ ✈➔ ▼➺♥❤ ✤➲ ✶✳✶✳✺ t❛ ❝â ✤à♥❤ ❧þ s❛✉✳ ✣à♥❤ ❧þ ❝❤♦ t❛

♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❝→❝ sè ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② ❋✐❜♦♥❛❝❝✐ ✈➔ ❞➣② ▲✉❝❛s✳

❚r♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ✤÷ñ❝ ❣å✐ ❜➡♥❣ ❝→❝❤ t❤❛② n = a(2) = b ✈➔ ✤➦t

x(1) = b − a(1), x(i) = a(i) − b, i = 3, 4, , m

❑❤✐ ✤â tø ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ t❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ s❛✉

Fb = Fx(1)+ F−x(3) + F−x(4)+ + F−x(m), ✭✷✳✷✮tr♦♥❣ ✤â

0 < x(1) < b; 0 < x(3) < x(4) < < x(m) ✭✷✳✸✮

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✷✳ P❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ rót ❣å♥

❝õ❛ ✭✷✳✶✮✳

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✸✳ ▼ët ♥❣❤✐➺♠ b, x(1), x(3), x(4), , x(m) ❝õ❛ ♣❤÷ì♥❣tr➻♥❤ ✭✷✳✷✮ ✤÷ñ❝ ❣å✐ ❧➔ 1 − t❤❛♠ sè ♥➳✉

❱➼ ❞ö ✷✳✶✳✼✳ ✣è✐ ✈î✐ ✤ç♥❣ ♥❤➜t t❤ù❝

Fb = Fb −1+ Fb −2,t❛ ❝â

y(1) = |b − x(1)| = |b − (b − 1)| = 1,y(3) = |b − x(3)| = |b − (b − 2)| = 2

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✽✳ ❚❛ ♥â✐ sè ❋✐❜♦♥❛❝❝✐ Fz ❧➔ ❧î♥✱ ♥➳✉ z > 2✳ ❚❛ ♥â✐ ♠ët

✤ç♥❣ ♥❤➜t t❤ù❝ ❧➔ ❧î♥ ♥➳✉ ♠➔ ❝→❝ ✈➳ ❝õ❛ ♥â ❧➔ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛

❝→❝ sè ❋✐❜♦♥❛❝❝✐ ❝â t➜t ❝↔ ❝→❝ ❝❤➾ sè ✤➲✉ ❧î♥ ❤ì♥ 2✳ ▼ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣tr➻♥❤ ✭✷✳✷✮ ✤÷ñ❝ ❣å✐ ❧➔ ❧î♥ ♥➳✉ b > 2 ✈➔ x(i) > 2 ✈î✐ ♠å✐ i = 1, 3, 4, , m

j ∈J Fj t❤ä❛ ♠➣♥✿

❤♦➦❝ ❧➔ P

j ∈J Fj = 0✱ ❤♦➦❝ P

j ∈J Fj = Fb✱

j ∈J Fj = 0.

✷✳✷ ❚r÷í♥❣ ❤ñ♣ m = 3 ✈➔ m = 4

▼ö❝ ✤➼❝❤ ❝õ❛ ♣❤➛♥ ♥➔② ❧➔ t➻♠ ❧í✐ ❣✐↔✐ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ tr♦♥❣tr÷í♥❣ ❤ñ♣ m = 3 ✈➔ m = 4 ❱î✐ m = 3, ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ❝â ❞↕♥❣

Fb = Fx(1)+ F−x(3), 0 < x(1) < b, 0 < x(3)

❚r÷î❝ t✐➯♥ t❛ ❝➛♥ ❜ê ✤➲ s❛✉✳

❇ê ✤➲ ✷✳✷✳✶✳ ◆➳✉ {b, x(1), x(3)} ❧➔ ♠ët ♥❣❤✐➺♠ ❧î♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮

✈î✐ m = 3 t❤➻ x(1) = b − 1, x(3) = b − 2, b ❧➫✱ b ≥ 5 ❤♦➦❝ x(1) = b − 2,x(3) = b − 1, b ❝❤➤♥✱ b ≥ 5

❚✐➳♣ t❤❡♦ t❛ ①➨t tr÷í♥❣ ❤ñ♣ m = 4 ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ❝â ❞↕♥❣

