(Luận văn thạc sĩ) một số bài toán số học trong hình học phẳng

ĐẠI HỌC THÁI NGUYÊN
TRƢỜNG ĐẠI HỌC KHOA HỌC
---------------------------

LÊ PHƢƠNG THẢO

MỘT SỐ BÀI TỐN SỐ HỌC
TRONG HÌNH HỌC PHẲNG

LUẬN VĂN THẠC SĨ TOÁN HỌC

THÁI NGUYÊN - 2019


ĐẠI HỌC THÁI NGUYÊN
TRƢỜNG ĐẠI HỌC KHOA HỌC
---------------------------

LÊ PHƢƠNG THẢO

MỘT SỐ BÀI TỐN SỐ HỌC
TRONG HÌNH HỌC PHẲNG
Chun ngành: Phƣơng pháp Toán sơ cấp
Mã số: 8 46 01 13

LUẬN VĂN THẠC SĨ TOÁN HỌC

NGƯỜI HƯỚNG DẪN KHOA HỌC
PGS.TS NGUYỄN VIỆT HẢI

THÁI NGUYÊN - 2019

ì

Pữỡ

❙➮ ❇⑨■ ❚❖⑩◆ ❙➮ ❍➴❈
❚❘❖◆● ❍➐◆❍ ❍➴❈ P❍➃◆●

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✾




ì

Pữỡ

❙➮ ❇⑨■ ❚❖⑩◆ ❙➮ ❍➴❈
❚❘❖◆● ❍➐◆❍ ❍➴❈ P❍➃◆●
❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ t♦→♥ sì ❝➜♣
▼➣ sè✿ ✽✹✻✵✶✶✸

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆

ữớ ữợ ồ

P

✷✵✶✾


✐

▲í✐ ❝↔♠ ì♥

✣➸ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♠ët ❝→❝❤ tổ ổ ữủ
sỹ ữợ ú ✤ï ♥❤✐➺t t➻♥❤ ❝õ❛ P●❙✳❚❙✳ ◆❣✉②➵♥ ❱✐➺t ❍↔✐✱ ●✐↔♥❣
✈✐➯♥ ❝❛♦ ❝➜♣ ❚r÷í♥❣ ✤↕✐ ❤å❝ ❍↔✐ P❤á♥❣✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❜➔② tä ❧á♥❣
❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛② ✈➔ ①✐♥ ❣û✐ ❧í✐ tr✐ ➙♥ ♥❤➜t ❝õ❛ tỉ✐ ✤è✐ ✈ỵ✐ ♥❤ú♥❣
✤✐➲✉ t❤➛② ✤➣ ❞➔♥❤ ❝❤♦ tỉ✐✳
❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ỏ t qỵ t
ổ ợ ồ rữớ ồ ❦❤♦❛ ❤å❝ ✲ ✣↕✐
❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ t ỳ tự qỵ ụ
ữ t ❝❤♦ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳
❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ỡ t t tợ ỳ
ữớ ✤➣ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ ❤é trđ ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ tr♦♥❣ s✉èt
q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳
❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦
❍↔✐ P❤á♥❣✱ t❤→♥❣ ✳✳✳ ♥➠♠ ✷✵✳✳✳
◆❣÷í✐ ✈✐➳t ▲✉➟♥ ✈➠♥

▲➯ P❤÷ì♥❣ ❚❤↔♦


✐✐

❉❛♥❤ ♠ö❝ ❤➻♥❤
✶✳✶
✶✳✷
✶✳✸
✶✳✹
✶✳✺

❚❛♠ ❣✐→❝ P②t❤❛❣♦r❡✿ BC 2 = AB 2 + AC 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳
❚❛♠ ❣✐→❝ ❍❡r♦♥ [c, e, b + d]✱ ✤÷í♥❣ ❝❛♦ a ✳ ✳ ✳ ✳ ✳ ✳ ✳
❚❛♠ ❣✐→❝ ❍❡r♦♥ t❤❡♦ sü t➠♥❣ ❞➛♥ ❝õ❛ ❝↕♥❤ ❧ỵ♥ ♥❤➜t ✳
❚❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ❝ì ❜↔♥ ✈➔ ❝→❝ ❜→♥ ❦➼♥❤ r, ra , rb , rc
❚➼♥❤ ❝❤➜t ❝→❝ ❝❡✈✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳
✳
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✳
✳
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✳
✳

✳ ✹
✳ ✽
✳ ✶✷
✳ ✶✸
✳ ✶✾

✷✳✶ ❍❛✐ ♥❣❤✐➺♠ ❧➔ t❛♠ ❣✐→❝ ✈✉ỉ♥❣ ✈ỵ✐ m = 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✷✳✷ ❍❛✐ ♥❣❤✐➺♠ ❧➔ t❛♠ ❣✐→❝ tị ✈ỵ✐ m = 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✷✳✸ ❚❛♠ ❣✐→❝ ❝↕♥❤ tü ♥❤✐➯♥ ♥❣♦↕✐ t✐➳♣ ✤÷í♥❣ trá♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✶
✸✳✷
✸✳✸
✸✳✹
✸✳✺
✸✳✻

❚ù ❣✐→❝ ❤ú✉ t✛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❚ù ❣✐→❝ ❤ú✉ t✛ ❝õ❛ ❇r❛❤♠❛❣✉♣t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✣ë ❞➔✐ ✷ ✤÷í♥❣ ❝❤➨♦✱ ❝❤✉ ✈✐✱ ❞✐➺♥ t➼❝❤ tù ❣✐→❝ ✳
❉ü♥❣ tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛ tø t❛♠ ❣✐→❝ ❍❡r♦♥
❍❛✐ ✤÷í♥❣ ❝❤➨♦ AB, BC ∈ P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✭■▼❖ ✶✾✻✽✱ #1✮✱ ❈→❝❤ ❣✐↔✐ t❤ù ❜❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✹✷
✹✹
✹✺
✹✼
✺✷
✺✹


✐✐✐

❉❛♥❤ ♠ư❝ ❜↔♥❣
✶✳✶

❚r➼❝❤ ❞❛♥❤ s→❝❤ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❝ì ❜↔♥

✶✳✷

❍å t❛♠ ❣✐→❝ ❍❡r♦♥ ♣❤ö t❤✉ë❝

✷✳✶

❇❛ ❝↕♥❤ ❧➔ ❝➜♣ sè ❝ë♥❣

✷✳✷

❇➔✐ t♦→♥

✷✳✸

❇➔✐ t♦→♥

2

P = nS
P 2 = nS

✈ỵ✐
✈ỵ✐

λ✱

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

λ

✳ ✳ ✳ ✳ ✳ ✳

✷✸

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

✸✸

n = 31
n = 42

✈ỵ✐ ✶✵ ❣✐→ trà

✶✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✵


✐✈

▼ư❝ ❧ư❝
●✐ỵ✐ t❤✐➺✉ ❧✉➟♥ ✈➠♥

✶

✶ ❚❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ✈➔ t❛♠ ❣✐→❝ ❍❡r♦♥

✹

✶✳✶ ❇➔✐ t♦→♥ t➻♠ t❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ✈➔ t❛♠ ❣✐→❝ ❍❡r♦♥
✶✳✶✳✶ ❈→❝ ❜ë ❜❛ P②t❤❛❣♦r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✷ ❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷ ❇➔✐ t♦→♥ HG✿ ❚❛♠ ❣✐→❝ ❍❡r♦♥ ✈ỵ✐ r, ra , rb , rc ∈ N ✳ ✳
✶✳✸ ❍å ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ♣❤ö t❤✉ë❝ λ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷ ❚❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ ❤➺ t❤ù❝ ❣✐ú❛ S ✈➔ P
✷✳✶ ❚❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ S = m.P, m ∈ N ✳ ✳
✷✳✶✳✶ ❚❤✉➟t t♦→♥ ●♦❡❤❧ ✈➔ t❤✉➟t t♦→♥ ▼❛r❦♦✈
✷✳✶✳✷ ❍❛✐ tr÷í♥❣ ❤đ♣ t❤❛♠ sè ♥❣✉②➯♥ ✳ ✳ ✳ ✳
✷✳✷ ❚❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ P 2 = nS, n ∈ N ✳ ✳
✷✳✷✳✶ ❚r÷í♥❣ ❤đ♣ n ❧➔ sè ♥❣✉②➯♥ tè ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✷ ❚r÷í♥❣ ❤ñ♣ n ❧➔ ❤ñ♣ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✸ ❚r÷í♥❣ ❤đ♣ r✐➯♥❣✿ ❚❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ✳

✸

▼ët sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥

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✳ ✹
✳ ✹
✳ ✼
✳ ✶✵
✳ ✶✽
✳
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✳
✳
✳
✳

✳

✷✹
✷✹
✷✺
✸✷
✸✺
✸✻
✸✽
✸✾

✹✶

✸✳✶ ❚ù ❣✐→❝ ❝â ❝↕♥❤ ✈➔ ✤÷í♥❣ ❝❤➨♦ ❤ú✉ t✛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✸✳✷ ❳→❝ ✤à♥❤ ❝→❝ ②➳✉ tè ❝õ❛ tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✸✳✸ ●✐ỵ✐ t❤✐➺✉ ♠ët sè ❜➔✐ t♦→♥ t❤✐ ❖❧②♠♣✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✺✽


