Open As frações contínuas e os números metálicos

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚

❆s ❋r❛çõ❡s ❈♦♥tí♥✉❛s ❡ ♦s

◆ú♠❡r♦s ▼❡tá❧✐❝♦s

♣♦r

❏♦sé ❏ú♥✐♦r ❱❡❧♦s♦ ❞❡ ❆r❛ú❥♦

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚

❆s ❋r❛çõ❡s ❈♦♥tí♥✉❛s ❡ ♦s

◆ú♠❡r♦s ▼❡tá❧✐❝♦s

♣♦r

❏♦sé ❏ú♥✐♦r ❱❡❧♦s♦ ❞❡ ❆r❛ú❥♦

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡✲ s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚ ✲ ❉▼ ✲ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜✲ t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠á✲ t✐❝❛✳

❆❣♦st♦✴✷✵✶✺ ❏♦ã♦ P❡ss♦❛ ✲ P❇

†_{❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡}

A663f Araújo, José Júnior Veloso de.

As frações contínuas e os números metálicos / José Júnior Veloso de Araújo.- João Pessoa, 2015.

50f. : il.

Orientador: Napoleón Caro Tuesta Dissertação (Mestrado) - UFPB/CCEN

1. Matemática. 2. Fração contínua. 3. Número metálico. 4. Número de ouro.

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ ❉❡✉s✱ ♣♦r s❡ ❢❛③❡r ♣r❡s❡♥t❡ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❛ ♠✐♥❤❛ ✈✐❞❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♥♦s ♠❛✐s ❞✐❢í❝❡✐s ❡ ❛tr✐❜✉❧❛❞♦s✳

➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ à ♠✐♥❤❛ ❡s♣♦s❛✱ ♣❡❧♦ ❛♣♦✐♦ r❡❝❡❜✐❞♦ ❡♠ t✉❞♦ q✉❡ ♣r❡❝✐s❡✐✳

❆♦s ♠❡✉s ❛♠✐❣♦s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♠❡✉s ❛❧✉♥♦s✱ q✉❡ s❡♠♣r❡ s♦✉❜❡r❛♠ ♠❡ ❛♣♦✐❛r✱ ♠♦str❛♥❞♦✲♠❡ q✉❡ é ♥❡❝❡ssár✐♦ ❢❛③❡r ❛❧❣✉♥s s❛❝r✐❢í❝✐♦s ❡ r❡♥ú♥❝✐❛s ♣❛r❛ ❝♦♥s❛❣r❛r ♠❛✐s ✉♠❛ ✈✐tór✐❛✳

❆♦s ♣r♦❢❡ss♦r❡s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛✱ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ q✉❡ ❞❡s❡♠♣❡♥❤❛r❛♠ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ♥❛ r❡❛❧✐③❛çã♦ ❞❡st❡ ♦❜❥❡t✐✈♦✱ ❡st✐♠✉❧❛♥❞♦✲♠❡ ♣❛r❛ q✉❡ ❡✉ ❜✉s❝❛ss❡ ❝❛❞❛ ✈❡③ ♠❛✐s ❞❡❞✐❝❛r✲♠❡ ❛♦ ❝✉rs♦✳

❆ t♦❞❛s ❛s ♣❡ss♦❛s q✉❡ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ s♦♥❤♦✳

❊✱ ✜♥❛❧♠❡♥t❡✱ ❛♦ P❘❖❋▼❆❚ ❡ à ❈❆P❊❙✱ ♣♦r ♣r♦♣♦r❝✐♦♥❛r ❡ss❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ q✉❛❧✐✜❝❛çã♦ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ❇r❛s✐❧✳

▼✉✐t♦ ♦❜r✐❣❛❞♦✳

❉❡❞✐❝❛tór✐❛

➚q✉❡❧❡ q✉❡ s❡♠♣r❡ ❣✉✐♦✉ ♠✐♥❤❛ ✈✐❞❛✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♠✐♥❤❛ tr❛❥❡tór✐❛ ❛❝❛✲ ❞ê♠✐❝❛✳ ◗✉❡ ♠❡ ❞❡✉ s❛❜❡❞♦r✐❛ ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦s ❝♦♥❤❡❝✐♠❡♥t♦s q✉❡ ❛❞q✉✐r✐✳ ◗✉❡ s❡♠♣r❡ ❛❣✐✉ ♣♦r ♠✐♠ ❡ ❡♠ ♠✐♠ ♥♦s ♠♦♠❡♥t♦s ❞❡❝✐s✐✈♦s✳ ❊♠ q✉❡♠ ♠❡ r❡❢✉❣✐❡✐ ♥♦s ♠♦♠❡♥t♦s ❞❡ ♣r♦✈❛çã♦ ♣❡❧♦s q✉❛✐s ♣❛ss❡✐ ❡ q✉❡ ♥ã♦ ♠❡ ❞❡✐①♦✉ ❛❜❛❧❛r✳ ◗✉❡ ❞❡s✈✐♦✉ ❞❡ ♠✐♠ ♦s ♦❧❤♦s ❞♦s ♠❡✉ ❛❞✈❡rsár✐♦s✳ ◗✉❡ ❡st❡♥❞❡✉ s✉❛s ♠ã♦s s♦❜r❡ ♠✐♠ ♥♦s ♠♦♠❡♥t♦s q✉❡ ♦❢❡r❡❝✐❛♠ ♣❡r✐❣♦✳ ◗✉❡ ❢♦✐ ♠✐s❡r✐❝♦r❞✐♦s♦ ❝♦♠✐❣♦ ❡ ♠❡ ❛❝♦❧❤❡✉ ❛té ♠❡s♠♦ q✉❛♥❞♦ ♥ã♦ ❢✉✐ ✜❡❧ ❛ ❡❧❡✳ ◗✉❡ ❢♦✐ ❝❛♣❛③ ❞❡ ♦❢❡r❡❝❡r ❛ ✈✐❞❛ ❞♦ s❡✉ ♣ró♣r✐♦ ✜❧❤♦ ❡♠ ❡①♣✐❛çã♦ ❞♦s ♠❡✉s ♣❡❝❛❞♦s✳ ❆ ❉❊❯❙