Fb = Fx(1) + F−x(3) + F−x(4),tr♦♥❣ ✤â b > x(1) > 0 ✈➔ 0 < x(3) < x(4)

❚÷ì♥❣ tü ♥❤÷ tr♦♥❣ tr÷í♥❣ ❤ñ♣ m = 3 t❛ ❝â ✤à♥❤ ❧þ s❛✉✳

✣à♥❤ ❧þ ✷✳✷✳✹✳ ❑❤✐ m = 4 t❤➻ b, x(1), x(3), x(4) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣tr➻♥❤ ✭✷✳✷✮ ♥➳✉ ♥â ❧➔ ♠ët tr♦♥❣ ✶✵ ♥❣❤✐➺♠ ❦❤→❝ ♥❤❛✉ ✤÷ñ❝ tr➻♥❤ ❜➔② tr♦♥❣

❜↔♥❣ ✷✳✶✱ ✷✳✷ ✈➔ ✷✳✸ ❞÷î✐ ✤➙②✳

❍❛✐ ♥❤â♠ ❣✐è♥❣ ♥❤❛✉ ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ♥❣✉②➯♥ tè 1−t❤❛♠ sè ❝õ❛ ♣❤÷ì♥❣tr➻♥❤ ✭✷✳✷✮ ✈î✐ m = 4 ✤÷ñ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❜↔♥❣ ✷✳✶ ❞÷î✐ ✤➙②✳

◆❤➟♥ ①➨t ✷✳✷✳✺✳ ▼÷í✐ ♥❤â♠ ♥❣❤✐➺♠ rí✐ ♥❤❛✉ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ❦❤✐

m = 4 ✤÷ñ❝ tr➻♥❤ ❜➔② ð tr➯♥✱ ✤➦t ❝❤♦ t❛ ❝➙✉ ❤ä✐ ✈➲ ✈✐➺❝ ①→❝ ✤à♥❤ ♠➟t ✤ë

❝õ❛ ❝→❝ ♥❣❤✐➺♠✳ ❚❛ ❝â t❤➸ t✐➳♥ ❤➔♥❤ ✈✐➺❝ ✤â ♥❤÷ s❛✉✿ ▼é✐ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✷✮t❤ä❛ ♠➣♥ ✭✷✳✸✮ ❧➔ ♠ët ❜ë ❜è♥✳

❚❛ q✉② ÷î❝ ❦➼ ❤✐➺✉ o, o′, o′′ t÷ì♥❣ ù♥❣ ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ ❧➫ tò② þ✳

◆➳✉ J ❧➔ t➟♣ ❝♦♥ ❝→❝ sè ♥❣✉②➯♥ t❤➻ a ≤ J ≤ b ❝â ♥❣❤➽❛ ❧➔ a ≤ j ≤ b ✈î✐

♠å✐ j ∈ J✳ ❚÷ì♥❣ tü J ❧➔ ❝❤➤♥ ✭❧➫✮ ♥❣❤➽❛ ❧➔ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû ❝õ❛ t➟♣

❤ñ♣ J ❧➔ ❝❤➤♥ ✭❧➫✮✳ ❑❤✐ ✤â ❦➳t q✉↔ s❛✉ ❧➔ ✤à♥❤ ❧þ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔②✳

✭✶✮ Fb = Fb −o−3 + F−(b−o−2)+ F−(b−o)+ + F−(b−1).

✭✷✮ Fb = Fb −2 + F−b + F−(b−1)

✭✸✮ Fb = Fb −2+ F−b+ F−(b+2)+ F−(b+4)+ + F−(b+o′′ +1) + F−(b+o ′′ +2)

✭✹✮ Fb = Fb −0−3 + F−(b−0−1)+ F−(b−0)+ + F−(b−1) + F−b + F−(b+1)

✭✺✮ Fb = Fb −0−3 + F−(b−0−1)+ F−(b−0)+ + F−(b−1) + F−b + F−(b+2)+ F−(b+4) + + F−(b+o′′ +1) + F−(b+o ′′ +2)

✭✻✮ Fb = Fb −o−3 + F−(b−o−2)+ F−(b−o)+ + F−(b−3)+ F−b+ F−(b+1)

✭✼✮ Fb = Fb −o−3 + F−(b−o−2)+ F−(b−o)+ + F−(b−3)+ F−b+ F−(b+2)+ F−(b+4) + + F−(b+o′′ +1) + F−(b+o ′′ +2)