✶

●✐ỵ✐ t❤✐➺✉ ❧✉➟♥ ✈➠♥
✶✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥
◆❤✐➲✉ ❜➔✐ t♦→♥✱ ❦❤→✐ ♥✐➺♠ tr♦♥❣ ❤➻♥❤ ❤å❝ ❧✐➯♥ q✉❛♥ ✤➳♥ sè ❤å❝✳ ✣➦❝ ❜✐➺t
❝â ♥❤ú♥❣ ❜➔✐ t♦→♥ ❤♦➔♥ t♦➔♥ t❤✉ë❝ ❧➽♥❤ ✈ü❝ sè ❤å❝ ♥❤÷ ❜ë ❜❛ P②t❤❛❣♦r❡✱
t❛♠ ❣✐→❝ ❍❡r♦♥✱✳✳✳✣➸ ❣✐↔✐ q✉②➳t ♥❤ú♥❣ ❜➔✐ t♦→♥ ♥➔② t❤÷í♥❣ ♣❤↔✐ ❣✐↔✐
♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡✱ ♣❤÷ì♥❣ tr➻♥❤ P②t❤❛❣♦r❡✱ ♣❤÷ì♥❣ tr➻♥❤ P❡❧❧✱✳✳✳✈➔

♥❤✐➲✉ ❦✐➳♥ t❤ù❝ s➙✉ ✈➲ sè ♥❣✉②➯♥ tè ♥â✐ r✐➯♥❣ ✈➔ sè ❤å❝ ♥â✐ ❝❤✉♥❣✳ ✣➲ t➔✐
♥➔② tr➻♥❤ ❜➔② ♥❤✐➲✉ ✈➜♥ ✤➲ ❝õ❛ sè ❤å❝ →♣ ❞ö♥❣ ✈➔♦ ❤➻♥❤ ❤å❝✱ ♠❛♥❣ ❧↕✐
♥❤ú♥❣ ❦➳t q✉↔ s➙✉ s➢❝ ✈➲ ❜➔✐ t♦→♥ ❤➻♥❤ ❤å❝ ❣✐↔✐ ❜➡♥❣ ❦✐➳♥ t❤ù❝ sè ❤å❝✳
▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧➔✿
✲ ❚r➻♥❤ ❜➔② ❤❛✐ ❜➔✐ t♦→♥✿ t➻♠ ❝→❝ t❛♠ ❣✐→❝ P②t❤❛❣♦r❡✱ t➻♠ ❝→❝ t❛♠ ❣✐→❝
❍❡r♦♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t✳ ◆➯✉ r❛ ❝→❝ t❤✉➟t t♦→♥ t➻♠ ♥❣❤✐➺♠ ❝õ❛
❝→❝ ❜➔✐ t♦→♥ ✤➦t r❛✳ ❈→❝ tr÷í♥❣ ❤đ♣ r✐➯♥❣ ①→❝ ✤à♥❤ t❛♠ ❣✐→❝ ❍❡r♦♥✿ ❇➔✐
t♦→♥ HG t➻♠ t❛♠ ❣✐→❝ ❍❡r♦♥ ✈ỵ✐ r, ra , rb , rc ∈ N✱ t❛♠ ❣✐→❝ ❝â ❝→❝ ❝↕♥❤ ❧➟♣
t❤➔♥❤ số ở ữợ t r
ỷ ử tự ừ số ồ ữ ỵ tt ❝❤✐❛ ❤➳t✱ sü ♣❤➙♥ t➼❝❤
♠ët sè tü ♥❤✐➯♥ t❤➔♥❤ ❝→❝ sè ♥❣✉②➯♥ tè✱ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡✱
❝→❝ ❧➟♣ ❧✉➟♥ sè ❤å❝ ♥â✐ ❝❤✉♥❣✱✳✳✳✤➸ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè tr÷í♥❣ ❤đ♣ r✐➯♥❣
q✉❛♥ trå♥❣ ❝õ❛ ❜➔✐ t♦→♥ t➻♠ t❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ t❤ä❛ ♠➣♥ ♠ët tr♦♥❣
❜❛ ✤✐➲✉ ❦✐➺♥ s❛✉
S = mP ; P 2 = nS ❤❛② R/r = N ∈ N.

✲ ◆➯✉ r❛ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✈➔ ❝→❝❤ ❣✐↔✐ q✉②➳t ú ự ỳ
t tự rt ỗ ữù ❧ü❝ ❞↕② ❝→❝ ❝❤✉②➯♥ ✤➲ ❦❤â ð
tr÷í♥❣ ❚❍❈❙ ✈➔ ❚❍P❚ ❣â♣ ♣❤➛♥ ✤➔♦ t↕♦ ❤å❝ s✐♥❤ ❣✐→✐ ♠æ♥ ❍➻♥❤ ❤å❝✳


✷

✷✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐✱ ♥❤ú♥❣ ✈➜♥ ✤➲ ❝➛♥ ❣✐↔✐ q✉②➳t
❉ü❛ ✈➔♦ ❝→❝ t➔✐ ❧✐➺✉ ❬✷❪✱ ❬✸❪✱ ❬✹❪✱ ❬✻❪ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ♠ët sè ❜➔✐
t♦→♥ ❤❛② ✈➲ t❛♠ ❣✐→❝ ♥❣✉②➯♥ ✈➔ ❝ô♥❣ ❧➔ ♥❤ú♥❣ ❜➔✐ t♦→♥ ❦❤â ❤❛② tr
ý t ồ s tr ữợ ✈➔ q✉è❝ t➳✳ ◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥
❝❤✐❛ ❧➔♠ ✸ ❝❤÷ì♥❣✿

❈❤÷ì♥❣ ✶✳ ❚❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ✈➔ t❛♠ ❣✐→❝ ❍❡r♦♥

❇➔✐ t♦→♥ t➻♠ ❜ë ❜❛ P②t❤❛❣♦r❡ ❧➔ ❜➔✐ t♦→♥ sè ❤å❝ q✉❡♥ t❤✉ë❝✱ t✉②
♥❤✐➯♥ ❦❤æ♥❣ t❤➸ ❦❤æ♥❣ ♥❤➢❝ ❧↕✐ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ ♥❤✐➲✉ ❝ỉ♥❣ tr➻♥❤✳
❱✐➺❝ ❧➔♠ ♥➔② ❝ơ♥❣ ❝♦✐ ❧➔ ❜ê s✉♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✤➛✉ t✐➯♥ ❝õ❛ ❜➔✐
t♦→♥ ✤➦t r❛✳ ❇➔✐ t♦→♥ t➻♠ t❛♠ ❣✐→❝ ❍❡r♦♥ ❞➝♥ tợ trữớ ủ r
tú t tú ♠ët ❦➳t q✉↔ tê♥❣ q✉→t✿ ❍å ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ử
tở t số ữỡ ỗ
t t t❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ✈➔ t❛♠ ❣✐→❝ ❍❡r♦♥
✶✳✷✳ ❇➔✐ t♦→♥ HG✿ ❚❛♠ ❣✐→❝ ❍❡r♦♥ ✈ỵ✐ r, ra , rb , rc ∈ N
✶✳✸✳ ❍å ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ♣❤ư t❤✉ë❝ λ✳

❈❤÷ì♥❣ ✷✳ ❚❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ ❤➺ t❤ù❝ ❣✐ú❛ S ✈➔ P
◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ✤➲ ❝➟♣ ✤➳♥ ❤❛✐ ❜➔✐ t♦→♥ ✈➲ t➻♠ t❛♠ ❣✐→❝ ❝↕♥❤
♥❣✉②➯♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ♣❤ư✿ ❚➻♠ t❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ S = mP
✈➔ t➻♠ t❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ P 2 = nS ✳ ❈→❝ ❦ÿ t❤✉➟t sè ❤å❝ ✤÷đ❝ ✈➟♥
❞ư♥❣ ❣✐↔✐ ❝→❝ ữỡ tr t t tợ tt
t ❣✐↔✐ ❜➔✐ t♦→♥ ❜➡♥❣ ❝→❝ ♣❤➛♥ ♠➲♠ t✐♥ ❤å❝✳ ❈❤÷ì♥❣ ỗ
ử s
ợ S = mP, m ∈ N
✷✳✷✳ ❚❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ợ P 2 = nS, n N

ữỡ ởt sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥
❈❤÷ì♥❣ ✸ ①➨t ❜➔✐ t♦→♥ t❛♠ rở tự ỳ t ợ
ữỡ ♣❤→♣ t✐➳♣ ❝➟♥ t÷ì♥❣ tü ✷ ❝❤÷ì♥❣ ✶ ✈➔ ✷✳ P❤➨♣ ❞ü♥❣ tù ❣✐→❝ ❤ú✉