❘❡s✉♠♦

❆ ❢❛♠í❧✐❛ ❞♦s ♥ú♠❡r♦s ♠❡tá❧✐❝♦s ❢♦✐ ✐♥tr♦❞✉③✐❞❛ ♣❡❧❛ ♠❛t❡♠át✐❝❛ ❛r❣❡♥t✐♥❛ ❱❡r❛ ❞❡ ❙♣✐♥❛❞❡❧✱ ❡♠1994✳ ❖s ◆ú♠❡r♦s ▼❡tá❧✐❝♦s sã♦ ♣♦✉❝♦ ❝♦♥❤❡❝✐❞♦s✱ ❝♦♠ ❡①❝❡çã♦ ❞♦ ◆ú♠❡r♦ ❞❡ ❖✉r♦✳ P♦ré♠✱ ♦✉tr♦s ♥ú♠❡r♦s ♠❡tá❧✐❝♦s t❛♠❜é♠ ♣♦ss✉❡♠ ♣r♦♣r✐❡❞❛❞❡s ❡ ❛♣❧✐❝❛çõ❡s ✐♠♣♦rt❛♥t❡s✳ ❆s ❋r❛çõ❡s ❈♦♥tí♥✉❛s ♣♦ss✐❜✐❧✐t❛♠ ✉♠❛ ♦✉tr❛ ♠❛♥❡✐r❛ ❞❡ r❡♣r❡s❡♥t❛r ❡ss❡s ♥ú♠❡r♦s✱ q✉❡ sã♦ ✐rr❛❝✐♦♥❛✐s✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❋r❛çã♦ ❈♦♥tí♥✉❛✳ ◆ú♠❡r♦ ▼❡tá❧✐❝♦✳ ◆ú♠❡r♦ ❞❡ ❖✉r♦✳

❆❜str❛❝t

❚❤❡ ❢❛♠✐❧② ♦❢ ♠❡t❛❧❧✐❝ ♠❡❛♥s ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② t❤❡ ❆r❣❡♥t✐♥❡ ♠❛t❤❡♠❛t✐❝s ❱❡r❛ ❙♣✐♥❛❞❡❧✱ ✐♥ ✶✾✾✹✳ ❚❤❡ ♠❡t❛❧❧✐❝ ♠❡❛♥s ❛r❡ ✉♥❦♥♦✇♥✱ ❡①❝❡♣t ❢♦r t❤❡ ●♦❧❞❡♥ ▼❡❛♥✳ ❍♦✇❡✈❡r✱ ♦t❤❡r ♠❡t❛❧❧✐❝ ♠❡❛♥s ❛❧s♦ ❤❛✈❡ ♣r♦♣❡rt✐❡s ❛♥❞ ✐♠♣♦rt❛♥t ❛♣♣❧✐❝❛t✐♦♥s✳ ❚❤❡ ❈♦♥t✐♥✉❡❞ ❋r❛❝t✐♦♥s ❡♥❛❜❧❡ ❛♥♦t❤❡r ✇❛② t♦ r❡♣r❡s❡♥t t❤❡s❡ ♥✉♠❜❡rs✱ ✇❤✐❝❤ ❛r❡ ✐r✲ r❛t✐♦♥❛❧✳

❑❡②✇♦r❞s✿ ❈♦♥t✐♥✉❡❞ ❋r❛❝t✐♦♥✳ ▼❡t❛❧❧✐❝ ▼❡❛♥✳ ●♦❧❞❡♥ ▼❡❛♥✳

❙✉♠ár✐♦

✶ ❋r❛çõ❡s ❈♦♥tí♥✉❛s ✶

✶✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ◆ú♠❡r♦s r❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✹ ◆ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✺ ❈♦♥✈❡r❣❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✻ ❋r❛çõ❡s ♣❡r✐ó❞✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✷ ◆ú♠❡r♦s ▼❡tá❧✐❝♦s ✷✶

✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸ ▼❡tá❧✐❝♦s ✐♥t❡✐r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✹ ▼❡tá❧✐❝♦s ❞♦ t✐♣♦ ✭σp,1✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✺ ❖ ◆ú♠❡r♦ ❞❡ ❖✉r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✻ ❖ ◆ú♠❡r♦ ❞❡ Pr❛t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✼ ❖ ◆ú♠❡r♦ ❞❡ ❇r♦♥③❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✸ ❖ ◆ú♠❡r♦ ❞❡ ❖✉r♦ ♥❛ ❍✐stór✐❛ ✸✶

✸✳✶ ❊❣✐t♦ ❆♥t✐❣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷ ❇❛❜✐❧ô♥✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✸ P✐t❛❣ór✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✹ ❘❡♥❛s❝✐♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

❆♣ê♥❞✐❝❡ ✸✽

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✸✾

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✷✳✶ ❚❡♦r❡♠❛ ✷✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✶ ❖ ❝â♥♦♥❡ ❞❡ ❑❤❡s✐✲❘❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷ P❛♣✐r♦ ❞❡ ❆❤♠❡s ♦✉ P❛♣✐r♦ ❞❡ ❘❤✐♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✸ ❍✐❡ró❣❧✐❢♦ ✭❧❡tr❛ ❤✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✹ ❚❛❜✉❧❡t❛ ❝✉♥❡✐❢♦r♠❡ ❜❛❜✐❧ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✺ ❙ó❧✐❞♦s P❧❛tô♥✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✻ P❡♥t❛❣r❛♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✼ ▲❡♦♥❛r❞♦ ❋✐❜♦♥❛❝❝✐ ✭✶✶✼✺✲✶✷✹✵✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✽ ▲❡♦♥❛r❞♦ ❞❛ ❱✐♥❝✐ ✭✶✹✺✷✲✶✺✶✾✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✾ ❖ ❍♦♠❡♠ ❱✐tr✉✈✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

