Số frobenius và giống của nửa nhóm số với chiều nhúng bằng 3

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ◆●❯❨➍◆ ◗❯❆◆● P❍Ó❈ ❙➮ ❋❘❖❇❊◆■❯❙ ❱⑨ ●■➮◆● ❈Õ❆ ◆Û❆ ◆❍➶▼ ❙➮ ❱❰■ ❈❍■➋❯ ◆❍Ó◆● ❇➀◆● ✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆❣❤➺ ❆♥ ✲ ✷✵✶✽ ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ◆●❯❨➍◆ ◗❯❆◆● P❍Ĩ❈ ❙➮ ❋❘❖❇❊◆■❯❙ ❱⑨ ●■➮◆● ❈Õ❆ ◆Û❆ ◆❍➶▼ ❙➮ ❱❰■ ❈❍■➋❯ ◆❍Ó◆● ❇➀◆● ✸ ❈❤✉②➯♥ ♥❣➔♥❤ ✿ ✣❸■ ❙➮ ❱⑨ ▲Þ ❚❍❯❨➌❚ ❙➮ ▼➣ sè✿ ✽ ✹✻ ✵✶ ✵✹ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ữớ ữợ P ◆●❯❨➍◆ ❚❍➚ ❍➬◆● ▲❖❆◆ ◆❣❤➺ ❆♥ ✲ ✷✵✶✽ ✸ ▼Ö❈ ▲Ư❈ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ✸ ✶ ✽ ✹ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➲ ♥û❛ ♥❤â♠ sè ✶✳✶✳ ❑❤→✐ ♥✐➺♠ ♥û❛ ♥❤â♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳ ❈❤✐➲✉ ♥❤ó♥❣ ✈➔ sè ❜ë✐ ❝õ❛ ♥û❛ ♥❤â♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✸✳ ❙è ❋r♦❜❡♥✐✉s ✈➔ ❣✐è♥❣ ❝õ❛ ♥û❛ ♥❤â♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹✳ ❙è ❣✐↔ ❋r♦❜❡♥✐✉s ✈➔ ❦✐➸✉ ❝õ❛ ♥û❛ ♥❤â♠ sè ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ◆û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸ ✷✷ ✷✳✶✳ ❈→❝ t❤❛♠ sè ①→❝ ✤à♥❤ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷✳ ❙è ❣✐↔ ❋r♦❜❡♥✐✉s ✈➔ ❣✐è♥❣ ❝õ❛ ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸ ✸✵ ❑➳t ❧✉➟♥ ✸✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✻ ✹ ▼Ð ✣❺❯ ❈❤♦ S ❧➔ ♠ët t➟♣ ❝♦♥ ❝❤ù❛ ❝õ❛ t➟♣ ❤ñ♣ ❝→❝ sè tü ♥❤✐➯♥ N✳ ●✐↔ sû S ✤â♥❣ ❦➼♥ ✤è✐ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ ✈➔ N \ S ❧➔ t➟♣ ❤đ♣ ❤ú✉ ❤↕♥ t❤➻ S ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥û❛ ♥❤â♠ sè✳ ▼é✐ ♥û❛ ♥❤â♠ sè S ✤➲✉ õ ởt s ỳ tỗ t n1 , , np ∈ S s❛♦ ❝❤♦ S =< n1 , , np >= {n1 λ1 + + np λp | λ1 , , λp ∈ N} ❍ì♥ ♥ú❛✱ ♠é✐ ♥û❛ ♥❤â♠ sè ✤➲✉ ❝â ❞✉② ♥❤➜t ♠ët ❤➺ s✐♥❤ tè✐ t✐➸✉ ✈➔ ❤➺ s✐♥❤ tè✐ t✐➸✉ ♥➔② ❧✉ỉ♥ ❤ú✉ ❤↕♥✳ ▲ü❝ ❧÷đ♥❣ ❝õ❛ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ ♥û❛ ♥❤â♠ sè S ✤÷đ❝ ❣å✐ ❧➔ ❝❤✐➲✉ ú S ỵ e(S) sỷ S ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ✈➔ {n1 , n2 , , np } ❧➔ ♠ët ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ S ❉♦ N \ S ❧➔ ♠ët t➟♣ ❤ñ♣ ❤ú✉ ❤↕♥ ♥➯♥ tr♦♥❣ Z \ S ❝â ♣❤➛♥ tû ❧ỵ♥ ♥❤➜t✳ P❤➛♥ tû ❧ỵ♥ ♥❤➜t ❝õ❛ Z \ S ✤÷đ❝ ❣å✐ ❧➔ sè ❋r♦❜❡♥✐✉s ❝õ❛ ♥û❛ õ số S ỵ F(S) ữ F(S) ❝❤➼♥❤ ❧➔ sè ♥❣✉②➯♥ ❧ỵ♥ ♥❤➜t ❦❤ỉ♥❣ ❜✐➸✉ t❤à t✉②➳♥ t➼♥❤ ✤÷đ❝ q✉❛ {n1 , n2 , , np } ✈ỵ✐ ❝→❝ ❤➺ sè ♥❣✉②➯♥ ❦❤ỉ♥❣ ➙♠✳ ❚➟♣ ❤đ♣ G(S) = N \ S ✤÷đ❝ ❣å✐ ❧➔ ❦❤♦↔♥❣ ❤ð ❝õ❛ ♥û❛ ♥❤â♠ sè S ✳ ❙è ❋r♦❜❡♥✐✉s F(S) ❝❤➼♥❤ ❧➔ ♣❤➛♥ tû ❧ỵ♥ ♥❤➜t tr♦♥❣ G(S)✳ ▲ü❝ ❧÷đ♥❣ ❝õ❛ G(S) ✤÷đ❝ ❣å✐ ❧➔ ❣✐è♥❣ ❝õ❛ ♥û❛ õ số S ỵ g(S) ổ ❝á♥ ✤÷đ❝ ❣å✐ ❧➔ ❜➟❝ ❦ý ❞à ❝õ❛ S ✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ t❤➻ sè ❋r♦❜❡♥✐✉s F(S) ❝õ❛ ♥û❛ ♥❤â♠ sè S ❧➔ sè ♥❣✉②➯♥ x ❞✉② ♥❤➜t t❤ä❛ ♠➣♥ x + n ∈ S ✈ỵ✐ ♠å✐ n ∈ N∗ ✳ ◆➳✉ tr♦♥❣ ✤✐➲✉ ❦✐➺♥ ♥➔② t❛ t❤❛② t❤➳ N∗ ❜ð✐ S ∗ = S \ {0} t❤➻ t❛ ❝â ❦❤→✐ ♥✐➺♠ sè ❣✐↔ ❋r♦❜❡♥✐✉s✿ sè ✺ ♥❣✉②➯♥ x ∈ Z \ S ✤÷đ❝ ❣å✐ ❧➔ sè ❣✐↔ ❋r♦❜❡♥✐✉s ♥➳✉ x + n ∈ S ✈ỵ✐ ♠å✐ n ∈ S ∗ ❚✉② ♥❤✐➯♥✱ sè ❣✐↔ ❋r♦❜❡♥✐✉s ❝õ❛ ♥û❛ ♥❤â♠ sè S ❧➔ ❦❤ỉ♥❣ ❞✉② ♥❤➜t ♥❤÷ ❧➔ sè ❋r♦❜❡♥✐✉s✳ ❚➟♣ ❤ñ♣ ❝→❝ sè ❣✐↔ ❋r♦❜❡♥✐✉s ❝õ❛ ♥û❛ ♥❤â♠ sè S ữủ ỵ PF(S) ỹ ữủ t ❤đ♣ PF(S) ✤÷đ❝ ❣å✐ ❧➔ ❦✐➸✉ ❝õ❛ ♥û❛ ♥❤â♠ sè S ỵ t(S) ó r F(S) PF(S) ✈➔ F(S) ❝❤➼♥❤ ❧➔ sè ❧ỵ♥ ♥❤➜t tr♦♥❣ PF(S) ◆➳✉ S ❧➔ ♥û❛ ♥❤â♠ sè s✐♥❤ ❜ð✐ ❤❛✐ ♣❤➛♥ tû {n1 , n2 }✱ ♥➠♠ ✶✽✽✷ ✈➔ ♥➠♠ ✶✽✽✹✱ tr♦♥❣ ❤❛✐ ❜➔✐ ❜→♦ ❦❤→❝ ♥❤❛✉✱ ♥❤➔ t♦→♥ ❤å❝ ❏✳ ❏✳ ❙②❧✈❡rst❡r ✤➣ ✤÷❛ r❛ ❝ỉ♥❣ t❤ù❝ ①→❝ ✤à♥❤ sè ❋r♦❜❡♥✐✉s F(S) ✈➔ ❣✐è♥❣ g(S) ♥❤÷ s❛✉✿ F(S) = n1 n2 − n1 − n2 , g(S) = (n1 − 1)(n2 − 1) ❚✉② ♥❤✐➯♥✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ❝❤✐➲✉ ♥❤ó♥❣ ❝õ❛ S ❧ỵ♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ ✸ t❤➻ ❦❤æ♥❣ ❝â ❝æ♥❣ t❤ù❝ ①→❝ ✤à♥❤ F(S) ✈➔ g(S) q✉❛ ❝→❝ ♣❤➛♥ tû s✐♥❤ ❝õ❛ S ♥❤÷ tr÷í♥❣ ❤đ♣ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✷ ♠➔ ❏✳ ❏✳ ❙②❧✈❡rst❡r ✤➣ ❧➔♠ ✤÷đ❝ ♥❤÷ tr➯♥✳ ◆❣❛② ❝↔ tr÷í♥❣ ❤đ♣ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸ ♠➔ ✈➝♥ ❝❤÷❛ ❝â ❧í✐ ❣✐↔✐ ❝❤♦ ❜➔✐ t♦→♥ ♥➔②✳ ❉♦ ✤â✱ ✤➣ ❝â ♠ët sè ♥❤➔ t♦→♥ ❤å❝ ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ ♥➔② ❝❤♦ ♠ët sè ♥û❛ ♥❤â♠ sè ❝ö t❤➸✳ ❈❤♦ S ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ❝â ❤➺ s✐♥❤ tè✐ t✐➸✉ ❧➔ {n1 , n2 , , np } ú ỵ r t ổ ❝â gcd{n1 , n2 , , np } = 1✳ ●å✐ d = gcd{n1 , , np−1 }✱ ♥➠♠ ✶✾✻✵ ♥❤➔ t♦→♥ ❤å❝ ❙✳ ▼✳ ❏♦❤♥s♦♥ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ n1 np−1 , , , np >) + (d − 1)np d d ✈➔ ♥➠♠ ✶✾✼✽✱ ♥❤➔ t♦→♥ ❤å❝ ❖✳ ❏✳ ❘o❞s❡t❤ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❝ỉ♥❣ t❤ù❝ F(S) = dF(< n1 np−1 , , , np >) + (d − 1)(np − 1) d d ◆❤í ❝→❝ ❦➳t q✉↔ ♥â✐ tr➯♥✱ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ sè ❋r♦❜❡♥✐✉s F(S) ✈➔ ❣✐è♥❣ g(S) = dg(< g(S)✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t ❝→❝ sè ♥❣✉②➯♥ ni tr♦♥❣ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ S ❧➔ ✤ỉ✐ ♠ët ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉✱ ♥❣❤➽❛ ❧➔✱ gcd{ni , nj } = 1, ∀i = j ✻ ❚r♦♥❣ ❜➔✐ ❜→♦ ❬✺❪ r❛ ♥➠♠ ✷✵✵✹✱ tr➯♥ t↕♣ ❝❤➼ ❆r❝❤✳ ▼❛t❤✳✱ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ❏✳ ❈✳ ❘♦s❛❧❡s ❛♥❞ P✳ ❆✳ ●❛r❝➼❛✲❙❛♥❝❤➨③ ✤➣ ♥❣❤✐➯♥ ❝ù✉ sè ❋r♦❜❡♥✐✉s ✈➔ ❣✐è♥❣ ❝õ❛ ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸ ♠➔ ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ ♥û❛ ♥❤â♠ sè ♥➔② ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✻ sè ♥❣✉②➯♥ ❞÷ì♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♠ët ❤➺ ✸ ♣❤÷ì♥❣ tr➻♥❤ ✤❛ t❤ù❝✳ Ð ✤â ❤å ✤➣ ✤÷❛ r❛ ❝ỉ♥❣ t❤ù❝ ①→❝ ✤à♥❤ sè ❋r♦❜❡♥✐✉s ✈➔ ❣✐è♥❣ t❤ỉ♥❣ q✉❛ ✻ sè ♥❣✉②➯♥ ❞÷ì♥❣ ♥➔②✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ❧↕✐ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❜➔✐ ❜→♦ ♥â✐ tr➯♥ ❝õ❛ ❏✳ ❈✳ ❘♦s❛❧❡s ❛♥❞ P✳ ❆✳ ●❛r❝➼❛✲❙❛♥❝❤➨③✳ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ♥ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➲ ♥û❛ ♥❤â♠ sè✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✈➲ ❦❤→✐ ♥✐➺♠ ♥û❛ ♥❤â♠ sè ✈➔ ♠ët sè ❜➜t ❜✐➳♥ ❝õ❛ ♥û❛ ♥❤â♠ sè ♥❤➡♠ ❧➔♠ ❝ì sð ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ð ❈❤÷ì♥❣ ✷✳ ❈❤÷ì♥❣ ✶ ✤÷đ❝ ✈✐➳t ❞ü❛ ✈➔♦ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪✱ ❬✷❪✱ ❬✸❪✱ ❬✹❪✳ ❈❤÷ì♥❣ ✷✿ ❱➲ ♥û❛ ♥❤â♠ sè ❝â ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ tr♦♥❣ ỗ ❝→❝ t❤❛♠ sè ①→❝ ✤à♥❤ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ ♥û❛ ♥❤â♠ sè ✈➔ sè ❋r♦❜❡♥✐✉s ✈➔ ❣✐è♥❣ ❝õ❛ ♥û❛ ♥❤â♠ sè ❝â ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ t t rữớ ữợ sỹ ữợ P ỗ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ ổ ữớ t t ữợ ộ ✤ë♥❣ ✈✐➯♥ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ◆❤➙♥ ❞à♣ ♥➔②✱ t→❝ ❣✐↔ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣ ❇ë ♠ỉ♥ ✣↕✐ sè✱ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ◆❣➔♥❤ ❚♦→♥✱ ❱✐➺♥ ❙÷ ♣❤↕♠ tü ♥❤✐➯♥✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ trü❝ t✐➳♣ ❣✐↔♥❣ ❞↕② ❧ỵ♣ ❈❛♦ ❤å❝ ✷✹ ❝❤✉②➯♥ ♥❣➔♥❤ ✣↕✐ sè ✈➔ ỵ tt số t rữớ ❣✐↔ ①✐♥ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ ✈➔ P❤á♥❣ ✣➔♦ t↕♦ ❙❛✉ ✤↕✐ ❤å❝ ❤❛✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✈➔ ✣↕✐ ❤å❝ ❦✐♥❤ t➳ ❦ÿ t❤✉➟t ▲♦♥❣ ❆♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ ✼ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ❚r➙♥ trå♥❣✦ ▲♦♥❣ ❆♥✱ t❤→♥❣ ✵✼ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ✽ ❈❍×❒◆● ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ❱➋ ◆Û❆ ◆❍➶▼ ❙➮ ✶✳✶ ❑❤→✐ ♥✐➺♠ ♥û❛ ♥❤â♠ sè ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ S ❧➔ ♠ët t➟♣ ❝♦♥ ❝❤ù❛ ✵ ❝õ❛ t➟♣ ❤ñ♣ ❝→❝ sè tü ♥❤✐➯♥ N✳ ●✐↔ sû S ✤â♥❣ ❦➼♥ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ ✈➔ N \ S ❤ú✉ ❤↕♥ t❤➻ S ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥û❛ ♥❤â♠ sè✳ ▼é✐ ♥û❛ ♥❤â♠ sè S õ ởt s ỳ tỗ t↕✐ n1 , , np ∈ S s❛♦ ❝❤♦ S =< n1 , , np >= {n1 λ1 + + np λp | λ1 , , λp ∈ N} ❈❤♦ A ❧➔ ♠ët t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ N õ ữợ ợ t A ỵ (A) = ữợ ợ ♥❤➜t ❝õ❛ ♠å✐ t➟♣ ❝♦♥ ❤ú✉ ❤↕♥ ❝õ❛ A ✤➲✉ ❜➡♥❣ ✶✳ ▼➺♥❤ ✤➲ s❛✉ ❝❤♦ t❤➜② ♥û❛ ♥❤â♠ ❝♦♥ ❝õ❛ N s✐♥❤ ❜ð✐ A ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❣❝❞(A) = 1✳ ✶✳✶✳✷ ▼➺♥❤ ✤➲✳ ❈❤♦ A ❧➔ ♠ët t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ N \{0}✳ ❑❤✐ ✤â✱ < A > ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❣❝❞(A) = ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t d = gcd(A)✳ ❘ã r➔♥❣ ♥➳✉ s ∈ A t❤➻ d | s✳ ❉♦ A ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ♥➯♥ N \ A ❧➔ t➟♣ ❤ú✉ ❤↕♥ ✈➔ tỗ t ởt số ữỡ x s ❝❤♦ d | x ✈➔ d | (x + 1)✳ ❙✉② r❛ d = 1✳ ◆❣÷đ❝ ❧↕✐✱ t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ N \ A ❧➔ ♠ët t➟♣ ❤ñ♣ ❤ú✉ ❤↕♥✳ t = gcd(A) tỗ t sè ♥❣✉②➯♥ z1 , z2 , , zn ✈➔ a1 , a2 , , an ∈ A s❛♦ ❝❤♦ z1 a1 + z2 a2 + + zn an = ✾ ❇➡♥❣ ❝→❝❤ ❝❤✉②➸♥ zi s❛♥❣ ✈➳ ♣❤↔✐✱ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ i1 , , ik , j1 , , jl ∈ 1, , n s❛♦ ❝❤♦ zi1 ai1 + + zik aik = − zj1 aj1 − − zjl ajl ❉♦ õ tỗ t s A s s + ❝ơ♥❣ t❤✉ë❝ A ✳ ❈❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ♥➳✉ n ≥ (s − 1)s + (s − 1) t❤➻ n ∈ A ✳ ❚❤➟t ✈➟②✱ ❧➜② q ✈➔ r ❧➔ ❝→❝ sè ♥❣✉②➯♥ s❛♦ ❝❤♦ n = qs + r ✈ỵ✐ ≤ r < s✳ ❉♦ n ≥ (s − 1)s + (s − 1) t❛ s✉② r❛ q ≥ s − ≥ r✳ ❱➻ ✈➟②✱ n = (rs + r) + (q − r)s = r(s + 1) + (q − r)s ∈ A ❍➺ q✉↔ s❛✉ ❝❤♦ t❛ ♠ët sü ♣❤➙♥ ❧♦↕✐ ❝→❝ ✈à ♥❤â♠ ❝♦♥ ❝õ❛ ✈à ♥❤â♠ ❝ë♥❣ ❝→❝ sè tü ♥❤✐➯♥ N t❤æ♥❣ q✉❛ ❝→❝ ♥û❛ ♥❤â♠ sè✳ ✶✳✶✳✸ ❍➺ q✉↔✳ ❈❤♦ M ❧➔ ♠ët ✈à ♥❤â♠ ❝♦♥ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ ✈à ♥❤â♠ ❝ë♥❣ ❝→❝ sè tü ♥❤✐➯♥ N✳ ❑❤✐ ✤â M ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♠ët ♥û❛ ♥❤â♠ sè✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ d = gcd(M )✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶✳✷✱ t❛ ❝â S= m |m∈M d ❧➔ ♠ët ♥û❛ ♥❤â♠ sè✳ ❉➵ t❤➜② →♥❤ ①↕ f : M → S, f (m) = m d ❧➔ ♠ët ✤➥♥❣ ❝➜✉ ✈à ♥❤â♠✳ ❱➻ ✈➟② M ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♥û❛ ♥❤â♠ sè S ●✐↔ sû A ✈➔ B ❧➔ ❝→❝ t➟♣ ❝♦♥ t số tỹ N õ t ỵ ❤✐➺✉ A + B = a + b | a ∈ A, b ∈ B ◆❤÷ ✈➟②✱ ♥➳✉ ❝❤♦ S ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ✈➔ S ∗ = S \ t❤➻ S ∗ + S ∗ ❧➔ t S ỗ tỷ t ữủ ữợ tờ tỷ tr S ✳ ✶✵ ✶✳✶✳✹ ❇ê ✤➲✳ ❈❤♦ S ❧➔ ♠ët ✈à ♥❤â♠ ❝♦♥ ❝õ❛ N✳ ❑❤✐ ✤â S ∗ \ (S ∗ + S ∗) ❧➔ ♠ët ❤➺ s✐♥❤ ❝õ❛ S ✳ ❍ì♥ t❤➳ ♥ú❛✱ ♠å✐ ❤➺ s✐♥❤ ❝õ❛ S ✤➲✉ ❝❤ù❛ S ∗ \ (S ∗ + S ∗)✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ s ∈ S ∗✳ ◆➳✉ s ∈/ S \ (S + S ) t tỗ t↕✐ x, y ∈ S ∗ s❛♦ ❝❤♦ x + y = s✳ ❈❤ó♥❣ t❛ t✐➳♣ tư❝ ♥❤÷ t❤➳ ✤è✐ ✈ỵ✐ ❝→❝ ♣❤➛♥ tû x ✈➔ y ✳ ❙❛✉ ♠ët số ỳ ữợ t t ữủ s1 , , sn ∈ S ∗ \ (S ∗ + S ∗ ) s❛♦ ❝❤♦ s = s1 + + sn ✣✐➲✉ ✤â ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ S ∗ \ (S ∗ + S ∗ ) ❧➔ ♠ët ❤➺ s✐♥❤ ❝õ❛ S ✳ ❇➙② ❣✐í ❝❤♦ A ❧➔ ♠ët ❤➺ s✐♥❤ ❝õ❛ S ✳ ▲➜② x ∈ S ∗ \ (S ∗ + S ∗ ) õ tỗ t n N , , , λn ∈ N ✈➔ a1 , , an ∈ A s❛♦ ❝❤♦ x = λ1 a1 + + λn an ❱➻ x ∈ / S ∗ + S t s r tỗ t i 1, , n s❛♦ ❝❤♦ x = ✳ ❉♦ ✤â x ∈ A ❍❛② ♥â✐ ❝→❝❤ ❦❤→❝✱ ♠å✐ ❤➺ s✐♥❤ ❝õ❛ S ✤➲✉ ❝❤ù❛ S ∗ \ (S + S ) ỵ ▼é✐ ♥û❛ ♥❤â♠ sè ✤➲✉ ❝â ❞✉② ♥❤➜t ♠ët ❤➺ s tố t s tố t ỗ ỳ ❤↕♥ ♣❤➛♥ tû✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✶✳✶✳✹ t❤➻ S ∗ \ (S ∗ + S ∗) ❧➔ ♠ët ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ S ✳ ❚❤❡♦ ❬✷❪✱ ✈ỵ✐ ♠é✐ n ∈ S ∗ ✱ t❛ ❝â S = Ap(S, n) ∪ n ✳ ❉♦ Ap(S, n) ∪ n ❧➔ t➟♣ ❤ú✉ ❤↕♥✱ s✉② r❛ S ∗ \ (S ∗ + S ∗ ) ❝ô♥❣ ❧➔ t➟♣ ❤ú✉ ❤↕♥✳ ✶✳✷ ❈❤✐➲✉ ♥❤ó♥❣ ✈➔ sè ❜ë✐ ❝õ❛ ♥û❛ ♥❤â♠ sè ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ S ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❧➔ A = {a1 , , ae } t❤ä❛ ♠➣♥ a1 < a2 < < ae ✳ ●✐→ trà e ✤÷đ❝ ❣å✐ ❧➔ ❝❤✐➲✉ ♥❤ó♥❣ ❝õ❛ S ✈➔ ❦➼ ❤✐➺✉ ❧➔ ❡✭S ✮ ✈➔ ❣✐→ trà a1 ✤÷đ❝ ❣å✐ ❧➔ sè ❜ë✐ ❝õ❛ S ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ ♠✭S ✮✳ ▼➺♥❤ ✤➲ s❛✉ ✤➙② s➩ ❝❤♦ t❛ t❤➜② ✤÷đ❝ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ sè ❜ë✐ ✈➔ ❝❤✐➲✉ ♥❤ó♥❣ ❝õ❛ ♠ët ♥û❛ ♥❤â♠ sè✳ ✷✷ ❈❍×❒◆● ✷ ◆Û❆ ◆❍➶▼ ❙➮ ❱❰■ ❈❍■➋❯ ◆❍Ó◆● ❇➀◆● ✸ ❚r♦♥❣ ❜➔✐ ❜→♦ ❬✺❪✱ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ❏✳ ❈✳ ❘♦s❛❧❡s ❛♥❞ P✳ ❆✳ ●❛r❝➼❛✲❙❛♥❝❤➨③ ✤➣ ♥❣❤✐➯♥ ❝ù✉ sè ❋r♦❜❡♥✐✉s ✈➔ ❣✐è♥❣ ❝õ❛ ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸ ♠➔ ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ ♥û❛ ♥❤â♠ sè ♥➔② ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✻ sè ♥❣✉②➯♥ ❞÷ì♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♠ët ❤➺ ✸ ♣❤÷ì♥❣ tr➻♥❤ ✤❛ t❤ù❝✳ Ð ✤â ❤å ✤➣ ✤÷❛ r❛ ❝ỉ♥❣ t❤ù❝ ①→❝ ✤à♥❤ sè ❋r♦❜❡♥✐✉s ✈➔ ❣✐è♥❣ t❤ỉ♥❣ q✉❛ ✻ sè ♥❣✉②➯♥ ❞÷ì♥❣ ♥➔②✳ ❈❤÷ì♥❣ ♥➔② s➩ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ❝õ❛ ❜➔✐ ❜→♦ ❬✺❪ ✳ ✷✳✶ ❈→❝ t❤❛♠ sè ①→❝ ✤à♥❤ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ r t ổ ỵ S ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ❝â ❤➺ s✐♥❤ tè✐ t✐➸✉ ❧➔ {n1 , n2 , n3 }✱ ✈ỵ✐ gcd {ni , nj } = 1, ∀i = j ✳ ❱ỵ✐ {i, j, k} = {1, 2, 3} t❛ ✤➦t ci = {x ∈ N\ {0} | xni ∈ nj , nk } õ tỗ t số ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ r12 , r13 , r21 , r23 , r31 , r32 s❛♦ ❝❤♦✿ c1 n1 = r12 n2 + r13 n3 , c2 n2 = r21 n1 + r23 n3 , c3 n3 = r31 n1 + r32 n2 ❇ê ✤➲ s❛✉ ❝❤♦ t❤➜② rij ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ✷✳✶✳✶ ❇ê ✤➲✳ r12, r13, r21, r23, r31, r32 ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ✷✸ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû r13 = 0✳ ❑❤✐ ✤â c1n1 = r12n2✳ ❉♦ gcd {n1, n2} = ✈➔ c1 ≤ n2 ♥➯♥ t❛ ❝â c1 = n2 ✳ ▼➦t ❦❤→❝✱ tỗ t x {1, 2, , n − 1} s❛♦ ❝❤♦ xn1 ≡ n3 (modn2 )✳ ❉♦ ✤â xn1 = zn2 + n3 ✱ ✈ỵ✐ z ∈ Z✳ ❱➻ {n1 , n2 , n3 } ❧➔ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ S ♥➯♥ z ∈ N✳ ❙✉② r❛ c1 ≤ x < n2 ✈➔ ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥ ✈ỵ✐ c1 = n2 ✳ ❱➟② r13 ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ❚÷ì♥❣ tü t❛ ❝ơ♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r12 , r21 , r23 , r31 , r32 số ữỡ ợ i = j, t❛ ❝â ci > rji✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♣❤↔♥ ❝❤ù♥❣✳ ❈❤➥♥❣ ❤↕♥✱ t❛ ❣✐↔ sû r➡♥❣ r31 ≥ c1 ✳ ❑❤✐ ✤â r31 = qc1 + r✱ ✈ỵ✐ q ∈ N\ {0} ✈➔ ≤ r < c1 ❉♦ ✤â c3 n3 = r31 n1 + r32 n2 = (qc1 + r) n1 + r32 n2 = rn1 + q (c1 n1 ) + r32 n2 = rn1 + q (r12 n2 + r13 n3 ) + r32 n2 = rn1 + qr12 n2 + qr13 n3 + r32 n2 ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ (c3 − qr13 ) n3 = rn1 + (qr12 + r32 ) n2 ∈ n1 , n2 ❍ì♥ ♥ú❛ ❞♦ r ≥ ✈➔ qr12 + r32 > t❛ ❝â c3 − qr13 > 0✳ ❙✉② r❛ c3 > qr13 ✱ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ t➼♥❤ ♥❤ä ♥❤➜t ❝õ❛ c3 ✳ ❱➻ ✈➟②✱ ❜ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✶✳✸ ❇ê ✤➲✳ ❱ỵ✐ ♠å✐ {i, j, k} = {1, 2, 3}✱ t❛ ❝â ci = rji + rki ự ú ỵ r (r13 + r23) n3 = (c1 − r21) n1 + (c2 − r12) n2✱ ❞♦ ✤â r13 + r23 ≥ c3 ✳ ❚÷ì♥❣ tü t❛ ❝ô♥❣ ❝â r21 + r31 ≥ c1 ✈➔ r12 + r32 ≥ c2 ✳ ❇➯♥ ❝↕♥❤ ✤â✱ t❛ ❝â c1 n1 + c2 n2 + c3 n3 = (r21 + r31 ) n1 + (r12 + r32 ) n2 + (r13 + r23 ) n3 ✷✹ ❚ø ✤â s✉② r❛ c1 = r21 + r31 ; c2 = r12 + r32 ; c3 = r13 + r23 ✳ ❚r➯♥ S t❛ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ ♠ët q✉❛♥ tự tỹ ữ s ợ a, b S t❛ ✈✐➳t a ≤S b ♥➳✉ b − a ∈ S ú ỵ r q tự tỹ tổ tữớ tr Z ữủ ợ a, b Z✱ t❛ ✈✐➳t a ≤ b ♥➳✉ b − a ∈ N ✷✳✶✳✹ ❇ê ✤➲✳ ❱ỵ✐ ♠å✐ {i, j, k} = {1, 2, 3}✱ max (Ap (S, ni )) = {(cj − 1) nj + (rik − 1) nk , (ck − 1) nk + (rij − 1) nj } ≤S ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✈ỵ✐ i = 1✱ ❝→❝ tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü✳ rữợ t t ự (c3 1) n3 + (r12 − 1) n2 ∈ Ap (S, n1 )✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐✱ (c3 − 1) n3 + (r12 − 1) n2 = a1 n1 + a2 n2 + a3 n3 , ✈ỵ✐ a1 , a2 , a3 ∈ N, a1 = 0✳ ❉♦ t➼♥❤ ♥❤ä ♥❤➜t ❝õ❛ c3 ✈➔ c2 ✱ t❛ ❝â a3 < c3 − ✈➔ a2 < r12 − 1✳ ❙✉② r❛ a1 n1 = (c3 − − a3 ) n3 + (r12 − − a2 ) n2 , ✈ỵ✐ c3 −1−a3 , r12 −1−a2 ∈ N ✈➔ ❞♦ ✤â a1 ≥ c1 ✳ ●å✐ q ∈ N\ {0} ✈➔ ≤ r < c1 s❛♦ ❝❤♦ a1 = qc1 + r✳ ❑❤✐ ✤â (c3 − − a3 ) n3 + (r12 − − a2 ) n2 = rn1 + qr12 n2 + qr13 n3 ❉♦ ✤â (c3 − − a3 − qr13 ) n3 = rn1 + (qr12 − r12 + + a2 ) n2 ✈ỵ✐ r ∈ N, qr12 − r12 + + a2 ∈ N\ {0}✱ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ c3 ✳ ❚÷ì♥❣ tü t❛ ❝â (c2 − 1) n2 + (r13 − 1) n3 ∈ Ap (S, n1 ) ✷✺ ❇➙② ❣✐í t❛ ❧➜② an2 + bn3 ∈ Ap (S, n1 )✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❤♦➦❝ (a, b) ≤ (r12 − 1, c3 − 1) ❤♦➦❝ (a, b) (c2 1, r13 1) ỗ ✤➲ ✷✳✶✳✶ t❛ ❝â a < c2 ✈➔ b < c3 ✳ ◆➳✉ (a, b) > (r12 − 1, c3 − 1) t❤➻ a ≥ r12 ✳ ❇➡♥❣ ♣❤↔♥ ❝❤ù♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ b < r13 ✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ b ≥ r13 ✱ t❤➻ an2 + bn3 = r12 n2 + r13 n3 + (a − r12 ) n2 + (b − r13 ) n3 = c1 n1 + (a − r12 ) n2 + (b − r13 ) n3 , ♠➙✉ t❤✉➝♥ ✈ỵ✐ an2 + bn3 ∈ Ap (S, n1 )✳ ❉♦ ✤â (a, b) ≤ (c2 − 1, r13 − 1)✳ ❇ê ✤➲ s❛✉ ❝❤♦ t❛ ♠ët ❜✐➸✉ ❞✐➵♥ ❝õ❛ ❝→❝ ♣❤➛♥ tû s✐♥❤ tr♦♥❣ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ ♥û❛ ♥❤â♠ sè S t❤æ♥❣ q✉❛ rij ✳ ✷✳✶✳✺ ❇ê ✤➲✳ n1 = r12 r13 + r12 r23 + r13 r32 , n2 = r13 r21 + r21 r23 + r23 r31 , n3 = r12 r31 + r21 r32 + r31 r32 ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â Ap (S, n) = n✱ ✈ỵ✐ ♠å✐ n ∈ S \ {0}✳ ❉♦ t➼♥❤ ♥❤ä ♥❤➜t ❝õ❛ c2 ✱ c3 ✱ ♥➳✉ a2 n2 + a3 n3 = b2 n2 + b3 n3 ✈ỵ✐ , bi ∈ {0, , ci − 1} t❤➻ (a2 , a3 ) = (b2 , b3 )✳ ❉♦ ✤â t❤❡♦ ❇ê ✤➲ ✷✳✶✳✹✱ t❛ s✉② r❛ Ap (S, n1 ) = (a, b) ∈ N2 | (a, b) ≤ (r12 − 1, c3 − 1) ∨ (a, b) ≤ (c2 − 1, r13 − 1) = r12 c3 + c2 r13 − r12 r13 ❉♦ ✤â n1 = r12 c3 + c2 r13 − r12 r13 ✈➔ ❞♦ ✤â ❜ð✐ ❇ê ✤➲ ✷✳✶✳✸ t❛ ❝â n1 = r12 r13 + r12 r23 + r13 r32 ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ✤÷đ❝ n2 = r13 r21 + r21 r23 + r23 r31 ✈➔ n3 = r12 r31 + r21 r32 + r31 r32 ✷✻ ✷✳✶✳✻ ❇ê ✤➲✳ ❈❤♦ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ a12, a13, a21, a23, a31, a32 ✈➔ m1 = a12 a13 + a12 a23 + a13 a32 , m2 = a13 a21 + a21 a23 + a23 a31 , m3 = a12 a31 + a21 a32 + a31 a32 ❱ỵ✐ ♠å✐ {i, j, k} = {1, 2, 3}✱ t❛ ❝â✿ ✶✮ (aji + aki ) mi = aij mj + aik mk ✱ ✷✮ ◆➳✉ gcd {mi, mj } = t❤➻ mk ∈/ mi , mj ✳ ❈❤ù♥❣ ♠✐♥❤✳ ✶✮ ❚❛ ❝❤ù♥❣ ♠✐♥❤ (a21 + a31) m1 = a12m2 + a13m3✳ ❚❤➟t ✈➟②✱ t❛ ❝â (a21 + a31 ) m1 = a21 a12 a13 + a21 a12 a23 + a21 a13 a32 + a31 a12 a13 + a31 a12 a23 + a31 a13 a32 = a12 (a13 a21 + a21 a23 + a23 a31 ) + a13 (a12 a31 + a21 a32 + a31 a32 ) = a12 m2 + a13 m3 ❚÷ì♥❣ tü t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ (aji + aki ) mi = aij mj + aik mk , ✈ỵ✐ ♠å✐ {i, j, k} = {1, 2, 3}✳ ✷✮ ❉♦ t➼♥❤ ✤è✐ ①ù♥❣✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ gcd {m1 , m2 } = t❤➻ m3 ∈ / m1 , m2 ✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ m3 ∈ m1 , m2 t m3 = m1 + àm2 ợ λ, µ ∈ N✳ ❚❤❡♦ ❑❤➥♥❣ ✤à♥❤ ✶✮✱ t❛ ❝â (a21 + a31 ) m1 = a12 m2 + a13 m3 ❉♦ ✤â a13 m3 = (a21 + a31 ) m1 − a12 m2 ◆❤÷ ✈➟② a13 (λm1 + µm2 ) = (a21 + a31 ) m1 − a12 m2 ✷✼ ✈➔ (a13 µ + a12 ) m2 = (a21 + a31 − a13 λ) m1 ❉♦ gcd {m1 , m2 } = ♥➯♥ t❛ s✉② r❛ a21 + a31 − a13 λ = km2 , k ∈ N\ {0} ✣➦❝ ❜✐➺t✱ a21 + a31 ≥ km2 ≥ m2 ✱ ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥ ✈ỵ✐ m2 = (a21 + a31 ) a23 + a13 a21 , ❜ð✐ ✈➻ aij > 0, ∀i, j ♥➯♥ a21 + a31 < m2 ✳ ❚ø ❇ê ✤➲ ✷✳✶✳✸ ✈➔ ❇ê ✤➲ ✷✳✶✳✺ t❛ ❝â ♣❤→t ❜✐➸✉ s❛✉✳ ✷✳✶✳✼ ❇ê ✤➲✳ ❱ỵ✐ ♠å✐ {i, j, k} = {1, 2, 3}✱ ni = cj ck − rjk rkj ✳ ▼ët ♣❤➛♥ tû (x1 , , xp ) ∈ Zp ✤÷đ❝ ❣å✐ ❧➔ ❞÷ì♥❣ ♠↕♥❤ ✭str♦♥❣❧② ♣♦s✐t✐✈❡✮ ♥➳✉ xi > 0, ∀i = 1, , p ỵ m1, m2, m3 sè ♥❣✉②➯♥ ❞÷ì♥❣ s❛♦ ❝❤♦ gcd {mi, mj } = ợ i = j õ ữỡ tr m1 = x12 x13 + x12 x23 + x13 x32 m2 = x13 x21 + x21 x23 + x23 x31 m3 = x12 x31 + x21 x32 + x31 x32 ❝â ♥❣❤✐➺♠ ❞÷ì♥❣ ♠↕♥❤ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ {m1, m2, m3} ❧➔ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ ♠ët ♥û❛ ♥❤â♠ số ỡ ỳ ữ t tỗ t t❤➻ ♥â ❧➔ ❞✉② ♥❤➜t✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥✿ s✉② r❛ tø ❇ê ✤➲ ✷✳✶✳✻✳ ✣✐➲✉ ❦✐➺♥ ✤õ✿ ❧➔ ❤➺ q✉↔ ❝õ❛ ❇ê ✤➲ ✷✳✶✳✺✳ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ♥❣❤✐➺♠✳ ❱ỵ✐ ni = mi , i ∈ {1, 2, 3}✱ tø ❇ê ✤➲ ✷✳✶✳✺ t❛ ❝â (x12 , x13 , x21 , x23 , x31 , x32 ) = (r12 , r13 , r21 , r23 , r31 , r32 ) ❧➔ ♠ët ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ♠↕♥❤✳ ✷✽ ●✐↔ sû (a12 , a13 , a21 , a23 , a31 , a32 ) ❧➔ ♠ët ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ♠↕♥❤ ❦❤→❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥✳ õ tỗ t i, j s aij < rij ✳ ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❣✐↔ sû a12 < r12 ✳ ❑❤✐ ✤â c1 n1 = r12 n2 + r13 n3 = (a12 + λ) n2 + r12 n3 ✈ỵ✐ λ ∈ N\ {0}✳ ❉♦ (aji + aki ) mi = aij mj + aik mk ♥➯♥ t❛ ❝â a12 n2 = (a21 + a31 ) n1 − a13 n3 ✈➔ a21 + a31 ≥ c1 ✳ ❱➻ t❤➳ c1 n1 = (a21 + a31 ) n1 − a13 n3 + λn2 + r13 n3 ✳ ❉♦ ✤â (a13 − r13 ) n3 = (a21 + a31 − c1 ) n1 + λn2 ✈➔ ✈➻ ✈➟② a13 > c3 ✳ ❉♦ n1 = a12 a13 + a12 a23 + a13 a32 = a13 (a12 + a32 ) + a12 a23 ✈➔ tø ❇ê ✤➲ ✷✳✶✳✻✱ ✭✶✮ a12 + a32 ≥ c2 ✱ t❛ s✉② r❛ n1 > c3 c2 + a12 a23 > c3 c2 ✱ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❇ê ✤➲ ✷✳✶✳✼✳ ✷✳✶✳✾ ✣à♥❤ ♥❣❤➽❛✳ ▼❛ tr➟♥ A ✤÷đ❝ ❣å✐ ❧➔ 0✲♠❛ tr➟♥ ♥➳✉ A ❝â ❞↕♥❣ A= a12 a13 a21 a23 a31 a32 tr♦♥❣ ✤â a12 , a13 , a21 , a23 , a31 , a32 ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ s❛♦ ❝❤♦ gcd (Ai , Aj ) = ✈ỵ✐ A1 = a12 a13 + a12 a23 + a13 a32 , A2 = a13 a21 + a21 a23 + a23 a31 , A3 = a12 a31 + a21 a32 + a31 a32 ỵ A = a12 a13 a21 a23 a31 a32 ❧➔ ✵✲♠❛ tr➟♥ t❤➻ A1, A2, A3 ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸ ✈➔ tr♦♥❣ ❤➺ s✐♥❤ ✤â ❝→❝ ♣❤➛♥ tû ✷✾ ✤ỉ✐ ♠ët ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉✳ ❍ì♥ ♥ú❛✱ (aji + aki)Ai = aij Aj + aik Ak ✈➔ aji + aki = min{x ∈ N \ {0} | xAi ∈ Aj , Ak } ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ S ❧➔ ♥û❛ ♥❤â♠ sè ❝â ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸ ✈➔ ❝â ❤➺ s✐♥❤ ❧➔ ❝→❝ ♣❤➛♥ tû ✤æ✐ ởt tố ũ t õ tỗ t ✵✲♠❛ tr➟♥ A s❛♦ ❝❤♦ S = A1, A2, A3 ự ứ ỵ t õ A1, A2, A3 ❧➔♠ët ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✻✱ t❛ ❝â (aji + aki ) Ai = aij Aj + aik Ak ✳ ❚ø ỵ t❛ s✉② r❛ aji + aki = {x ∈ N\ {0} | xAi ∈ Aj , Ak } ❇➙② ❣✐í t❛ ❣✐↔ sû S ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ❝â ❤➺ s✐♥❤ tè✐ t✐➸✉ {n1 , n2 , n3 } s❛♦ ❝❤♦ gcd {ni , nj } = ✈ỵ✐ ♠å✐ i = j ✳ ❚❤➳ t❤➻ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✺✱ t❛ ❝â A= r12 r13 r21 r23 r31 r32 ❧➔ ♠ët ✵✲ ♠❛ tr➟♥ ✈ỵ✐ S = A1 , A2 , A3 ✳ A = 1 ❧➔ ♠ët ✵✲♠❛ tr➟♥ ✈ỵ✐ A1 = 5, A2 = 7, A3 = 9✳ ❉♦ ✤â 5, 7, ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸✳ ❍ì♥ ♥ú❛✱ ✷✳✶✳✶✶ ❱➼ ❞ư✳ × = × + × 9, × = × + × 9, ✷✳✶✳✶✷ ❱➼ ❞ö✳ A = 102✳ × = × + × ❧➔ ♠ët ✵✲♠❛ tr➟♥ ✈ỵ✐ A1 = 41, A2 = 63, A3 = 11 ❉♦ ✤â 41, 63, 102 ❧➔ ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸✳ ❍ì♥ ♥ú❛✱ × 41 = × 63 + × 102, 12 × 63 = × 41 + × 102, × 102 = × 41 + 11 × 63 ✸✵ ✷✳✷ ❙è ❣✐↔ ❋r♦❜❡♥✐✉s ✈➔ ❣✐è♥❣ ❝õ❛ ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸ ❚r♦♥❣ t♦➔♥ ❜ë ♠ư❝ ♥➔②✱ trø ♠ët ✈➔✐ tr÷í♥❣ ủ t õ r t ổ ỵ S ❧➔ ♠ët ♥û❛ ♥❤â♠ sè ❝â ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸ ✈ỵ✐ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❧➔ {n1 , n2 , n3 } s❛♦ ❝❤♦ gcd {ni , nj } = 1, ∀i = j ✳ ✷✳✷✳✶ ▼➺♥❤ ✤➲✳ ❚➟♣ ❝→❝ sè ❣✐↔ ❋r♦❜❡♥✐✉s ❝õ❛ ♥û❛ ♥❤â♠ sè S ❧➔ PF (S) = {(c3 − 1) n3 + (r12 − 1) n2 − n1 , (c2 − 1) n2 + (r13 − 1) n3 − n1 } ✈➔ ❦✐➸✉ ❝õ❛ S ❧➔ t (S) = 2✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✹ ✈➔ ▼➺♥❤ ✤➲ ✶✳✹✳✹ t❛ ❝â✿ PF (S) = {(c3 − 1) n3 + (r12 − 1) n2 − n1 , (c2 − 1) n2 + (r13 − 1) n3 − n1 } ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✹ ✈➔ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ c2 , c3 ✱ t❛ s✉② r❛ (c3 − 1) n3 + (r12 − 1) n2 − n1 = (c2 − 1) n2 + (r13 − 1) n3 − n1 ❉♦ ✤â t (S) = ❚ø ♠➺♥❤ ✤➲ tr➯♥ t❛ ❝â ❝æ♥❣ t❤ù❝ s❛✉ ✤➙② ①→❝ ✤à♥❤ sè ❋r♦❜❡♥✐✉s ❝õ❛ ♥û❛ ♥❤â♠ sè S ✳ ❈ỉ♥❣ t❤ù❝ ♥➔② ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ❏♦❤♥s♦♥ ♥➠♠ ✶✾✻✵✳ ✷✳✷✳✷ ❍➺ q✉↔✳ ❙è ❋r♦❜❡♥✐✉s ❝õ❛ ♥û❛ ♥❤â♠ sè S ❧➔ F (S) = c1 n1 + max {r23 n3 , r32 n2 } − (n1 + n2 + n3 ) ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â F (S) = max PF (S) = max {(c3 − 1) n3 + (r12 − 1) n2 − n1 , (c2 − 1) n2 + (r13 − 1) n3 − n1 } ✸✶ ●✐↔ t❤✐➳t r➡♥❣ c3 n3 + r12 n2 < c2 n2 + r13 n3 ✳ ❑❤✐ ✤â t❤❡♦ ❇ê ✤➲ ✷✳✶✳✸ t❛ s✉② r❛ r23 n3 < r32 n2 ✳ ❉♦ ✤â F (S) = (c2 − 1) n2 + (r13 − 1) n3 − n1 = c2 n2 + r13 n3 − (n1 + n2 + n3 ) = r12 n2 + r32 n2 + r13 n3 − (n1 + n2 + n3 ) = c1 n1 + r32 n2 − (n1 + n2 + n3 ) ▲÷✉ þ r➡♥❣ ♥➳✉ c3 n3 + r12 n2 > c2 n2 + r13 n3 t❤➻ F (S) = c1 n1 + r23 n3 − (n1 + n2 + n3 ) ◆❤÷ ✈➟②✱ F (S) = c1 n1 + max {r23 n3 , r32 n2 } − (n1 + n2 + n3 ) ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✹✳✼ t❤➻ ♠ët ♥û❛ ♥❤â♠ sè ❧➔ ✤è✐ ①ù♥❣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❦✐➸✉ ❝õ❛ ♥â ❜➡♥❣ ✶✱ ♥❣❤➽❛ ❧➔ ♥â ❝â ❞✉② ♥❤➜t ♠ët sè ❣✐↔ ❋r♦❜❡♥✐✉s ✤â ❝❤➼♥❤ ❧➔ sè ❋r♦❜❡♥✐✉s✳ ◆❤÷ ✈➟②✱ ▼➺♥❤ ✤➲ ✷✳✷✳✶ ❝❤♦ t❛ ♥❣❛② ❤➺ q✉↔ s❛✉ ✤➙②✳ ✷✳✷✳✸ ❍➺ q✉↔✳ ▼ët ♥û❛ ♥❤â♠ sè ❝â ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸ ✈ỵ✐ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❧➔ ❝→❝ sè ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉ tø♥❣ ✤ỉ✐ ♠ët ❦❤ỉ♥❣ ❧➔ ♥û❛ ♥❤â♠ sè ✤è✐ ①ù♥❣✳ ❍➺ q✉↔ s❛✉ ✤➙② ✤÷đ❝ s✉② r❛ ♥❣❛② tø ▼➺♥❤ ✤➲ ✷✳✷✳✶ ✈➔ ✣à♥❤ ỵ q A = a12 a13 a21 a23 a31 a32 ❧➔ ♠ët ✵✲♠❛ tr➟♥ ✈➔ S = A1 , A2 , A3 ❑❤✐ ✤â PF (S) = {g1, g2}✱ tr♦♥❣ ✤â g1 = −a12 a13 −a12 a23 −a12 a31 −a13 a21 −a13 a32 −a21 a23 −a21 a32 −a23 a31 −a31 a32 +a12 a13 a21 +a12 a21 a23 +a12 a13 a31 +2a12 a23 a31 +a13 a21 a32 +a21 a23 a32 +a13 a31 a32 +a23 a31 a32 , g2 = −a12 a13 −a12 a23 −a12 a31 −a13 a21 −a13 a32 −a21 a23 −a21 a32 −a23 a31 −a31 a32 ✳ ✸✷ +a12 a13 a21 +a12 a21 a23 +a12 a13 a31 +a12 a23 a31 +2a13 a21 a32 +a21 a23 a32 +a13 a31 a32 +a23 a31 a32 ỵ S ởt ỷ õ số ❝â ❤➺ s✐♥❤ tè✐ t✐➸✉ ❧➔ {n1, n2, n3} ✈ỵ✐ gcd {ni, nj } = 1✱ ∀i, j ∈ {1, 2, 3} ✈➔ i = j ✳ ❑❤✐ ✤â ∆= (c1 n1 + c2 n2 + c3 n3 )2 − (c1 n1 c2 n2 + c1 n1 c3 n3 + c2 n2 c3 n3 − n1 n2 n3 ) ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣ ✈➔ PF(S) ={ ((c1 − 2)n1 + (c2 − 2)n2 + (c3 − 2)n3 + ∆), ((c1 − 2)n1 + (c2 − 2)n2 + (c3 − 2)n3 − ∆)} ✣➦❝ ❜✐➺t✱ F(S) = 12 ((c1 − 2)n1 + (c2 − 2)n2 + (c3 − 2)n3 + ∆) ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ci, rij ✱ ❇ê ✤➲ ✷✳✶✳✺ ✈➔ ▼➺♥❤ ✤➲ ✷✳✷✳✶ t❛ s✉② r❛ (r23 n3 − r32 n2 )2 = (c1 n1 + c2 n2 + c3 n3 )2 − (c1 n1 c2 n2 + c1 n1 c3 n3 + c2 n2 c3 n3 − n1 n2 n3 ) ❉♦ ✤â ∆ ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✹ ✈➔ ▼➺♥❤ ✤➲ ✶✳✹✳✹ t❛ s✉② r❛ PF (S) = {c1 n1 + r23 n3 − (n1 + n2 + n3 ) , c3 n3 + r21 n1 − (n1 + n2 + n3 )} ❱➻ c3 n3 = r31 n1 + r32 n2 ♥➯♥ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✸ t❛ ❝â c3 n3 + r21 n1 = (r31 + r21 ) n1 + r32 n2 = c1 n1 + r32 n2 ❉♦ ✤â PF (S) = {c1 n1 + r23 n3 − (n1 + n2 + n3 ) , c1 n1 + r32 n2 − (n1 + n2 + n3 )} ✸✸ ❚❛ ❝â t❤➸ ✈✐➳t ❧↕✐ PF(S) ={c1 n1 + max{r23 n3 , r32 n2 } − (n1 + n2 + n3 ) , c1 n1 + min{r23 n3 , r32 n2 } − (n1 + n2 + n3 )} ={c1 n1 + (r23 n3 + r32 n2 + ∆) − (n1 + n2 + n3 ) , c1 n1 + (r23 n3 + r32 n2 − ∆) − (n1 + n2 + n3 )} ={ (c1 n1 + r12 n2 + r13 