✸

t✛ ♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥ ✭tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛✮ ✤÷đ❝ ❣✐↔✐ q✉②➳t trå♥ ✈➭♥✳ Ð
✤➙② ❝ô♥❣ tr➻♥❤ ❜➔② ♠ët ✈➔✐ ❜➔✐ t♦→♥ ❤➻♥❤ ❤å❝ ❝â ♥ë✐ ❞✉♥❣ sè ❤å❝ ✤➣ ❣➦♣
tr♦♥❣ ý t ồ s t ữợ

ở ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ✸ ♣❤➛♥✿
✸✳✶✳ ❚ù ❣✐→❝ ❝â ❝↕♥❤ ✈➔ ✤÷í♥❣ ❝❤➨♦ ❤ú✉ t✛
✸✳✷✳ ❳→❝ ✤à♥❤ ❝→❝ ②➳✉ tè ❝õ❛ tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛
✸✳✸✳ ●✐ỵ✐ t❤✐➺✉ ♠ët sè ❜➔✐ t♦→♥ t❤✐ ❖❧②♠♣✐❝✳


✹

❈❤÷ì♥❣ ✶
❚❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ✈➔ t❛♠ ❣✐→❝
❍❡r♦♥
✶✳✶

❇➔✐ t♦→♥ t➻♠ t❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ✈➔ t❛♠ ❣✐→❝
❍❡r♦♥

✶✳✶✳✶

❈→❝ ❜ë ❜❛ P②t❤❛❣♦r❡

❚r♦♥❣ ❤➻♥❤ ❤å❝ õ ởt ỵ q trồ q tở ỵ
Ptr ở ừ ỵ tr ởt t ❣✐→❝ ✈✉ỉ♥❣ ❜➻♥❤
♣❤÷ì♥❣ ❝↕♥❤ ❤✉②➲♥ ❜➡♥❣ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❤❛✐ ❝↕♥❤ ❣â❝ ✈✉ỉ♥❣✧✱ ❤➻♥❤ ✶✳✶
❱➻ ✈➟② ♠➔ ♣❤÷ì♥❣ tr➻♥❤ x2 + y 2 = z 2 ✈ỵ✐ x, y, z ❧➔ ❝→❝ sè tü ♥❤✐➯♥✱ ✤÷đ❝
❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ P②t❤❛❣♦r❡✳ ✈➔ ♥❣❤✐➺♠ tü ♥❤✐➯♥ (x, y, z) ❝õ❛ ♣❤÷ì♥❣

❍➻♥❤ ✶✳✶✿

❚❛♠ ❣✐→❝ P②t❤❛❣♦r❡✿ BC 2 = AB 2 + AC 2

✺

tr➻♥❤ ♥➔② ❣å✐ ❧➔ ❜ë ❜❛ P②t❤❛❣♦r❡✳
❚r♦♥❣ sè ❤å❝✱ t➟♣ ❤đ♣ ❝→❝ sè ♥❣✉②➯♥ tè ❝â t❤➸ ✤÷đ❝ ①❡♠ ❧➔ ♠ët ❜ë ❣❡♥
❤♦➔♥ ❝❤➾♥❤ ❞ò♥❣ ✤➸ ①➙② ❞ü♥❣ t♦➔♥ ❜ë ❝→❝ sè tü ♥❤✐➯♥✳ ●✐è♥❣ ♥❤÷ ♠é✐
❝♦♥ ♥❣÷í✐ ❝â ♥❤ú♥❣ ✤➦❝ ✤✐➸♠ r✐➯♥❣ ❜✐➺t ❞♦ ❝❤ó♥❣ t❛ ❝â ♥❤ú♥❣ ❜ë ❣❡♥ ❦❤→❝
♥❤❛✉✱ ❝→❝ sè ❝ô♥❣ ✈➟② ♠é✐ ❝♦♥ sè ❦❤→❝ ♥❤❛✉ sð ❤ú✉ ♠ët ❜ë ❣❡♥ ❦❤→❝ ♥❤❛✉✳
❱➻ 12 = 2.2.3 = 22 .3 t❛ ❝â t❤➸ ♥â✐ sè ✶✷ ❝â ❤❛✐ ❣❡♥ sè ✷ ✈➔ ♠ët ❣❡♥ sè ✸✱
tr♦♥❣ ❦❤✐ ✤â 90 = 2.3.3.5 = 2.32 .5 ❝â ♠ët ❣❡♥ sè ✷✱ ❤❛✐ ❣❡♥ sè ✸ ✈➔ ♠ët
❣❡♥ sè ✺✳
▼ët ❝→❝❤ tê♥❣ q✉→t✱ ❦❤✐ sè tü ♥❤✐➯♥ n ✤÷đ❝ ♣❤➙♥ t➼❝❤ r❛ t❤ø❛ sè ♥❣✉②➯♥
tè ♥❤÷ s❛✉
n = pα1 1 · pα2 2 . . . pαk k

t❤➻ t❛ ♥â✐ n ❝â α1 ❣❡♥ p1 , α2 ❣❡♥ p2 , . . . , αk ❣❡♥ pk ✳
❚❛ ♥❤➢❝ ❧↕✐ ♠ët t➼♥❤ ❝❤➜t sè ❤å❝✱ ❝â t❤➸ ✤➦t t➯♥ ❧➔ t➼♥❤ ✧t→❝❤ ✤÷đ❝✧✿♥➳✉
a.b = A2 ✭sè ❝❤➼♥❤ ♣❤÷ì♥❣✮✱ a, b ∈ N t❤➻ a ✈➔ b ♣❤↔✐ ❝â ❞↕♥❣ a = u2 ·w, b =
v 2 .w ✈ỵ✐ u, v, w ∈ N✳ ❈â t❤➸ ❣✐↔✐ t❤➼❝❤ ♥❤÷ s❛✉✿ ♥➳✉ sè ❧÷đ♥❣ ❣❡♥ p tr♦♥❣
a ❧➔ ❧➫ t❤➻ sè ❧÷đ♥❣ ❣❡♥ p ❝â tr♦♥❣ b ❝ơ♥❣ s➩ ❧➔ ❧➫✱ ✈➻ ✈➟② sè ❧÷đ♥❣ ❣❡♥ tr♦♥❣
a, b ♣❤↔✐ ❧➔ sè ❝❤➤♥✳ ❇➡♥❣ ❝→❝❤ t➟♣ ❤ñ♣ ❝→❝ ❧♦↕✐ ❣❡♥ ❧➫ ♥➔② ❧↕✐ t❤➔♥❤ sè w
t❤➻ t❛ s➩ ❝â a = u2 w, b = v 2 w✳
●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ P②t❤❛❣♦r❡ x2 + y 2 = z 2
2
2
2
ữợ õ x = z − y = (z − y)(z + y)✳ ❚❤❡♦ t➼♥❤ ❝❤➜t t→❝❤
t❤➻ z + y = u2 w, z − y = v 2 w ✈➔ x = uvw✳ ❱➟② ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
❝❤➼♥❤ ❧➔



x = uvw




u2 v 2 w
y=
2 2

2


u
+
v w

z=
2
ữợ ự ♠✐♥❤ y, z ∈ N✳
❚❤➟t ✈➟②✱ ♥➳✉ w = 2m + 1 t❤➻ z + y = u2 (2m + 1); z − y = v 2 (2m + 1)✳
❚ø ✷ ✤➥♥❣ t❤ù❝ s✉② r❛ 2z = u2 + v 2 (2m+1) ✈➔ 2y = u2 − v 2 (2m+1)✳
◆❤÷ ✈➟②✱ u2 + v 2 ✈➔ u2 − v 2 ✤➲✉ ❝❤➤♥✱ tù❝ y, z ❧➔ ❝→❝ sè tü ♥❤✐➯♥✳ ❈á♥
♥➳✉ w ❝❤➤♥ t❤➻ ❤✐➸♥ ♥❤✐➯♥ y, z ∈ N✳ ❚r÷í♥❣ ❤đ♣ w ❝❤➤♥ t❛ ✤➦t w = 2s✱
♥❣❤✐➯♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ x = 2uvs, y = u2 − v 2 s, z = u2 + v 2 s✳