◆♦t❛çõ❡s

◆♦t❛çõ❡s ●❡r❛✐s

• φ é ♦ ◆ú♠❡r♦ ❞❡ ❖✉r♦✳

• σp,q é ♦ ♥ú♠❡r♦ ♠❡tá❧✐❝♦ ❛ss♦❝✐❛❞♦ à ❡q✉❛çã♦x2−px−q= 0✳

■♥tr♦❞✉çã♦

❆♣❡s❛r ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s s❡r❡♠ ♦❜❥❡t♦ ❞❡ ♣❡sq✉✐s❛ ♥❛ ❛t✉❛❧✐❞❛❞❡✱ ♦s ◆ú♠❡r♦s ▼❡tá❧✐❝♦s sã♦ ♣♦✉❝♦ ❝♦♥❤❡❝✐❞♦s✱ ❝♦♠ ❡①❝❡çã♦ ❞❡ s❡✉ ♠❡♠❜r♦ ❢❛♠♦s♦✱ ♦ ◆ú♠❡r♦ ❞❡ ❖✉r♦✳ ❆❧é♠ ❞❡❧❡✱ t❡♠♦s ♦ ◆ú♠❡r♦ ❞❡ Pr❛t❛✱ ♦ ◆ú♠❡r♦ ❞❡ ❇r♦♥③❡✱ ❛❧é♠ ❞❡ ♦✉tr♦s q✉❡ tr❛t❛r❡♠♦s ♥❡st❡ tr❛❜❛❧❤♦✳

❆ ❢❛♠í❧✐❛ ❞♦s ♥ú♠❡r♦s ♠❡tá❧✐❝♦s✱ ✐♥tr♦❞✉③✐❞❛ ♣❡❧❛ ♠❛t❡♠át✐❝❛ ❛r❣❡♥t✐♥❛ ❱❡r❛ ❞❡ ❙♣✐♥❛❞❡❧✱ ❡♠ ✶✾✾✹✱ é ❢♦r♠❛❞❛ ♣❡❧❛s r❛í③❡s ♣♦s✐t✐✈❛s ❞❛s ❡q✉❛çõ❡s ❞❛ ❢♦r♠❛ x2₋

px+q= 0✱ ♦♥❞❡p ❡ q sã♦ ♥ú♠❡r♦s ♣♦s✐t✐✈♦s✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ❛s ❢r❛çõ❡s ❝♦♥tí♥✉❛s✱ t❡♠❛ ❛✐♥❞❛ ♠✉✐t♦ ♣❡sq✉✐s❛❞♦ ♥♦s ❞✐❛s ❞❡ ❤♦❥❡✱ ♣♦ss✐❜✐❧✐t❛♠ ✉♠❛ ♦✉tr❛ ♠❛♥❡✐r❛ ❞❡ r❡♣r❡s❡♥t❛r ♥ú♠❡r♦s r❡❛✐s✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✳ P♦r ❡①❡♠♣❧♦✱ ♦ ◆ú♠❡r♦ ❞❡ ❖✉r♦✱ q✉❡ é ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧✱ ❛ss✐♠ ❝♦♠♦ ❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞♦s ♥ú♠❡r♦s ♠❡tá❧✐❝♦s t❛♠❜é♠ ♦ sã♦✱ ❝✉❥♦ ✈❛❧♦r é 1+√5

2 ✱

♥❛ s✉❛ ❢♦r♠❛ ❞❡❝✐♠❛❧✱ ♣♦ss✉✐ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❝♦♠ ✐♥✜♥✐t❛s ❝❛s❛s ♥ã♦ ♣❡r✐ó❞✐❝❛s✱ ❝✉❥♦ ✈❛❧♦r ❛♣r♦①✐♠❛❞♦ é 1,618✳ ◆♦ ❡♥t❛♥t♦✱ ❛ s✉❛ ❡①♣❛♥sã♦ ❡♠ ❢r❛çã♦ ❝♦♥tí♥✉❛ é✱ s✐♠♣❧❡s♠❡♥t❡✱ [¯1]✳ P♦r ❡ss❡ ♠♦t✐✈♦✱ ♦♣t❛♠♦s ♣♦r ✉t✐❧✐③❛r ❛s ❢r❛çõ❡s ❝♦♥tí♥✉❛s ♥❡ss❡ ❡st✉❞♦✱ ♦♥❞❡ ♣♦❞❡♠♦s r❡❧❛❝✐♦♥❛r ❛s ❞✉❛s t❡♦r✐❛s✳

◆♦ ✶♦ _{❝❛♣ít✉❧♦✱ ❢❛r❡♠♦s ✉♠ r❡s✉♠♦ ❞❛ t❡♦r✐❛ ❞❛s ❋r❛çõ❡s ❈♦♥tí♥✉❛s ❛♣r❡s❡♥✲}

t❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ✶✺ ❞♦ ❧✐✈r♦ ❊❧❡♠❡♥t❛r② ◆✉♠❜❡r ❚❤❡♦r② ❬✷❪ ❡ ♥♦ ❧✐✈r♦ ❈♦♥t✐♥✉❡❞ ❋r❛❝t✐♦♥s ❬✼❪✳

◆♦ ✷♦ _{❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦s ◆ú♠❡r♦s ▼❡tá❧✐❝♦s✱ ✉t✐❧✐③❛♥❞♦ ❛s ✐❞❡✐❛s ❞❡}

❱❡r❛ ❙♣✐♥❛❞❡❧ ❬✽❪✱ ❡①♣❧♦r❛♥❞♦ ❡ ❛♣r♦❢✉♥❞❛♥❞♦ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳

◆♦ ✸♦ _{❝❛♣ít✉❧♦✱ tr❛r❡♠♦s ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❞♦ ◆ú♠❡r♦ ❞❡ ❖✉r♦✱ ❞❡s❞❡ ♦}