✭✽✮ Fb = Fb −o−4−o ′+F−(b−o−3−o′ )+F−(b−o−1−o′ )+ +F(b−o−4)+F−(b−o−1)+

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❞↕♥❣ ✾ tr♦♥❣ ✤à♥❤ ❧þ ♥❣➝✉ ♥❤✐➯♥ ✷✳✸✳✶ ❧➔

♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✱ ❝→❝ ❞↕♥❣ ❦❤→❝ s➩ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣tü✳

❚r÷î❝ ❤➳t✱ t❛ s➩ ❝❤➾ r❛ r➡♥❣✿

F(b+1) = F−(b+2)+ F−(b+4)+ · · · + F−(b+o ′′ +1)+ F−(b+o′′ +2)

❚❤➟t ✈➟② ❞♦ b ❧➔ ❝❤➤♥✱ ✈➔ t❤❡♦ ❜ê ✤➲ ✷✳✸✳✹ t❛ ❝â✿

F−(b+o′′ +1)+ F−(b+o′′ +2) = F−(b+o′′ )

F−(b+o′′ )+ F−(b+o′′ −1) = F−(b+o′′ −2)

· · ·

❈ù t✐➳♣ tö❝ q✉→ tr➻♥❤ ♥❤÷ ✈➟② t❛ ❝â

F−(b+5) + F−(b+4) = F−(b+3)

F−(b+3) + F−(b+2) = F−(b+1)

Fv − Fv −1 + Fv −2 − Fv −3+ · · · < Fv −1.

0 < u(1) < z, u(3) < u(4) < < u(n) < z, n ≥ 3.

❑❤✐ ✤â t❛ ❝â ♠ët ❜ê ✤➲ q✉❛♥ trå♥❣ s❛✉✳

0 = F−u(3) + + F−u(p) + Fz −o,

✈î✐ u(i) ≤ z − o + 1✳ ❑❤✐ ✤â u(p) = z − o + 1

✭❜✮ ❈❤♦ sè ♥❣✉②➯♥ ❞÷ì♥❣ p ≥ 3 t❤ä❛ ♠➣♥

Fz−(o+1) = F−u(3) + + F−u(p),

✈î✐ u(i) ≤ z − (o + 1) + 1✳ ❑❤✐ ✤â u(p) = z − (o + 1) + 1✳

●✐↔ sû ∀i ∈ K✱ t❛ ❧✉æ♥ ❝â u(i) 6 z − o − 1.

❑❤✐ ✤â t❤❡♦ ❜ê ✤➲ ✷✳✸✳✻ t❛ ❝â✿ X

0 = F−u(3) + F−u(4) + + F−u(n−1)+ Fz −3 ✭✷✳✺✮

❱î✐ 3 ≤ i ≤ n − 1✱ t❛ ❝â u(i) ≤ z − 2 ✈➻ u(i) ≤ u (n − 1) < u(n) = z − 1✳

⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✹✳✷ ✭❛✮ t❛ ❝â u (n − 1) = z − 2 ❉♦ ✤â✱ tø t➼♥❤ ❝❤➤♥ ❝õ❛

z s✉② r❛

F−(z−3)+ F−u(n−1) = Fz −3 − Fz −2 = −Fz −4

✤➲ ✷✳✹✳✶✭❜✮ ✈➔ ✷✳✹✳✸✱ j 6= 1✳ ❉♦ ✤â j = n✳ ❱➟② t❛ ❝â u(n) = z − 1

✭❜✮ ❚❤❡♦ ❝→❝ ❇ê ✤➲ ✷✳✹✳✺✭❛✮ ✈➔ ✷✳✹✳✸✱ u(1) 6= z − 1✳ ●✐↔ sû u(1) = z − 2,t❤❡♦ ❇ê ✤➲ ✷✳✹✳✺✭❛✮ ✈➔ ✷✳✹✳✶ ✱ t❛ ❝â Fz = Fu(1) + F−u(n) = Fz −2 + Fz −1 ❧➔

♠ët ♥❤➙♥ tè t❤ü❝ sü ❝õ❛ ✭✷✳✹✮✱ ♠➙✉ t❤✉➝♥ ✈î✐ ❣✐↔ t❤✐➳t✱ ✈➻ n > 3✳ ❉♦ ✤âu(1) < z − 2