n3 + r23 n3 + r32 n2 + ∆) − (n1 + n2 + n3 ) , (c1 n1 + r12 n2 + r13 n3 + r23 n3 + r32 n2 − ∆) − (n1 + n2 + n3 )} ={ (c1 n1 + r12 n2 + r13 n3 + r23 n3 + r32 n2 + ∆) − (n1 + n2 + n3 ) , (c1 n1 + r12 n2 + r13 n3 + r23 n3 + r32 n2 − ∆) − (n1 + n2 + n3 )} ={ (c1 n1 + c2 n2 + c3 n3 + ∆) − (n1 + n2 + n3 ) , (c1 n1 + c2 n2 + c3 n3 − ∆) − (n1 + n2 + n3 )} ❱➻ F(S) = max PF(S) ♥➯♥ t❛ ❝â F(S) = ((c1 − 2)n1 + (c2 − 2)n2 + (c3 − 2)n3 + ∆) ◆❤➢❝ ❧↕✐ r➡♥❣ t➟♣ ❤đ♣ G(S) = N \ S ✤÷đ❝ ❣å✐ ❧➔ ❦❤♦↔♥❣ ❤ð ❝õ❛ ♥û❛ ♥❤â♠ sè S ✳ ❙è ❋r♦❜❡♥✐✉s F(S) tỷ ợ t tr G(S) ỹ ữủ ố ỷ õ số S ỵ g(S)✱ ✤ỉ✐ ❦❤✐ ❝á♥ ✤÷đ❝ ❣å✐ ❧➔ ❜➟❝ ❦ý ❞à ❝õ❛ S ✳ ❝õ❛ G(S) ✤÷đ❝ ❣å✐ ❧➔ ✷✳✷✳✻ ✣à♥❤ ỵ S ởt ỷ õ số õ s✐♥❤ tè✐ t✐➸✉ ❧➔ {n1, n2, n3}✱ ✈ỵ✐ gcd {ni, nj } = 1✱ ∀i, j ∈ {1, 2, 3} ✈➔ i = j ✳ ❑❤✐ ✤â g (S) = ((c1 − 1) n1 + (c2 − 1) n2 + (c3 − 1) n3 − c1 c2 c3 + 1) ✸✹ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❇ê ✤➲ ✷✳✶✳✺✱ t❛ ❝â Ap (S, n1 ) = an2 + bn3 | (a, b) ≤ (r12 − 1, c3 − 1) ❤♦➦❝ (a, b) ≤ (c2 − 1, r13 − 1) ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✸✳✹✱ t❛ ❝â g (S) = = n1 w− w∈Ap(S,n1 ) n1 − ((c1 − 1) n1 + (c2 − 1) n2 + (c3 − 1) n3 − c1 c2 c3 + 1) ứ ỵ ỵ t õ q s ❍➺ q✉↔✳ ❈❤♦ A = ❑❤✐ ✤â a12 a13 a21 a23 a31 a32 ❧➔ ♠ët ✵✲♠❛ tr➟♥ ✈➔ S = A1 , A2 , A3 g(S) = (1 − a12 a13 − a12 a23 − a12 a31 − a13 a21 − a13 a32 − a21 a23 − a21 a32 − a23 a31 − a31 a32 + a12 a13 a21 + a12 a21 a23 + a12 a13 a31 + 2a12 a23 a31 + 2a13 a21 a32 + a21 a23 a32 + a13 a31 a32 + a23 a31 a32 ) ✸✺ ❑➌❚ ▲❯❾◆ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ t➻♠ ❤✐➸✉ ✈➲ sè ❋r♦❜❡♥✐✉s ✈➔ ❣✐è♥❣ ❝õ❛ ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸ ❞ü❛ tr➯♥ ❜➔✐ ❜→♦ ❬✺❪ ❝õ❛ ❏✳❈✳ ❘♦s❛❧❡s✱ P✳❆✳ ●❛r❝➼❛✲ ❙→♥ ❝❤❡③ r❛ ♥➠♠ ✷✵✵✹ tr➯♥ t↕♣ ❝❤➼ ❆r❝❤✳ ▼❛t❤✳✳ ❈ư t❤➸ ❧➔ ❝❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② ❝→❝ ♥ë✐ ❞✉♥❣ s❛✉ ✤➙②✳ ✶✳ ❑❤→✐ ♥✐➺♠ ✈➔ ♠ët sè ❜➜t ❜✐➳♥ ❝õ❛ ♥û❛ ♥❤â♠ sè✳ ✷✳ ❱➲ ❝→❝ t❤❛♠ sè ①→❝ ✤à♥❤ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸✳ ✸✳ ❙è ❋r♦❜❡♥✐✉s ✈➔ ❣✐è♥❣ ❝õ❛ ♥û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ ❜➡♥❣ ✸ ✸✻ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❚✐➳♥❣ ❱✐➺t ❬✶❪ P❤❛♥ ❱➠♥ ❆♥❤ ✭✷✵✶✼✮✱ ◆û❛ ♥❤â♠ sè ✤è✐ ①ù♥❣ ✈➔ ✈➔♥❤ ●♦r❡♥st❡✐♥✱ ▲✉➟♥ ✈➠♥ t❤↕❝ s➽ t♦→♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✳ ❬✷❪ ▲÷ì♥❣ ◆❣å❝ ◆❤➟t ✭✷✵✶✼✮✱ ▼ët sè ❜➜t ❜✐➳♥ ❝õ❛ ♥û❛ ♥❤â♠ sè✱ ▲✉➟♥ ✈➠♥ t❤↕❝ s➽ t♦→♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✳ ❬✸❪ ▲÷ì♥❣ ❍ú✉ ❚❤❛♥❤ ✭✷✵✶✼✮ ◆û❛ ♥❤â♠ sè ❜➜t ❦❤↔ q✉②✱ ▲✉➟♥ ✈➠♥ t❤↕❝ s➽ t♦→♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✳ ❬✹❪ ▲÷ì♥❣ ✣➻♥❤ ❚r✉♥❣ ✭✷✵✶✼✮ ◆û❛ ♥❤â♠ sè ✈ỵ✐ ❝❤✐➲✉ ♥❤ó♥❣ tè✐ ✤↕✐✱ ▲✉➟♥ ✈➠♥ t❤↕❝ s➽ t♦→♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✳ ❚✐➳♥❣ ❆♥❤ ❬✺❪ ❏✳❈✳ ❘♦s❛❧❡s✱ P✳❆✳ ●❛r❝➼❛✲ ❙→♥ ❝❤❡③ ✭✷✵✵✹✮✱ ◆✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ✇✐t❤ ❡♠❜❡❞❞✐♥❣ ❞✐♠❡♥s✐♦♥ t❤r❡❡✱ ❆r❝❤✳ ▼❛t❤✳ ✽✸✱ ✹✽✽✕✹✾✻✳ ❬✻❪ ❏✳❈✳ ❘♦s❛❧❡s✱ P✳❆✳ ●❛r❝➼❛✲ ❙→♥❝❤❡③ ✭✷✵✵✵✮✱ ◆✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s✱ ❉❡✲ ✈❡❧♦♣♠❡♥t ✐♥ ♠❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✷✵✱ ❙♣r✐♥❣❡r✳ ... a21 a 23 +a12 a 13 a31 +2a12 a 23 a31 +a 13 a21 a32 +a21 a 23 a32 +a 13 a31 a32 +a 23 a31 a32 , g2 = −a12 a 13 −a12 a 23 −a12 a31 −a 13 a21 −a 13 a32 −a21 a 23 −a21 a32 −a 23 a31 −a31 a32 ✳ ✸✷ +a12 a 13 a21... (a21 + a31 ) m1 = a21 a12 a 13 + a21 a12 a 23 + a21 a 13 a32 + a31 a12 a 13 + a31 a12 a 23 + a31 a 13 a32 = a12 (a 13 a21 + a21 a 23 + a 23 a31 ) + a 13 (a12 a31 + a21 a32 + a31 a32 ) = a12 m2 + a 13 m3 ❚÷ì♥❣... r12 r31 + r21 r32 + r31 r32 ✷✻ ✷✳✶✳✻ ❇ê ✤➲✳ ❈❤♦ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ a12, a 13, a21, a 23, a31, a32 ✈➔ m1 = a12 a 13 + a12 a 23 + a 13 a32 , m2 = a 13 a21 + a21 a 23 + a 23 a31 , m3 = a12 a31 + a21 a32