❊❣✐t♦ ❆♥t✐❣♦ ❛té ❛ é♣♦❝❛ ❞♦ ❘❡♥❛s❝✐♠❡♥t♦✳

❊s♣❡r❛♠♦s q✉❡ ❡ss❡ tr❛❜❛❧❤♦ t❡♥❤❛ r❡❧❡✈â♥❝✐❛ ❡❞✉❝❛❝✐♦♥❛❧✱ ❛✉①✐❧✐❛♥❞♦ ♣r♦❢❡ss♦✲ r❡s ❡ ❡st✉❞❛♥t❡s q✉❡ ❞❡s❡❥❡♠ ❝♦♥❤❡❝❡r ❡✴♦✉ s❡ ❛♣r♦❢✉♥❞❛r ♥♦ ❡st✉❞♦ ❞♦s ◆ú♠❡r♦s ▼❡tá❧✐❝♦s✱ ❝♦♠ s✉♣♦rt❡ ❞❡ ✉♠❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ♣❡sq✉✐s❛ ❝✐❡♥tí✜❝❛✳

❈♦♥✜❛♠♦s ❞❡✐①❛r ✉♠ ♣❡q✉❡♥♦ ❡ s✐❣♥✐✜❝❛♥t❡ ❝♦♥tr✐❜✉t♦ ❛♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠á✲ t✐❝❛ ❡ ♥♦ss❛ ♣ró♣r✐❛ ❢♦r♠❛çã♦✱ s♦❜r❡t✉❞♦ q✉❛♥❞♦ ❝♦♥❢r♦♥t❛❞♦s ♥♦ss♦ ❡♥t❡♥❞✐♠❡♥t♦ ✐♥t❡❧❡❝t♦ ❛♦ ✐♥í❝✐♦ ❡ ✜♥❛❧ ❞❡st❛ r❡❛❧✐③❛çã♦✳

❈❛♣ít✉❧♦ ✶

❋r❛çõ❡s ❈♦♥tí♥✉❛s

✶✳✶ ■♥tr♦❞✉çã♦

❯♠ t❡♠❛ ♠✉✐t♦ ❝♦♠✉♠ ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦✱ ❛ ❡q✉❛çã♦ q✉❛❞rát✐❝❛ s❡♠♣r❡ ❞❡s♣❡rt♦✉ ❛ ❝✉r✐♦s✐❞❛❞❡ ❞❡ ♠✉✐t♦s ♠❛t❡♠át✐❝♦s ♣♦r ✈ár✐♦s sé❝✉❧♦s✳

❱❛♠♦s ❝♦♠❡ç❛r ❡st❡ ❝❛♣ít✉❧♦ ❛ ♣❛rt✐r ❞❛ r❡s♦❧✉çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ q✉❛❞rát✐❝❛✱

x2₋x₋1 = 0 ✭✶✳✶✮ ♠❛s ✈❛♠♦s r❡s♦❧✈ê✲❧❛ ❞❡ ✉♠❛ ❢♦r♠❛ ❞✐❢❡r❡♥t❡ ❞❛ ❝♦♥✈❡♥❝✐♦♥❛❧✳

➱ ❢á❝✐❧ ✈❡r q✉❡ ③❡r♦ ♥ã♦ é r❛✐③ ❞❡ss❛ ❡q✉❛çã♦✱ ♦✉ s❡❥❛✱ x₆= 0✳

❆ss✐♠✱ ♣♦❞❡♠♦s ❞✐✈✐❞✐r ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❡q✉❛çã♦ ♣♦rx✱ ♦❜t❡♥❞♦

x= 1 + 1

x

❆❣♦r❛✱ ✈❛♠♦s s✉❜st✐t✉✐r ♦ ✈❛❧♦r ❞❡ x♥❛ ♣ró♣r✐❛ ❡q✉❛çã♦✱ ♦❜t❡♥❞♦

x= 1 + 1 1 + 1

x

❊st❡♥❞❡♥❞♦ ❡st❡ ♣r♦❝❡ss♦ ✐♥ú♠❡r❛s ✈❡③❡s✱ ❝❤❡❣❛♠♦s à s❡❣✉✐♥t❡ r❡❧❛çã♦✿

x= 1 + 1 1 + 1

1 + 1 1 + 1

1 +...

✭✶✳✷✮

❆ ♣r✐♦r✐✱ ❡ss❛ ♥ã♦ ♣❛r❡❝❡ s❡r ❛ s♦❧✉çã♦ ❞❡ ✶✳✶✳ ▼❛s✱ ❛♥❛❧✐s❛♥❞♦ ❛ s✉❝❡ssã♦ ❞❡ ❢r❛çõ❡s ♦❜t✐❞❛s ❛♦ ✜♥❛❧ ❞❡ ❝❛❞❛ r❡♣❡t✐çã♦✱ ♣❡r❝❡❜❡♠♦s q✉❡ s✉r❣❡♠ ♠❡❧❤♦r❡s ❛♣r♦①✐✲ ♠❛çõ❡s✳ ❱❡❥❛♠♦s✿

✶✳✷✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

1 + 1 1= 2❀

1 + 1 1 + 1

1

= 1,5❀

1 + 1 1 + 1

1 + 1 1

= 1,666...