✻

❑❤✐ u2 + v 2 ✈➔ u2 − v 2 ❧➔ ❝→❝ sè ❝❤➤♥ t❤➻ t❛ ✤➦t u = v + 2k ✱ s✉② r❛


2


x = (v + 2k)vw = v + 2kv w

✈✐➳t ❧↕✐ t❤➔♥❤

y = 2kv + 2k 2 w


z = v 2 + 2kv + 2k 2 w

x = (x + k)2 − k 2 w, y = 2(v + k)kw, z = (v + k)2 + k 2 w

❚r♦♥❣ ❝↔ ❤❛✐ tr÷í♥❣ ❤đ♣ tr➯♥ t❛ ❝â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
P②t❤❛❣♦r❡ ❧➔✿


 x = c.(2ab)
y = c. a2 − b2 , a, b, c ∈ N, a > b

 z = c. a2 + b2

P❤÷ì♥❣ tr➻♥❤ P②t❤❛❣♦r❡ ❝â ✈ỉ sè ♥❣❤✐➺♠ ♣❤ư t❤✉ë❝ ✸ t❤❛♠ sè✳ ❚✉②
♥❤✐➯♥ ✈ỵ✐ n ≥ 3✱ ❋❡r♠❛t ❦❤➥♥❣ ✤à♥❤ r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ xn + y n = z n
❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥ ❦❤→❝ ✵✳
❱ỵ✐ c = 1 t❛ ❝â ❜ë ❜❛ P②t❤❛❣♦r❡ x = 2ab, y = a2 − b2 , z = a2 + b2
❞♦ ❊✉❝❧✐❞❡ t➻♠ r❛ ✭❦❤♦↔♥❣ trữợ ổ ở
ử ✈➲ ♠ët ❜ë ❜❛ P②t❤❛❣♦r❡ ❝ì ❜↔♥ ✭❝→❝ ❝❛♥❤ t÷ì♥❣ ù♥❣ ❝õ❛ ❝❤ó♥❣ ❧➔

a = 2mn, b = m2 − n2 , c = m2 + n2 tr♦♥❣ ✤â m, n ❧➔ ♥❣✉②➯♥ tè ❝ị♥❣
♥❤❛✉✱ ❝❤➾ ❝â ♠ỉt sè ❧➫✮✳ ❚❛ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ❝õ❛ ❜ë ❜❛ P②t❤❛❣♦r❡ ❝ì
❜↔♥ (a, b, c) ✭①❡♠ tr♦♥❣ ❬✶❪✮✿
✭✐✮ ❍❛✐ ❝❛♥❤ ❣â❝ ✈✉æ♥❣ m2 −n2 ✈➔ 2mn✱ ❝↕♥❤ 2mn ❣å✐ ❧➔ ❝↕♥❤ ❣â❝ ✈✉æ♥❣

❝❤➤♥❀ c = m2 + n2 ❧➔ ❝↕♥❤ ❤✉②➲♥✳

(c − a)(c − b)
❧➔ sè ❝❤➼♥❤ ♣❤÷ì♥❣✳
2

✭✐✐✮ ❚r♦♥❣ ✸ sè a, b, c ❝â ♥❤✐➲✉ ♥❤➜t ♠ët sè ❝❤➼♥❤ ♣❤÷ì♥❣✳ ỗ t ổ số
ở Ptr ỡ ❤✉②➲♥ ✭❤♦➦❝ ❝↕♥❤ ❣â❝ ✈✉ỉ♥❣✮ ❧➔ ❝❤➼♥❤
♣❤÷ì♥❣✳ ❚ê♥❣ ❝õ❛ ❝↕♥❤ ❤✉②➲♥ ✈➔ ❝↕♥❤ ❣â❝ ✈✉ỉ♥❣ ❝❤➤♥ ❝õ❛ ❜ë ❜❛ P②t❤❛❣♦r❡
❝ì ❜↔♥ ❧✉ỉ♥ ❧➔ sè ❝❤➼♥❤ ♣❤÷ì♥❣✳
✭✐✐✐✮ ❉✐➺♥ t➼❝❤ S =

ab
2

❧➔ sè tü ♥❤✐➯♥ ❝❤➤♥✳ ❚r♦♥❣ ❤❛✐ sè a, b ❝â ✤ó♥❣

♠ët sè ❧➫❀ ✈➔ c ❧➔ sè ❧➫✳
✭✐✈✮ ❚r♦♥❣ ✸ sè a, b, c ❝â ✤ó♥❣ ♠ët sè ❝❤✐❛ ❤➳t ❝❤♦ ✺✳


✼

✭✈✮ ❚r♦♥❣ ✹ sè a, b, a + b, b − a ❝â ✤ó♥❣ ♠ët sè ❝❤✐❛ ❤➳t ❝❤♦ ✼❀ tr♦♥❣ ✹
sè a + c, b + c, c − a, c − b ❝â ✤ó♥❣ ♠ët sè ❝❤✐❛ ❤➳t ❝❤♦ ✽ ✭❝❤♦ ✾✮❀ tr♦♥❣ ✻

sè a, b, 2a + b, 2a − b, 2b + a, 2b − a ❝â ✤ó♥❣ ♠ët sè ❝❤✐❛ ❤➳t ❝❤♦ ✶✶✳

✶✳✶✳✷ ❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥
❈â ♠ët sè ❝→❝❤ ①→❝ ✤à♥❤ ❦❤→✐ ♥✐➺♠ ✧t❛♠ ❣✐→❝ ❍❡r♦♥✧✳ ❚r♦♥❣ ✤➲ t➔✐ ♥➔②
t❛ ❝❤å♥ ❝→❝❤ ①→❝ ✤à♥❤ s❛✉✿

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❚❛♠ ❣✐→❝ ❍❡r♦♥ ❧➔ t❛♠ ❣✐→❝ ♠➔ ✤ë ❞➔✐ ❜❛ ❝↕♥❤ a, b, c

✈➔ ❞✐➺♥ t➼❝❤ ❙ ❝õ❛ ♥â ❧➔ ❝→❝ sè tü ♥❤✐➯♥✳

❚❛♠ ❣✐→❝ ❍❡r♦♥ ✤÷đ❝ ỵ ỳ ỵ HT [x, y, z; S] ✤÷đ❝
❤✐➸✉ ❧➔ t❛♠ ❣✐→❝ ❍❡r♦♥ ❝↕♥❤ x, y, z ✱ ❞✐➺♥ t➼❝❤ S ✳ HT [x, y, z; S] ✤÷đ❝ ❣å✐
❧➔ ❝ì ❜↔♥ ✭❤❛② ♥❣✉②➯♥ t❤õ②✮ ♥➳✉ (x, y, z) = 1✳ ❚r♦♥❣ ❦❤✐ t➻♠ ❝→❝ t❛♠ ❣✐→❝
❍❡r♦♥ ❝â t❤➸ ❝❤♦ ❦➳t q✉↔ ❝↕♥❤ ✈➔ ❞✐➺♥ t➼❝❤ ❧➔ sè ❤ú✉ t✛✱ ❜➡♥❣ ❝→❝❤ ♥❤➙♥
t➜t ❝↔ ✈ỵ✐ ❜ë✐ sè ❝❤✉♥❣ ♥❤ä ♥❤➜t ❝õ❛ ♠➝✉ t❛ ✈➝♥ ✤÷đ❝ ♥❣❤✐➺♠ tü
t t r ỳ t ỵ RT ✱ ❤♦➔♥ t♦➔♥ ①→❝ ✤à♥❤ ✤÷đ❝
t❛♠ ❣✐→❝ ❍❡r♦♥ ✭♥❣✉②➯♥✮ ✈➔ ♥❣÷đ❝ ❧↕✐✳ ❱✐➺❝ t➻♠ ❝→❝ ❝ỉ♥❣ t❤ù❝ ❝❤♦ t❛♠ ❣✐→❝
❍❡r♦♥ ❝ì ❜↔♥ ❜➡♥❣ ❤➻♥❤ ❤å❝ t❤✉➛♥ tó② ❣➦♣ ♥❤✐➲✉ õ
sỷ ử ỵ tt số t❛ ❦❤ỉ♥❣ ♥❤ú♥❣ tr→♥❤ ✤÷đ❝ ♥❤ú♥❣ ❦❤â
❦❤➠♥ ✤â ♠➔ ❝á♥ t➻♠ ✤÷đ❝ ❝→❝ ❝ỉ♥❣ t❤ù❝ ❜✐➸✉ ❞✐➵♥ ✤ì♥ ❣✐↔♥✳
❚❛♠ ❣✐→❝ ❍❡r♦♥ ✤÷đ❝ ✤➦t t❤❡♦ t➯♥ ❝õ❛ ♥❤➔ t♦→♥ ❤å❝ ❍② ▲↕♣ ✧❍❡r♦♥ ♦❢
❆❧❡①❛♥❞r✐❛✧ ✈➻ ♥â ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ❝æ♥❣ t❤ù❝ t➼♥❤ ❞✐➺♥ t➼❝❤
S=

s(s − a)(s − b)(s − c),

✈ỵ✐ s =

a+b+c
2

ợ ữ t t ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè t➼♥❤ ❝❤➜t
❝õ❛ t❛♠ ❣✐→❝ ❍❡r♦♥✱ ❝â t❤❛♠ ❦❤↔♦ ✈➔ ❤➺ t❤è♥❣ tr♦♥❣ ❬✸❪✳

❚➼♥❤ ❝❤➜t ✶✳✶✳✶✳ ❇➜t ❦➻ ♠ët t❛♠ ❣✐→❝ ♥➔♦ ❝â ✤ë ❞➔✐ ❜❛ ❝↕♥❤ t↕♦ t❤➔♥❤
♠ët ❜ë ❜❛ P②t❤❛❣♦r❡ ✤➲✉ ❧➔ ♠ët HT [x, y, z; S]✳

❈❤ù♥❣ ♠✐♥❤✳ ❱➻ ❜ë ❜❛ sè P②t❤❛❣♦r❡ ❧➔ ❝→❝ sè tü ♥❤✐➯♥ ✈➔ ❞✐➺♥ t➼❝❤ ❝õ❛ ♥â
❜➡♥❣ ♠ët ♥û❛ t➼❝❤ ❤❛✐ ❝↕♥❤ ❣â❝ ✈✉æ♥❣✱ tr♦♥❣ ✤â ✶ ❝↕♥❤ ❣â❝ ✈✉æ♥❣ ♣❤↔✐ ❧➔
sè ❝❤➤♥✳