❈❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✹✳✺ ✭❛✮ ✈î✐ n = m, z = b, u(i) = x(i) t❛t❤➜② x(m) = z − 1✳

❚❤❡♦ ❇ê ✤➲ ✷✳✹✳✺✱ x(i) < b − 2 ✈î✐ ♠å✐ i✳ ❉♦ ✤â ✈➻ m − 1 ≥ 3 →♣ ❞ö♥❣ ♠ët

❧➛♥ ♥ú❛ ❇ê ✤➲ ✷✳✹✳✺ ✭❛✮ ✈î✐ n = m − 1, z = b − 2, u(i) = x(i) ❝❤♦ t❛

x (m − 1) = z − 1 = b − 3

◆➳✉ m − 2 ≥ 3 ❝❤ó♥❣ t❛ ❝â t❤➸ trø ❝↔ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❝❤♦ Fb −3 ✈➔

→♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✹✳✺ t❛ ❝â x(i) < b−4✳ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✹✳✺ ✭❛✮ ✈î✐ z = b−

4, u(i) = x(i), n = m − 2 ❑❤✐ ✤â x (m − 2) = b − 5✳ ❚✐➳♣ tö❝ q✉→ tr➻♥❤ ♥➔②✱

❜➡♥❣ q✉② ♥↕♣ t❛ ❝â Fb = Fx(1)+ F−(b−o)+ F−(b−o+2)+ + F−(b−3)+ F−(b−1)

●✐↔ sû n, z, u(k), k = 1, 3, 4, , ❧➔ ❝→❝ sè ♥❣✉②➯♥ s❛♦ ❝❤♦

Fz = Fu(1) + F−u(3)+ F−u(4) + + F−u(n) ✭✷✳✻✮

❧➔ ♠ët ✤ç♥❣ ♥❤➜t t❤ù❝ ❧î♥✱ ♥❣✉②➯♥ tè✱ ❝❤➤♥ ✈î✐ u(1) < z, u(3) < u(4) < < u(n), n ≥ 3 ✈➔ ✈î✐ u(i) ≥ z ✈î✐ i ≥ 3 ♥➔♦ ✤â✳

tr➻♥❤ ✭✷✳✻✮ t❛ ❝â u(j) < z + o − 1 ✈î✐ ♠å✐ j ≤ n − 1✳ ❱➻ z ❝❤➤♥✱ ❝❤➾ sè ❝❤➤♥

❧î♥ ♥❤➜t ❝õ❛ ✈➳ ❜➯♥ ♣❤↔✐ ❧➔ z + o − 3✳ ❉♦ ✤â✱ s❛✉ ❦❤✐ ❝❤✉②➸♥ ❝→❝ sè ➙♠s❛♥❣ ❜➯♥ tr→✐ ✈➔ sû ❞ö♥❣ ❣✐↔ t❤✐➳t o ≥ 3 t❛ ❝â

♥➔♦ ✤â✳ ❉♦ ✤â tø ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✱ t❛ ❝â i = k ✈➔ x(j) ≤ b − 3 ✈î✐ j < k✳

♠➙✉ t❤✉➝♥ ✈î✐ ❍➺ q✉↔ ✷✳✹✳✹✳

✭✐✈✮ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ♥➳✉ i ∈ S1 t❤➻ x(i) < b✳ ❚❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✱x(i) ❧➔ ❧î♥ ♥❤➜t ①↔② r❛ ❦❤✐ i = k ❤♦➦❝ i = 1✳ ❉♦ ✤â tø ✭✐✮✱ ✭✐✐✮ ✈➔ ✭✐✐✐✮ t❛ ❝âx(k) = b − 1✳

❚❤❡♦ ❇ê ✤➲ ✷✳✻✳✷ ✭✐✈✮ ❝â ♠ët sè ♥❣✉②➯♥ ❧î♥ ♥❤➜t p✱ ✈î✐ 1 ≤ p ≤ k − 2s❛♦ ❝❤♦

❍ì♥ ♥ú❛✱ ♥➳✉ x (k − p) = b − (p + 2) ❧➔ ✤ó♥❣ t❤➻ ✤ç♥❣ ♥❤➜t t❤ù❝ ✭✷✳✶✻✮s➩ ❦❤æ♥❣ ✤ó♥❣✱ ✈➔ ✈➻ ✈➟② ✤ç♥❣ ♥❤➜t t❤ù❝ ✭✷✳✶✵✮ s➩ ❦❤æ♥❣ ①↔② r❛✳ ❱➻ t❤➳x(j) ≤ b − (p + 3)✳