Pr♦ss❡❣✉✐♥❞♦ ♥❡ss❛ ❛♥á❧✐s❡✱ ♦❜t❡♠♦s ❛ s❡q✉ê♥❝✐❛

(2; 1,5; 1,666...; 1,6; 1,625; 1,615; 1,619;...) ✭✶✳✸✮ ❆ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❛ ❡q✉❛çã♦ ✶✳✶ é ♦ ❝♦♥❤❡❝✐❞♦ ♥ú♠❡r♦ ❞❡ ♦✉r♦ φ = 1+√5

2 ∼=

1,618✳

❉✐❛♥t❡ ❞♦ ❡①♣♦st♦✱ ♣♦❞❡♠ s✉r❣✐r ❛❧❣✉♥s q✉❡st✐♦♥❛♠❡♥t♦s✿

• ❆ s❡q✉ê♥❝✐❛ ✶✳✸ ❝♦♥✈❡r❣❡ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ✶✳✶❄ • ◗✉❡ t✐♣♦ ❞❡ ❢r❛çã♦ é ❡ss❡ q✉❡ ❛♣❛r❡❝❡ ❡♠ ✶✳✷❄

❊st❡s ❡ ♦✉tr♦s q✉❡st✐♦♥❛♠❡♥t♦s s❡rã♦ ❡s❝❧❛r❡❝✐❞♦s ♥❡st❡ ❝❛♣ít✉❧♦✳

✶✳✷ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s

❉❡✜♥✐çã♦ ✶✳✶ ❚♦❞❛ ❡①♣r❡ssã♦ ❞❛ ❢♦r♠❛

a0+

b₀

a1+

b₁

a₂+ b2

a3+...

✭✶✳✹✮

é ❝❤❛♠❛❞❛ ❞❡ ❢r❛çã♦ ❝♦♥tí♥✉❛✱ ♦♥❞❡ a1, a2, a3, ..., b1, b2, b3, ...sã♦ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ ◆❡ss❛ ❞✐ss❡rt❛çã♦✱ ♥♦ ❡♥t❛♥t♦✱ ✈❛♠♦s r❡str✐♥❣✐r ♥♦ss❛ ❞✐s❝✉ssã♦ ♥❛s ❢r❛çõ❡s ❝♦♥✲ tí♥✉❛s s✐♠♣❧❡s✱ q✉❡ tê♠ ❛ ❢♦r♠❛

a0+

1

a₁+ 1

a₂+ 1

a₃+...

✭✶✳✺✮

✶✳✸✳ ◆Ú▼❊❘❖❙ ❘❆❈■❖◆❆■❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

♦♥❞❡ a0 é ✉♠ ✐♥t❡✐r♦ q✉❛❧q✉❡r ❡ a1, a2, a3, ... sã♦ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✳

❈♦♠ ♦ ✐♥t✉✐t♦ ❞❡ s✐♠♣❧✐✜❝❛r ❛ ❡s❝r✐t❛✱ ❛ ❡①♣r❡ssã♦ ✶✳✺ t❛♠❜é♠ ♣♦❞❡ s❡r r❡♣r❡✲ s❡♥t❛❞❛ ❝♦♠♦

[a0;a1, a2, a3, ...]

❉❡ss❛ ❢♦r♠❛✱ ❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❛ ❡q✉❛çã♦ ✶✳✶ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r[1; 1,1,1, ...]✱

♦✉✱ s✐♠♣❧❡s♠❡♥t❡✱ [¯1]✳

❖s t❡r♠♦s a0, a1, a2, a3, ... sã♦ ❝❤❛♠❛❞♦s ❞❡ q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s ❞❛ ❢r❛çã♦ ❝♦♥tí✲ ♥✉❛✳

❙❡ ❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ ♣♦ss✉✐ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ q✉♦❝✐❡♥t❡ ♣❛r❝✐❛✐s✱ ❡❧❛ é ❝❤❛♠❛❞❛ ❞❡ ❢r❛çã♦ ❝♦♥tí♥✉❛ ✜♥✐t❛✳ ❙❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s é ✐♥✜♥✐t❛✱ ❡❧❛ é ❝❤❛♠❛❞❛ ❞❡ ❢r❛çã♦ ❝♦♥tí♥✉❛ ✐♥✜♥✐t❛✳

❆ ❢r❛çã♦ ❝♦♥tí♥✉❛ q✉❡ ❛♣❛r❡❝❡ ♥♦ ✷♦ _{♠❡♠❜r♦ ❞❛ r❡❧❛çã♦ ✶✳✷ é ✉♠ ❡①❡♠♣❧♦ ❞❡}

❢r❛çã♦ ❝♦♥tí♥✉❛ ✐♥✜♥✐t❛✳ ❆❧é♠ ❞✐ss♦✱ ❡❧❛ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ s❡r ♣❡r✐ó❞✐❝❛ ❝♦♠ ♣❡rí♦❞♦ ✐❣✉❛❧ ❛ ✶✳

✶✳✸ ◆ú♠❡r♦s r❛❝✐♦♥❛✐s

❯♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ é ✉♠❛ ❢r❛çã♦ ❞❛ ❢♦r♠❛ p

q✱ ♦♥❞❡ p ❡ q sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡ q₆= 0✳

❊❢❡t✉❛♥❞♦ ❛ ❞✐✈✐sã♦ ❞❡p ♣♦r q✱ ❡♥❝♦♥tr❛♠♦s ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡❝✐♠❛❧ ❞❡ p_q✳

❖✉tr❛ ❢♦r♠❛ ❞❡ r❡♣r❡s❡♥t❛r ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ é ❛tr❛✈és ❞❡ ✉♠❛ ❢r❛çã♦ ❝♦♥tí♥✉❛✳ P❛r❛ ✐❧✉str❛r✱ ✈❛♠♦s t♦♠❛r ♦ ♥ú♠❡r♦ 67

29✳

❉✐✈✐❞✐♥❞♦ 67♣♦r 29❡♥❝♦♥tr❛♠♦s 2 ❝♦♠♦ q✉♦❝✐❡♥t❡ ❡ 9 ❝♦♠♦ r❡st♦✳ ❆ss✐♠✱

67

29 = 2 + 9 29

= 2 + 1 29

9

Pr♦❝❡❞❡♥❞♦ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❝♦♠ 29

9✱ ❡♥❝♦♥tr❛♠♦s✱ 67

29= 2 + 1

3 + 2 9

❊ss❡ ♣r♦❝❡ss♦ é r❡♣❡t✐❞♦ ❛té ♦ s✉r❣✐♠❡♥t♦ ❞❡ ✉♠ r❡st♦ ♥✉❧♦✳ ▲♦❣♦✱ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ 67