✽

▼ët ✈➼ ❞ư ❝❤♦ ♠ët t❛♠ ❣✐→❝ ❍❡r♦♥ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ t❛♠ ❣✐→❝ ✈✉æ♥❣ ❧➔
t❛♠ ❣✐→❝ ❝â a = 5, b = 5, c = 6 ✈ỵ✐ ❞✐➺♥ t➼❝❤ ❧➔ 12❀ t❛♠ ❣✐→❝ ♥➔② t❤✉ ✤÷đ❝
❜➡♥❣ ❝→❝❤ ❣❤➨♣ ❤❛✐ t❛♠ ❣✐→❝ ❝â ✤ë ❞➔✐ ❜❛ ❝↕♥❤ ❧➔ ✸✱ ✹✱ ✺ ❞å❝ t❤❡♦ ❝↕♥❤ ❝â
✤ë ❞➔✐ ❜➡♥❣ ✹✳ P❤÷ì♥❣ ♣❤→♣ tê♥❣ q✉→t ❝❤♦ ❝→❝❤ ❧➔♠ ♥➔② ✤÷đ❝ ♠✐♥❤ ❤å❛
ð ❤➻♥❤ ✶✳✷✿ ▲➜② ♠ët t❛♠ ❣✐→❝ ✈ỵ✐ ✤ë ❞➔✐ ❜❛ ❝↕♥❤ ❧➔ ♠ët ❜ë ❜❛ P②t❤❛❣♦r❡
a, b, c ✭c ❧➔ sè ❧ỵ♥ ♥❤➜t✮❀ ♠ët t❛♠ ❣✐→❝ ❦❤→❝ ❝â ✤ë ❞➔✐ ❜❛ ❝↕♥❤ ❧➔ ♠ët ❜ë
❜❛ sè P②t❤❛❣♦r❡ a, d, e ✭e ❧➔ sè ❧ỵ♥ ♥❤➜t✮✱ ❣❤➨♣ ❝❤ó♥❣ ❧↕✐ ❞å❝ t❤❡♦ ❝↕♥❤
❝â ✤ë ❞➔✐ ❧➔ ❛ ✤➸ ✤÷đ❝ ♠ët t❛♠ ❣✐→❝ ❝â ✤ë ❞➔✐ ❜❛ ❝↕♥❤ ❧➔ ❝→❝ sè tü ♥❤✐➯♥
1
c, e, b + d✱ ✈➔ ❝â ❞✐➺♥ t➼❝❤ ❧➔ ♠ët sè ❤ú✉ t✛✿ S = (b + d) · a ✭♠ët ♥û❛
2
❝↕♥❤ ✤→② ♥❤➙♥ ✈ỵ✐ ❝❤✐➲✉ ❝❛♦✮✳ ▼ët ❝➙✉ ❤ä✐ t❤ó ✈à ✤➦t r❛ ❧➔ ❧✐➺✉ t➜t ❝↔ ❝→❝

❍➻♥❤ ✶✳✷✿

❚❛♠ ❣✐→❝ ❍❡r♦♥ [c, e, b + d]✱ ✤÷í♥❣ ❝❛♦ a

t❛♠ ❣✐→❝ ❍❡r♦♥ ✤➲✉ ❝â t❤➸ ✤÷đ❝ t↕♦ r❛ ❜➡♥❣ ❝→❝❤ ❣❤➨♣ ❤❛✐ t❛♠ ❣✐→❝ ✈✉ỉ♥❣
✭✈ỵ✐ ✤ë ❞➔✐ ❝→❝ ❝↕♥❤ ❧➔ ❝→❝ sè tü ♥❤✐➯♥ ✭❜ë ❜❛ P②t❤❛❣♦r❡✮✮ ♥❤÷ tr➻♥❤ ❜➔②
ð tr➯♥ ❦❤ỉ♥❣❄ ❈➙✉ tr↔ ❧í✐ ❧➔ ❦❤ỉ♥❣✳ ◆➳✉ t❛ ❧➜② ♠ët t❛♠ ❣✐→❝ ❍❡r♦♥ ✈ỵ✐
✤ë ❞➔✐ ❜❛ ❝↕♥❤ ❧➔ ✵✱ ✺❀ ✵✱ ✺ ✈➔ ✵✱ ✻ t❤➻ rã r➔♥❣ ♥â ❦❤æ♥❣ t❤➸ ữủ tứ
t ợ ở ✤➲✉ tü ♥❤✐➯♥✳ ❍♦➦❝ ♠ët ✈➼ ❞ư ❦❤→❝ t÷í♥❣
♠✐♥❤ ❤ì♥✱ ❧➔ ❧➜② ♠ët t❛♠ ❣✐→❝ ✈ỵ✐ ✤ë ❞➔✐ ❝→❝ ❝↕♥❤ ✺✱ ✷✾✱ ✸✵ ✈ỵ✐ ❞✐➺♥ t➼❝❤
✼✷✱ t❤➻ s➩ ❦❤ỉ♥❣ ❝â ✤÷í♥❣ ❝❛♦ ♥➔♦ ❝õ❛ ♥â ❧➔ ♠ët sè tü ♥❤✐➯♥✳
❈â t❤➸ ❝❤✐❛ ♠ët t❛♠ ❣✐→❝ ❍❡r♦♥ t❤➔♥❤ ❤❛✐ t❛♠ ❣✐→❝
✈✉æ♥❣ ♠➔ ✤ë ❞➔✐ ❝→❝ ❝↕♥❤ ❝õ❛ ❝❤ó♥❣ t↕♦ t❤➔♥❤ ♥❤ú♥❣ ❜ë ❜❛ P②t❤❛❣♦r❡ ❤ú✉
❚➼♥❤ ❝❤➜t ✶✳✶✳✷✳


✾

t➾ ✭✸ ❝↕♥❤ ❧➔ ❝→❝ sè ❤ú✉ t✛ t❤ä❛ ♠➣♥ ữỡ tr Ptr

t ợ c, e, b + d ✈➔ ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝ S ❧➔ ♥❤ú♥❣
sè ❤ú✉ t ú t õ t ồ ỵ s ❝❤♦ ✤ë ❞➔✐ ❝↕♥❤ b + d
❧➔ ❧ỵ♥ ♥❤➜t✱ ❦❤✐ ✤â ✤÷í♥❣ ✈✉ỉ♥❣ ❣â❝ ❤↕ tø ✤➾♥❤ ✤è✐ ❞✐➺♥ ①✉è♥❣ ❝↕♥❤ ♥➔②
♥➡♠ ❜➯♥ tr♦♥❣ ❝↕♥❤✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❜ë ❜❛ (a, b, c) ✈➔ (a, d, e) ❧➔ ❝→❝
❜ë ❜❛ P②t❤❛❣♦r❡✱ t❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ a, b ✈➔ d ❧➔ ♥❤ú♥❣ sè ❤ú✉ t➾✳ ❚❤➟t
✈➟②✱ ✈➻ ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝ ❧➔✿ S = 21 (b + d)ad✳ ❘ót a t❛ ✤÷đ❝ a = b 2S
+d
❧➔ ♠ët sè ❤ú✉ t➾✱ ✈➻ S ✈➔ b + d ✤➲✉ ❧➔ ♥❤ú♥❣ sè ❤ú✉ t➾✳ P❤➛♥ ❝á♥ ❧↕✐ ❝➛♥
❝❤ù♥❣ ♠✐♥❤ b ✈➔ d ỳ t
ử ỵ Ptr ố ợ t ❣✐→❝ ✈✉æ♥❣✱ t❛ ❝â a2 +b2 = c2
✈➔ a2 + d2 = e2✳ ❚rø ✈➳ t❤❡♦ ✈➳ ❤❛✐ ✤➥♥❣ t❤ù❝
❈❤ù♥❣ ♠✐♥❤✳

b2 − d 2 = c2 − e 2

⇔(b − d)(b + d) = c2 − e2
c2 − e 2
⇔b − d =
b+d

❱➳ ♣❤↔✐ ❧➔ ❤ú✉ t➾✱ ❜ð✐ ✈➻ t❤❡♦ ❣✐↔ t❤✐➳t c, e ✈➔ b + d ❧➔ ♥❤ú♥❣ sè ❤ú✉ t➾✳
❉♦ ✤â✱ b − d ❧➔ ❤ú✉ t➾✳ ❚❛ ❧↕✐ ❝â (b + d) ❧➔ ❤ú✉ t➾ t❤❡♦ ❣✐↔ t❤✐➳t✱ s✉② r❛
(b + d) + (b − d) ❧➔ ❤ú✉ t➾✳ ❍❛② 2b ❧➔ ❤ú✉ t➾✳ ❙✉② r❛ b ❤ú✉ t➾ ✈➔ d ❝ô♥❣ ♣❤↔✐
❧➔ sè ❤ú✉ t➾✳
❇➔✐ t t t r tữỡ ữỡ ợ
t ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡✱ tr♦♥❣ ✤â✱ S, s ❧➔ ❞✐➺♥ t➼❝❤✱ ❝❤✉ ✈✐ t❛♠
❣✐→❝ ABC
❚➼♥❤ ❝❤➜t ✶✳✶✳✸✳