· · · ,x(k − p) = (b − p − 2) − 1

❳→❝ ✤✐♥❤ o ✈➔ o′ tø ❝→❝ ♣❤÷ì♥❣ tr➻♥❤

(b − p − 2) − q = b − o − 4 − o′

✈➔ b − o − 4 = b − q − 3✳ ❱➻ b✱ q✱ ✈➔ p ❝❤➤♥ ♥❣❛② ❝↔ ❦❤✐ o ✈➔ o′ ❧➔ ❧➫✳ ❑➳t

❤ñ♣ ♥❤ú♥❣ ❦➳t q✉↔ ♥➔② ✈î✐ ✭✷✳✶✸✮ t❛ ❝â ✤ç♥❣ ♥❤➜t t❤ù❝ ✭✷✳✶✻✮✳

✭✐✐✐✮ ●✐↔ sû ♥❣÷ñ❝ ❧↕✐ r➡♥❣ ✭✷✳✶✸✮ ❧➔ ✤ó♥❣ ✈î✐ p ≥ 1✱ p ❧➫✳ ✣➸ t❤✉➟♥ t✐➺♥ ❝❤♦tr➻♥❤ ❜➔②✱ tr÷î❝ t✐➯♥ t❛ ❣✐↔ sû p ≥ 3✳ ❑❤✐ ✤â

0 = 

F−x(k) − Fb −2 + F−x(k−1)+ + F−x(k−(p−1))

+ F−x(k−p)+ F−x(k−p−1) + + F−x(3) + Fx(1), ( t❤❡♦ ✭✷✳✶✵✮✮

❚✐➳♣ tö❝ q✉→ tr➻♥❤ ✤➳♥ ❦❤✐ ✤↕t ✤÷ñ❝ ♠ët sè r s❛♦ ❝❤♦

x(m) = b + o + 1, x(m − 1) = b + ox(m − 2) = b + o − 2, · · · , x(m − r) = b + 1

✲ ◆➳✉ tç♥ t↕✐ i✱ s❛♦ ❝❤♦ x(i) ≥ b✱ t❤❡♦ ✤à♥❤ ❧þ ✷✳✺✳✷ t❛ ❝â ❤♦➦❝ x(m) =

b + 1✱ ❤♦➦❝ x(m) = b+o′′+ 2, x(m −1) = b+o′′+ 1, x(m −2) = b+o′′−1, ,x(m − j) = b + 2✱ ✈î✐ j ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣ ♥➔♦ ✤â✳

✲ ⑩♣ ❞ö♥❣ ❤➺ q✉↔ ✷✳✸✳✺✱ t❛ t❤➜② tç♥ t↕✐ j0 s❛♦ ❝❤♦ x(j0) = b✱ ✈➔ ✈î✐ t➟♣{x(i) : i > j0} , ❤♦➦❝ t➟♣ {x(m) = b + 1} , ❤♦➦❝ ❧➔ t➟♣

{x(j0 + 1), x(j0 + 2), , x(m − 1), x(m)}

∪ {, , b − o − 4, b − o − 1, b − o, , b − 1}

❱➻ ✈➟② ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ s➩ ❧➔ ♠ët tr♦♥❣ ❝→❝ ❞↕♥❣ tø ✭✷✮ ✤➳♥

✭✾✮✳

✲ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ b ❧➔ sè ❧➫✱ →♣ ❞ö♥❣ ✤à♥❤ ❧þ ✷✳✼✳✻ t❛ t❤➜② ♣❤÷ì♥❣tr➻♥❤ ✭✷✳✷✮ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥ tè ❧î♥✳

✹✳ ❚r➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ tr♦♥❣ tr÷í♥❣ ❤ñ♣ m = 3, 4✳

✺✳ ❚r➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ tr♦♥❣ tr÷í♥❣ ❤ñ♣ tê♥❣ q✉→t✳