29 ♥❛ ❢♦r♠❛ ❞❡ ❢r❛çã♦ ❝♦♥tí♥✉❛ é

✶✳✸✳ ◆Ú▼❊❘❖❙ ❘❆❈■❖◆❆■❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

67

29 = 2 + 1

3 + 1 4 + 1

2 = [2; 3,4,2]

❖ ♣ró①✐♠♦ t❡♦r❡♠❛ ❣❛r❛♥t❡ q✉❡ q✉❛❧q✉❡r ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛ ❞❡ ❢r❛çã♦ ❝♦♥tí♥✉❛✳

❚❡♦r❡♠❛ ✶✳✶ ◗✉❛❧q✉❡r ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ♣♦❞❡ s❡r ❡①♣r❡ss♦ ❝♦♠♦ ✉♠❛ ❢r❛çã♦ ❝♦♥✲ tí♥✉❛ s✐♠♣❧❡s ✜♥✐t❛✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ q✉❛❧q✉❡r ❢r❛çã♦ ❝♦♥tí♥✉❛ s✐♠♣❧❡s ✜♥✐t❛ r❡♣r❡✲ s❡♥t❛ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ p

q ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ q✉❛❧q✉❡r✳ P❡❧♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✱ ♦❜t❡♠♦s✿

p

q =a0+

r₀

q✱ ♦♥❞❡ 0≤r0 < q✳

❙❡r₀ = 0✱ ❡♥tã♦ p_q é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✳ ❆ss✐♠✱ ♦ ♣r♦❝❡ss♦ t❡r♠✐♥❛ ❡ ❛ ❡①♣❛♥sã♦

❞❡ p

q ❡♠ ❢r❛çã♦ ❝♦♥tí♥✉❛ é[a0]✳ P♦ré♠✱ s❡r_{0 6}= 0✱ ❢❛③❡♠♦s✿

p

q=a0+

1

q r0

❡ r❡♣❡t✐♠♦s ♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✱ ❞✐✈✐❞✐♥❞♦ q ♣♦r r₀✱ ♦❜t❡♥❞♦

q

r₀ =a1+

r₁

r₀, ♦♥❞❡ 0≤r1 < r0✳

❙❡r1 = 0✱ ♦ ♣r♦❝❡ss♦ t❡r♠✐♥❛ ❡ ❛ ❡①♣❛♥sã♦ ❞❡ p_q ❡♠ ❢r❛çã♦ ❝♦♥tí♥✉❛ é [a0;a1]✳ P♦ré♠✱ s❡r_{1 6}= 0✱ ❢❛③❡♠♦s✿

p

q =a0+

1

a1+

1

r0

r1

✶✳✸✳ ◆Ú▼❊❘❖❙ ❘❆❈■❖◆❆■❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

❡ r❡♣❡t✐♠♦s ♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✱ ❞✐✈✐❞✐♥❞♦ r0 r1✳

◆♦t❡ q✉❡ ♦ ♣r♦❝❡ss♦ t❡r♠✐♥❛ q✉❛♥❞♦rn= 0 ♣❛r❛ ❛❧❣✉♠n✱ ♦ q✉❡ s❡♠♣r❡ ♦❝♦rr❡✱

♣♦✐s (q, r₀, r₁, r₂, ..., rn) é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡❝r❡s❝❡♥t❡ ❞❡ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✳ ❈❛s♦

❝♦♥trár✐♦✱ t❡rí❛♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s q > r0 > r1 > ... q✉❡ ♥ã♦ ♣♦ss✉✐ ♠❡♥♦r ❡❧❡♠❡♥t♦✱ ♦ q✉❡ ♥ã♦ é ♣♦ssí✈❡❧ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✳

❆ss✐♠✱ ♣♦r ❞✐✈✐sõ❡s s✉❝❡ss✐✈❛s ♦❜t❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❡q✉❛çõ❡s✿

p

q =a0+

r₀

q, 0< r0 < q

q

r₀ =a1+

r₁

r₀, 0< r1 < r0

r₀ r1

=a2+

r₂ r1

, 0< r2 < r1

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

r(n−2)

r(n−1)

=an+ 0

r(n−1)

, rn = 0

q✉❡ t❡r♠✐♥❛ ❞❡♣♦✐s ❞❡ ✉♠ ❝❡rt♦ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❞✐✈✐sõ❡s✱ ❝♦♠ ❛ ❡q✉❛çã♦ ❡♠ q✉❡ ♦ r❡st♦ r é ✐❣✉❛❧ ❛ 0✳

P♦rt❛♥t♦✱ ❛ ❡①♣❛♥sã♦ ❞❡ p

q ❡♠ ❢r❛çã♦ ❝♦♥tí♥✉❛ é ✜♥✐t❛ ❡ s✉❛ r❡♣r❡s❡♥t❛çã♦ é [a₀;a₁, a₂, ..., an]✳

❆ ❞❡♠♦♥str❛çã♦ ❞❛ r❡❝í♣r♦❝❛ é ✐♠❡❞✐❛t❛✱ ♣♦✐s s❡ ❛ ❡①♣❛♥sã♦ é ✜♥✐t❛✱ s❡♠♣r❡ ♣♦❞❡♠♦s✱ ❢❛③❡♥❞♦ ♦ ❝❛♠✐♥❤♦ ✐♥✈❡rs♦✱ ♦❜t❡r ✉♠❛ ❢r❛çã♦ r❛❝✐♦♥❛❧✳

❯♠❛ ❢♦r♠❛ ♣rát✐❝❛ ❞❡ ❡♥❝♦♥tr❛r ❛ ❡①♣❛♥sã♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ❡♠ ❢r❛çã♦ ❝♦♥tí♥✉❛ é ✉t✐❧✐③❛♥❞♦ ♦ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ❞❡ ❢♦r♠❛ s❡♠❡❧❤❛♥t❡ à ❞❡t❡r♠✐♥❛çã♦ ❞♦ ♠❞❝✳