S 2 = s(s − a)(s − b)(s − c)

❈æ♥❣ t❤ù❝ tê♥❣ q✉→t ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ✤➣ ✤÷đ❝ ❝ỉ♥❣ ❜è ❜ð✐ ❇r❛❤✲
♠❛❣✉♣t❛ ✈➔ ❈❛r♠✐❝❤❛❡❧ ♥➠♠ ✶✾✺✷ ✭t❤❡♦ ❉✐❝❦s♦♥ ✷✵✵✺✱ ♣✳ ✶✾✸✮✱ ✤â ❧➔
a = n m2 + k 2
✭✶✳✶✮
b = m n2 + k 2
✭✶✳✷✮
✭✶✳✸✮
c = (m + n) m.n − k 2
s = m.n(m + n)
✭✶✳✹✮

✶✵

✭✶✳✺✮
✣➙② ❧➔ ♠ët ❦✐➸✉ tr♦♥❣ ❧ỵ♣ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ✈ỵ✐ ♠å✐ m, n, k ∈ N s❛♦ ❝❤♦
S = kmn(m + n) mn − k 2

m2 · n
(m, n, k) = 1, m.n > k ≥
(2m + n)
2

✈➔ m ≤ n ≤ 1.

❚❤❡♦ ✤â t❛ ❝â t❤➸ ❧✐➺t ❦➯ ♠ët sè t❛♠ ❣✐→❝ ❍❡r♦♥ s➢♣ ①➳♣ t❤❡♦ sü t➠♥❣
❝õ❛ ❝↕♥❤ ❧ỵ♥ ♥❤➜t tr♦♥❣ t❛♠ ❣✐→❝✿
(3, 4, 5), (5, 5, 6), (5, 5, 8),
(6, 8, 10), (10, 10, 12), (5, 12, 13),
(9, 12, 15), (4, 13, 15), (13, 14, 15),
(10, 13, 13), (10, 10, 16), ...

❝â ❞✐➺♥ t➼❝❤ ❧➛♥ ❧÷đt 6, 12, 12, 24, 48, 30, 60, 54, ....
❚r➼❝❤ ❞❛♥❤ s→❝❤ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❝ì ❜↔♥ ①➳♣ t❤❡♦ ❞✐➺♥
t➼❝❤ t➠♥❣ ❞➛♥✱ ♥➳✉ ❝ò♥❣ ❞✐➺♥ t➼❝❤ t❤➻ ①➳♣ t❤❡♦ ❝❤✉ ✈✐ t➠♥❣ ❞➛♥✳
❱➼ ❞ö ✶✳✶✳✶✳

◆➠♠ ✶✾✾✹✱ tr♦♥❣ ▼❛t❤♠❛t✐❝❛❧ ◆♦t❡s✱ ❱♦❧✳ ✺✺✱ ◆✵ ✷✱ ❙✳ ❙❤✳ ❑♦③❤❡❣❡❧✬❞✐♥♦✈
✭◆❣❛✮ ✤➣ ❝æ♥❣ ❜è ❦➳t q✉↔ ❜➔✐ t♦→♥ t➻♠ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❝ì ❜↔♥ ✈ỵ✐ ✻
❦✐➸✉ ❜✐➸✉ ❞✐➵♥ ❦❤→❝ ♥❤❛✉ ✈➔ ✈✐➺❝ t➻♠ t❛♠ ❣✐→❝ ❍❡r♦♥ ❝ì ❜↔♥ ✤÷đ❝ ❝♦✐ ❧➔
❤♦➔♥ t❤➔♥❤✳ ❙❛✉ ✤➙② t❛ ①➨t ♠ët sè ❜➔✐ t♦→♥ t➻♠ t❛♠ ❣✐→❝ ❍❡r♦♥ ❦➧♠ t❤❡♦
♠ët sè ✤✐➲✉ ❦✐➺♥ ✤➦❝ ❜✐➺t✳

✶✳✷

❇➔✐ t♦→♥

HG✿

❚❛♠ ❣✐→❝ ❍❡r♦♥ ✈ỵ✐

r, ra, rb, rc ∈ N

❚❛♠ ❣✐→❝ ❍❡r♦♥ HT [a, b, c, S]✱ ♥❣♦➔✐ t➯♥ ❣å✐ t❛♠ ❣✐→❝ ❍❡r♦♥ ❝ì ❜↔♥✱ ❝á♥
✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ♥➳✉ ✸ ✤÷í♥❣ ❝❛♦ ha, hb, hc ∈/ N✱ tr÷í♥❣
❤đ♣ tr→✐ ❧↕✐ t❛♠ ❣✐→❝ ✤÷đ❝ ❣å✐ ❧➔ ♣❤➙♥ t➼❝❤ ✤÷đ❝✱ tù❝ ❧➔ ➼t ♥❤➜t ✶ ✤÷í♥❣
❝❛♦ ❧➔ sè tü ♥❤✐➯♥✳ ❚✐➳♣ t❤❡♦ t ỵ t ữớ trỏ ở t
t✐➳♣ ❧➛♥ ❧÷đt ❧➔ I, Ia, Ib, Ic✳ Ð ✤➙② t❛ ①➨t ♠ët ❜➔✐ t♦→♥ t➻♠ t❛♠ ❣✐→❝
❍❡r♦♥ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❝❤➦t ❤ì♥ ✈➲ ❜→♥ ❦➼♥❤ r, ra, rb, rc✿
❇➔✐ t♦→♥

HG✿ ❚➻♠ t❛♠ ❣✐→❝ ❍❡r♦♥ s❛♦ ❝❤♦ r, ra , rb , rc ∈ N✱ tr♦♥❣ ✤â


✶✶

❉✐➺♥ t➼❝❤ ❚● ❈❤✉ ✈✐ ❚● ✣ë ❞➔✐ b + d ✣ë ❞➔✐ e ✣ë ❞➔✐ c
✻
✶✷
✺
✹
✸
✶✷

✶✻
✻
✺
✺
✶✷
✶✽
✽
✺
✺
✷✹
✸✷
✶✺
✶✸
✹
✸✵
✸✵
✶✸
✶✷
✺
✸✻
✸✻
✶✼
✶✵
✾
✸✻
✺✹
✷✻
✷✺
✸
✹✷

✹✷
✷✵
✶✺
✼
✻✵
✸✻
✶✸
✶✸
✶✵
✻✵
✹✵
✶✼
✶✺
✽
✻✵
✺✵
✷✹
✶✸
✶✸
✻✵
✻✵
✷✾
✷✺
✻
✻✻
✹✹
✷✵
✶✸
✶✶
✼✷

✻✹
✸✵
✷✾
✺
✽✹
✹✷
✶✺
✶✹
✶✸
✽✹
✹✽
✷✶
✶✼
✶✵
✽✹
✺✻
✷✺
✷✹
✼
✽✹
✼✷
✸✺
✷✾
✽
✾✵
✺✹
✷✺
✶✼
✶✷
✾✵

✶✵✽
✺✸
✺✶
✹
✶✶✹
✼✻
✸✼
✷✵
✶✾
✶✷✵
✺✵
✶✼
✶✼
✶✻
✶✷✵
✻✹
✸✵
✶✼
✶✼
✶✷✵
✽✵
✸✾
✷✺
✶✻
✶✷✻
✺✹
✷✶
✷✵
✶✸
✶✷✻

✽✹
✹✶
✷✽
✶✺
✶✷✻
✶✵✽
✺✷
✺✶
✺
✶✸✷
✻✻
✸✵
✷✺
✶✶
❇↔♥❣ ✶✳✶✿ ❚r➼❝❤ ❞❛♥❤ s→❝❤ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❝ì ❜↔♥


✶✷

❍➻♥❤ ✶✳✸✿ ❚❛♠ ❣✐→❝ ❍❡r♦♥ t❤❡♦ sü t➠♥❣ ❞➛♥ ❝õ❛ ợ t

r, ra , rb , rc ữủt ❧➔ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ✈➔ ❝→❝ ✤÷í♥❣ trá♥ ❜➔♥❣
t✐➳♣ ❝õ❛ t❛♠ ❣✐→❝✳

●✐↔✐ ✤➛② ✤õ ✈➔ ✤÷❛ r❛ t❤✉➟t t♦→♥ t➻♠ ❤➳t ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❍●
❧➔ ❝ỉ♥❣ ✈✐➺❝ ❦❤ỉ♥❣ ✤ì♥ ❣✐↔♥✳ ❈❤ó♥❣ tỉ✐ ❝❤➾ ❞ø♥❣ ❧↕✐ ð ✈✐➺❝ ✤÷❛ r❛ ❦➳t ❧✉➟♥
t÷í♥❣ ♠✐♥❤ tr♦♥❣ ♠ët sè trữớ ủ ử t
ã rữớ ủ t P②t❤❛❣♦r❡✳
●✐↔ sû ABC ❧➔ t❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ❝ì ❜↔♥ ✈ỵ✐ a2 + b2 = c2 ✳ ❚❛ t❤➜②
tr♦♥❣ ❤❛✐ sè a, b ♣❤↔✐ ❝â ♠ët sè ❧➫✱ c ❝ô♥❣ ❝➛♥ ♣❤↔✐ ❧➫✳ ❉♦ ✤â✱ ♥û❛ ❝❤✉