P❛r❛ ✐❧✉str❛r✱ ✈❛♠♦s r❡t♦♠❛r ♦ ♥ú♠❡r♦ 67

29✱ ❝✉❥❛ ❡①♣❛♥sã♦ ❥á s❛❜❡♠♦s q✉❡ é [2; 3,4,2]✳

✷ ✸ ✹ ✷ ✻✼ ✷✾ ✾ ✷ ✶

✾ ✷ ✶ ✵

❖s q✉♦❝✐❡♥t❡s ♦❜t✐❞♦s sã♦ ♦s q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s ❞❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ q✉❡ r❡♣r❡✲ s❡♥t❛ 67

29✳

❆❣♦r❛✱ ✈❛♠♦s s✐♠♣❧✐✜❝❛r ❛ ❢r❛çã♦ ❝♦♥tí♥✉❛[0; 2,3,4,2]✳ ◆♦t❡ q✉❡ a0 = 0✳

✶✳✸✳ ◆Ú▼❊❘❖❙ ❘❆❈■❖◆❆■❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

0 + 1 2 + 1

3 + 1 4 + 1

2

= 1 2 + 1

3 + 2 9

= 1 2 + 9

29

= 29 67

❆ss✐♠✱ 29

67= [0; 2,3,4,2]✳

❯♠❛ ❝♦♠♣❛r❛çã♦ ❞❛ ❡①♣❛♥sã♦ 67

29 = [2; 3,4,2] ❝♦♠ 29

67 = [0; 2,3,4,2] s✉❣❡r❡ ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿

❚❡♦r❡♠❛ ✶✳✷ ❙❡❥❛ p

q ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ♣♦s✐t✐✈♦ t❛❧ q✉❡ p > q✳

p

q= [a0;a1, ..., an] s❡✱ ❡ s♦♠❡♥t❡ s❡✱

q

p= [0;a0, a1, ..., an]✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ p

q ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ♣♦s✐t✐✈♦ ❝✉❥❛ ❡①♣❛♥sã♦ ❡♠ ❢r❛çã♦

❝♦♥tí♥✉❛ é [a0;a1, ..., an]✳ ❉❛í✱ ❢❛③❡♥❞♦ ❛❧❣✉♠❛s ♠❛♥✐♣✉❧❛çõ❡s ❛❧❣é❜r✐❝❛s✱ t❡♠♦s✿

p

q = [a0;a1, ..., an]

= a0+

1

a1+✳✳✳

1

an

▲♦❣♦✱✐♥✈❡rt❡♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s✱t❡♠♦s✿

✶✳✸✳ ◆Ú▼❊❘❖❙ ❘❆❈■❖◆❆■❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

q

p =

1

a0+

1

a₁+✳✳✳ 1

an

= 0 + 1

a0+

1

a₁+✳✳✳ 1

an = [0;a0, a1, ..., an]

❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ❞❡♠♦♥str❛✲s❡ ❛ r❡❝í♣r♦❝❛✳

❆❣♦r❛✱ ✈❛♠♦s ❛♥❛❧✐s❛r ❝♦♠♦ ♦❜t❡r ❛ ❡①♣❛♥sã♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ♥❡❣❛t✐✈♦ −pq✳ ▼❛s✱ ❛♥t❡s✱ ✈❛♠♦s ❞❡✜♥✐r ♦ q✉❡ é ❛ ♣❛rt❡ ✐♥t❡✐r❛ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ x✳

❉❡✜♥✐çã♦ ✶✳✷ ❆ ♣❛rt❡ ✐♥t❡✐r❛ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧x é ♦ ♠❛✐♦r ✐♥t❡✐r♦_⌊x_⌋q✉❡ ♥ã♦ é

♠❛✐♦r q✉❡ x✳ ❉❡✜♥✐♠♦s ❛ ♣❛rt❡ ❢r❛❝✐♦♥ár✐❛ _{x_} ❞❡ x ♣♦r _{x_}=x_{− ⌊}x_⌋✳✭❡①❡♠♣❧♦s✿

⌊3_⌋= 3✱_⌊3,5_⌋= 3 ❡ _⌊−4,7_⌋=₋5✳✮ ◆♦t❡ q✉❡ ♣❛r❛ p

q >0✱ a0 =⌊ p

q⌋✳ ❖ ♠❡s♠♦ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ p

q < 0✳ ❖✉ ♠❡❧❤♦r✱ ♣❛r❛ q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧ x✱ a0 =⌊x⌋✳

P❛r❛ ❡♥❝♦♥tr❛r ❛ ❡①♣❛♥sã♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ♥❡❣❛t✐✈♦−pq ♣♦❞❡♠♦s t❛♠❜é♠ ✉t✐❧✐③❛r ♦ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✳ ◆❡ss❡ ❝❛s♦✱ ❝♦♠♦ a0 = ⌊−p_q⌋✱ a0 < 0✳ P♦ré♠✱ ♦s ❞❡♠❛✐s q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s sã♦ t♦❞♦s ♣♦s✐t✐✈♦s✳