1
1
✈✐ P = (a + b + c) ∈ N ✈➔ ❞✐➺♥ t➼❝❤ S = ab ∈ N✳ ❈→❝ ❝↕♥❤ t❛♠ ❣✐→❝
2
2
P②t❤❛❣♦r❡ ❝ì ❜↔♥ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝ ❞↕♥❣ m2 − n2 ✈➔ 2mn ❧➔ ❤❛✐ ❝↕♥❤
❣â❝ ✈✉æ♥❣✱ m2 + n2 ❧➔ ❝↕♥❤ ❤✉②➲♥ ✈ỵ✐ m, n ∈ N✳ ●å✐ r, ra , rb , rc ❧➔ ❜→♥
❦➼♥❤ ❝→❝ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ✈➔ ❜➔♥❣ t✐➳♣ ✤è✐ ❞✐➺♥ ❝→❝ ❣â❝ A, B, C ✱ t÷ì♥❣

ù♥❣✳ ❚r♦♥❣ ♠é✐ ❜ë ❜❛ P②t❤❛❣♦r❡ ❝ì ❜↔♥ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ✈➔ ✸
❜→♥ ❦➼♥❤ ❝õ❛ ❜❛ ✤÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣ ❧➔ sè tü ♥❤✐➯♥✱ ❤➻♥❤ ✶✳✹✳ ◆❣÷đ❝ ❧↕✐
♥➳✉ t❛♠ ❣✐→❝ ✈✉æ♥❣ ABC ❝â ❜➜t ❦ý ✸ tr♦♥❣ ✹ sè r, ra , rb , rc ❧➔ sè tü ♥❤✐➯♥
t❤➻ ❞➵ t❤➜② ❜❛ sè ❛✱❜✱❝ ❧➔ sè tü ♥❤✐➯♥ ✈➻
a = r + ra = rc − rb ∈ N
b = r + rb = rc − ra ∈ N


✶✸

❍➻♥❤ ✶✳✹✿ ❚❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ❝ì ❜↔♥ ✈➔ ❝→❝ ❜→♥ ❦➼♥❤

r, ra , rb , rc

c = ra + rb = r c − r ∈ N

♥➯♥ ABC ❧➔ t❛♠ ❣✐→❝ P②t❤❛❣♦r❡✳ ❚ø ✤â t❛ ❝â✿

▼➺♥❤ ✤➲ ✶✳✶✳ ▼å✐ t❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ❝ì ❜↔♥ ✤➲✉ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥
❍●✳

• rữớ ủ t
ợ t ❝❤➥♥❣ ❤↕♥✱ (a, b, c) = (5, 5, 6) t❤➻ (S, r, ra , rb , rc ) =
3
10
15
12, , 4, 4, 6 ; (a, b, c) = (13, 13, 10) t❤➻ (S, r, ra , rb , rc ) = 60, , 12, 12,
✳
2
3
2

❚❛ ❝â ❦➳t q✉↔ tê♥❣ q✉→t s❛✉✿

▼➺♥❤ ✤➲ ✶✳✷✳ ❆❇❈ ❧➔ t❛♠ ❣✐→❝ ❍❡r♦♥ ❝ì ❜↔♥ ✈ỵ✐ a = b t❤➻ ra = rb ∈ N

❝á♥ r rc ổ ỗ tớ
c
c
c
ự õ P = a + , P − a = s − b = , P − c = a − ♥➯♥
2
2
2
r√ 2
2
2
2
S =
4a − c2 ✱ ❦➨♦ t❤❡♦ 4a − c = m ✈ỵ✐ m ∈ N ♥➔♦ ✤â✳ ❉♦ ✤â✱
4

−c2 ≡ m2 (mod4)✱ ♥❤÷ ✈➟② c = 2d, m = 2n ✈ỵ✐ d, n ∈ N ♥➔♦ ✤â t❤ä❛


✶✹

(d, n) = 1✳ ❚ø ✤â s✉② r❛✿ S = dn ✈➔ ra = rb =
r=

S
dn
=
;
s
a+d

rc =

S
= n✳ ❚÷ì♥❣ tü✱
P a

dn
ỗ tớ õ r, rc N t
ad

2d2
2d2 n
=
N
rc − r = 2

a − d2
n

tù❝ ❧➔ n ∈ {1, 2}✳ ❱➻ n2 = (a + d)(a − d) ♥➯♥ ❦❤æ♥❣ ❝â n = 1 ❤♦➦❝ n = 2
✤➸ d ∈ N✳

❚❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝✿ ▼å✐ t❛♠ ❣✐→❝ ❍❡r♦♥ ỡ ổ
ừ t
ãrữớ ủ ♥❣❤✐➺♠ ❧➔ t❛♠ ❣✐→❝ ❝â ❝↕♥❤ ❧➟♣ t❤➔♥❤ ❝➜♣ sè ở rữớ
ủ t r ợ t ❝➜♣ sè ❝ë♥❣ t❤÷í♥❣ ✤÷đ❝ ①➨t
✈➻ ❦➳t q✉↔ ❝ơ♥❣ t❤✉ ữủ ởt ợ t r t

✶✳✸✳ ❑❤✐ ✸ ❝↕♥❤ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧➔ ❝➜♣ sè ❝ë♥❣✱ ❜➔✐ t♦→♥ ❍● ❝â
❞✉② ♥❤➜t ♥❣❤✐➺♠

(a, b, c) = (3, 4, 5)✳

3b
b
b
b
, s − a = + d, s − b = , s − c = − d ♥➯♥
2
2
2
2
3 (b2 − 4d2 )✱ ❦➨♦ t❤❡♦ b2 − 4d2 = 3m2 ✈ỵ✐ m ∈ N ♥➔♦ ✤â✳ ❉♦

❈❤ù♥❣ ♠✐♥❤✳

❚❛ ❝â s =

b
4
2
✤â✱ b ≡ 3m2 (mod4)✱ ♥❤÷ ✈➟② b = 2e, m = 2n ✈ỵ✐ e, n ∈ N ♥➔♦ ✤â t❤ä❛
S
S
= 3n ✈➔ rb =
= 3n
(e, n) = 1✳ ❚ø ✤â s✉② r❛✿ S = 3en ✈➔ rb =
s−b
s−b
S
3en
S
3en
❚÷ì♥❣ tü✱ ra =
=
; rc =
=
s−a c+d
s−c c−d
2
2
●✐↔ sû ra , rc ∈ N✳ ❑❤✐ ✤â ❞♦ c − d = 3n2 ♥➯♥
S =

ra + rc =

6e2 n
2e2
=
∈N
c2 − d 2
n

❧➟♣ tù❝ s✉② r❛ n ∈ {1, 2}✳ ◆➳✉ n = 1 t❤➻ 3 = 3n2 = (e + d)(e − d) ♥➯♥
e = 2, d = 1✱ tù❝ ❧➔ (a, b, c) = (3, 4, 5)✳ ◆➳✉ n = 2 t❤➻ 12 = (e + d)(e − d)✳
❱➻ (e, d) = 1 ♥➯♥ ❦❤æ♥❣ t❤➸ ❝â (e + d, e − d) = (6, 2)✳ ❈❤➾ ❝á♥ ❧↕✐ ❦❤↔ ♥➠♥❣
(e + d, e − d) = (12, 1) ❤♦➦❝ ❜➡♥❣ ✭✹✱ ✸✮✱ ❦❤ỉ♥❣ t❤➸ ❝❤♦ d ∈ N✳
•

❍❛✐ ❤å ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❍●

❱ỵ✐ t❛♠ ❣✐→❝ ❍❡r♦♥ ❝ì ❜↔♥ ✭❦❤æ♥❣ ❧➔ t❛♠ ❣✐→❝ P②t❤❛❣♦r❡✮ ❝â t❤➸ ❝â t➜t
❝↔ r, ra , rb , rc ∈ N✳ ❈❤➥♥❣ ❤↕♥ (a, b, c) = (7, 15, 20) t❤➻ (S, r, ra , rb, rc ) =




(42, 2, 3, 7, 42) ú ỵ r ợ t ❣✐→❝ ♥➔②✱ ha =

2S
= 12 ♥➯♥ ♥â ♣❤➙♥
a

t➼❝❤ ✤÷đ❝✳ ❚❛ s➩ ❝❤➾ r❛ ❜➔✐ t♦→♥ ❍● ❝â ✈æ sè ♥❣❤✐➺♠ ♣❤➙♥ t➼❝❤ ✤÷đ❝✿

▼➺♥❤ ✤➲ ✶✳✹✳ ✭ ①❡♠ ❬✶❪✱ ❬✸❪✮

❈â ♠ët ❤å ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ✭❦❤ỉ♥❣ ❧➔
t❛♠ ❣✐→❝ P②t❤❛❣♦r❡✮ ❝ì ❜↔♥ ✈➔ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❍●✳
❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ n > 1✱ ❧➜②

a = 4n2
b = 4n3 − 2n2 + 1 = (2n + 1) 2n2 − 2n + 1 .
c = 4n3 + 2n2 − 1 = (2n − 1) 2n2 + 2n + 1

❱➻ b ❧➫ ✈➔ a + b − c = 2 ♥➯♥ t❛♠ ❣✐→❝ ❧➔ ❝ì ❜↔♥ ✈ỵ✐ ♠å✐ n > 1✳ ❈ô♥❣
✈➟②✱ tø c + 2 = a + b t❛ ❝â
c2 − a2 − b2 = 2(ab − 2c − 2) = 2(ab − 2a − 2b + 2) ≥ 0

❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ n = 1✳ ❉♦ ✤â ✈ỵ✐ ♠å✐ n > 1 ❝→❝ t❛♠
❣✐→❝ ❧➔ ♥❤å♥ ♥❤÷ ✈➟② ❦❤ỉ♥❣ ❧➔ t❛♠ ❣✐→❝ P②t❤❛❣♦r❡✳ ❱ỵ✐ ❣✐↔ t❤✐➳t ✤â✱
P = 4n3 + 2n2 = 2n2 (2n + 1)
P − a = 4n3 − 2n2 = 2n2 (2n − 1)

P − b = 4n2 − 1 = (2n − 1)(2n + 1)

P − c = 4n3 + 2n2 − 4n3 + 2n2 − 1 = 1
S = 2n2 (2n − 1)(2n + 1)
S
r = = 2n − 1
P
S
ra =
= 2n + 1
P −b
S

rb =
= 2n + 1
P −b
S
= 2n2 (2n − 1)(2n + 1) = S
rc =
P −c

▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤

▼➺♥❤ ✤➲ ✶✳✺✳ ❈â ♠ët ❤å ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ✭❦❤ỉ♥❣ ❧➔ t❛♠ ❣✐→❝ P②t❤❛❣♦r❡✮
❝ì ❜↔♥ ✈➔ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❍●✳


✶✻

❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ n > 1✱ ❣✐↔ sû

a = 25n2 + 5n − 5 = 5 5n2 + n − 1
b = 25n3 + 20n2 − 7n + 3 = (5n + 3) 5n2 − 4n + 1
c = 25n3 + 20n2 − 2n − 4 = (5n − 2) 5n2 + 6n + 2

❱➻ a ❧➫ ✈➔ a + b − c = 2 ♥➯♥ t❛♠ ❣✐→❝ ❧➔ ❝ì ❜↔♥ ✈ỵ✐ ♠å✐ n > 1✳ ❈ơ♥❣
✈➟②✱ tø c + 2 = a + b t❛ ❝â c2 − a2 − b2 > 0✳ ❍ì♥ ♥ú❛✳
P = 25n3 + 20n2 − 2n − 3 = (5n + 3) 5n2 + n − 1

P − a = 25n3 − 5n2 − 7n + 2 = (5n − 2) 5n2 + n − 1

❉➵ ❦✐➸♠ tr❛ ✤÷đ❝✿
ha =

2S
2(5n − 2)(5n + 3)
=
∈
/N
a
5

2(5n − 2) 5n2 + n − 1
2
2S
=
=
10n
+
6
−
∈
/N
hb =
b
5n2 − 4n + 1
5n2 − 4n + 1
2(5n + 3) 5n2 + n − 1
2S
2
hc =
=
=

10n
−
4
+
∈
/N
c
5n2 + 6n + 2
5n2 + 6n + 2

♥➯♥ t❛♠ ❣✐→❝ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝✳
❚r÷í♥❣ ❤đ♣ ✤➛✉ t✐➯♥✱ n = 2 t❤➻ (a, b, c) = (105, 169, 172)✳ ❙✉② r❛✿
S = 2184, ha = 15, r = 8, ra = 13, rb = 21, rc = 2184

❑➳t ❧✉➟♥✿ ❚❛ t❤✉ ✤÷đ❝ ❤❛✐ ❤å ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❍●✿ ❝â ✈ỉ sè t❛♠
❣✐→❝ ❍❡r♦♥ ❝ì ❜↔♥ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ✈➔ ✈ỉ sè t❛♠ ❣✐→❝ ❍❡r♦♥ ❝ì ❜↔♥ ❦❤ỉ♥❣
♣❤➙♥ t➼❝❤ ✤÷đ❝ t❤ä❛ ♠➣♥ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❝ị♥❣ ✸ ❜→♥ ❦➼♥❤
✤÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣ ❧➔ ❝→❝ sè tü ♥❤✐➯♥✳
▼ët sè ❜➔✐ t♦→♥ ♠ð✿
❇➔✐ t♦→♥ ❍● ❝â t❤➸ ❝â ✈æ sè ♥❣❤✐➺♠ ❧➔ ❝→❝ t❛♠ ❣✐→❝ ♥❤å♥
❤❛② ❦❤æ♥❣❄ ❈❤ù♥❣ ♠✐♥❤✳
❇➔✐ t♦→♥ ✶✳✶✳

❇➔✐ t♦→♥ ✶✳✷✳

❚❤✉➟t t♦→♥ ①→❝ ✤à♥❤ tt ừ t

ã ữợ ❝→❝ t❛♠ ❣✐→❝ ❍●✳
●å✐ I, Ia , Ib , Ic ❧➛♥ ❧÷đt ❧➔ t➙♠ ❝→❝ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣✱ ✤÷í♥❣ trá♥ ❜➔♥❣

t✐➳♣ tr♦♥❣ ❝→❝ ❣â❝ ❆✱❇✱❈✳ ❉ü❛ ✈➔♦ ❦➳t q✉↔ ✭✶✳✹✮ ✈➔ ✭✶✳✺✮ t❛ t❤û ①→❝ ✤à♥❤
tå❛ ✤ë ❝→❝ t➙♠ ❝õ❛ ✹ ✤÷í♥❣ trá♥✳


✶✼

❛✮ ✣è✐ ✈ỵ✐ ❤å ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ♣❤➙♥ t➼❝❤ ✤÷đ❝ t❤❡♦ ❦➳t q✉↔ ✭✶✳✹✮✿
a = 4n2 , b = (2n + 1) 2n2 − 2n + 1 , c = (2n − 1) 2n2 + 2n + 1 .

❚❛ ❝â t❤➸ ❝❤å♥ tå❛ ✤ë

C = (0, 0), A = (−2n(n − 1)(2n + 1), (2n − 1)(2n + 1); B = 4n2 , 0 ).

❑❤✐ ✤â tå❛ ✤ë ❝→❝ t➙♠ ✤÷í♥❣ trá♥✿

I = (s − c, r) = (1, 2n − 1)

Ia = (s − b, ra ) = ((2n − 1)(2n + 1), 2n + 1)
Ib = (a − s, rb ) = −2n2 (2n − 1), 2n2

Ic = (s, r) = 2n2 (2n + 1), 2n2 (2n − 1)(2n + 1)

❘ã r➔♥❣ ✹ t➙♠ ❝→❝ ✤÷í♥❣ trá♥ ✤➲✉ ❧➔ ❝→❝ ✤✐➸♠ ♥❣✉②➯♥✳ ●✐→ trà ✤➛✉
n = 2 ❝❤♦ t❛ t❛♠ ❣✐→❝ ❆❇❈ ✈➔ ❝→❝ t➙♠ ✤÷í♥❣ trá♥✿
A = (−20, 15), B(16, 0), C(0, 0)
I = (1, 3), Ia = (15, 5), Ib = (−24, 8), Ic = (40, 120)

❜✮ ❍å ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❦❤æ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝ t❤❡♦ ✭✶✳✺✮✿
a = 5 5n2 + n − 1
b = (5n + 3) 5n2 − 4n + 1

c = (5n − 2) 5n2 + 6n + 2

❚❛ ❝â t❤➸ ❝❤å♥ ❣è❝ tå❛ ✤ë✿ C = (0, 0) ✈➔

B = (−4(5n2 + n − 1), −3(5n2 + n − 1) = (−4rb , −3rb )
A = (2n(2n − 1)(5n + 3), (n − 1)(3n − 1)(5n + 3)
= (2n(2n − 1)ra , (n − 1)(3n − 1)ra

❑❤✐ ✤â t➼♥❤ ✤÷đ❝ tå❛ ✤ë ❝→❝ t➙♠ ✤÷í♥❣ trá♥✿
aA + bB + cC
= (3n − 1, 4n + 1)
a+b+c
−aA + bB + cC
Ia =
= (−4n + 1)ra , (−3n + 2)ra
−a + b + c
aA − bB + cC
= ((4n − 1)rb , (3n − 2)rb )
Ib =
a−b+c
aA + bB − cC
Ic =
= ((3n − 2)ra rb , (−4n + 1)ra rb )
a+b−c
I=