P❛r❛ ✐❧✉str❛r✱ ✈❛♠♦s ❡♥❝♦♥tr❛r ❛ ❡①♣❛♥sã♦ ❞❡−37

44 ❡ −

44 37✳ ❊①❡♠♣❧♦✿ ❊①♣❛♥❞✐♥❞♦ −3744 ❡♠ ❢r❛çã♦ ❝♦♥tí♥✉❛✳

✲✶ ✻ ✸ ✷ ✲✸✼ ✹✹ ✼ ✷ ✶

✼ ✷ ✶ ✵

▲♦❣♦✱ −3744 = [−1; 6,3,2]✳ ⋄

❊①❡♠♣❧♦✿ ❊①♣❛♥❞✐♥❞♦ −4437 ❡♠ ❢r❛çã♦ ❝♦♥tí♥✉❛✳

✲✷ ✶ ✹ ✸ ✷

✲✹✹ ✸✼ ✸✵ ✼ ✷ ✶

✸✵ ✼ ✷ ✶ ✵

✶✳✹✳ ◆Ú▼❊❘❖❙ ■❘❘❆❈■❖◆❆■❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

▲♦❣♦✱ −4437 = [−2; 1,4,3,2]✳ ⋄

◆♦t❡ q✉❡✱ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡|p_| <_|q_|✱ ♦ ♣r✐♠❡✐r♦ q✉♦❝✐❡♥t❡ ♣❛r❝✐❛❧ é ₋1✱ ♣❛r❛

q✉❛✐sq✉❡r p✱ q ✐♥t❡✐r♦s ❡ p_q < 0✳ ❏á ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ _|p_| > _|q_|✱ ♦ q✉♦❝✐❡♥t❡

♣❛r❝✐❛❧ é s❡♠♣r❡ ♥❡❣❛t✐✈♦✳ ❊♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ❛♣❡♥❛s ♦ ♣r✐♠❡✐r♦ q✉♦❝✐❡♥t❡ ♣❛r❝✐❛❧ é ♥❡❣❛t✐✈♦✳

✶✳✹ ◆ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s

❯♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ é ✉♠ ♥ú♠❡r♦ q✉❡ ♥ã♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ ❛ r❛③ã♦ ❞❡ ❞♦✐s ✐♥t❡✐r♦s✱ ♦✉ s❡❥❛✱ ♥❛ ❢♦r♠❛ p

q✱ ♦♥❞❡ p ❡q sã♦ ✐♥t❡✐r♦s ❡ q6= 0✳ ❖s ♥ú♠❡r♦s

√

2, √3, 1_±√2, 3±

√ 7 5

sã♦ t♦❞♦s ✐rr❛❝✐♦♥❛✐s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ q✉❛❧q✉❡r ♥ú♠❡r♦ ❞❛ ❢♦r♠❛

P _±√D

Q ,

♦♥❞❡ P, D, Qsã♦ ✐♥t❡✐r♦s ❡D é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ q✉❡ ♥ã♦ é ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦

é ❝❤❛♠❛❞♦ ❞❡ ■rr❛❝✐♦♥❛❧ ◗✉❛❞rát✐❝♦✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ♣♦❞❡♠♦s ❡①♣r❡ss❛r ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ❝♦♠♦ ❢r❛çõ❡s ❝♦♥tí♥✉❛s✳ P❛r❛ ❝♦♥str✉✐r ❛ ❡①♣❛♥sã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ ❡♠ ❢r❛çã♦ ❝♦♥tí♥✉❛ s✐♠♣❧❡s ✉t✐❧✐③❛r❡♠♦s s✉❜st✐t✉✐çõ❡s s✉❝❡ss✐✈❛s✱ ❞❛ ❢♦r♠❛ q✉❡ ❞❡s❝r❡✈❡r❡♠♦s ❛ s❡❣✉✐r✳

❙❡❥❛♠ x ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ q✉❛❧q✉❡r ❡ a₀ =_⌊x_⌋✳

P♦❞❡♠♦s ❡s❝r❡✈❡r x❝♦♠♦

x=a₀+ 1

x1✱ ♦♥❞❡

0< 1 x1

<1

❊♥tã♦✱ x1 = 1

x₋a₀ >1 é ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧✳

❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

x₁ =a₁+ 1

x2✱ ♦♥❞❡

a₁ =_⌊x_{1⌋ ≥}1, 0< 1

x2

<1

❡ ♦❜t❡♠♦s x2 = 1

x₁₋a₁ >1✱ q✉❡ é✱ t❛♠❜é♠✱ ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧✳

❘❡♣❡t✐♥❞♦✲s❡ ❡ss❡ ♣r♦❝❡ss♦✱ ♦❜t❡♠♦s✱ s✉❝❡ss✐✈❛♠❡♥t❡✱ ❛s ❡q✉❛çõ❡s✿

✶✳✹✳ ◆Ú▼❊❘❖❙ ■❘❘❆❈■❖◆❆■❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

x=a₀+ 1

x1

, x₁ >1

x1 =a1+

1

x₂, x2 >1, a1 ≥1

✳✳✳ ✳✳✳ ✳✳✳

xn=an+

1

xn+1

, xn+1 >1, an≥1

✳✳✳ ✳✳✳ ✳✳✳

♦♥❞❡ a0, a1, ..., an, ... sã♦ ✐♥t❡✐r♦s ❡ x1, x2, ..., xn, ... sã♦ ✐rr❛❝✐♦♥❛✐s✳

◆♦t❡♠♦s q✉❡ ❡st❡ ♣r♦❝❡ss♦ ♥ã♦ t❡r♠✐♥❛✱ ♣♦✐s ✐st♦ só ♦❝♦rr❡r✐❛ s❡ xn = an ♣❛r❛ ❛❧❣✉♠ n✱ ♦ q✉❡ é ✐♠♣♦ssí✈❡❧✱ ♣♦✐s xn é ✐rr❛❝✐♦♥❛❧ ♣❛r❛ t♦❞♦ n✳

❋❛③❡♥❞♦ s✉❜st✐t✉✐çõ❡s ❛♣r♦♣r✐❛❞❛s✱ ♦❜t❡♠♦s ❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ s✐♠♣❧❡s ✐♥✜♥✐t❛✿

x = a₀+ 1

x1

= a₀+ 1

a1+

1

x₂

= a₀+ 1

a₁+ 1

a2+

1

x₃

= a₀+ 1

a₁+ 1

a2+✳✳✳+

1

an+✳✳✳

q✉❡ t❛♠❜é♠ ❞❡♥♦t❛♠♦s ♣♦r [a₀;a₁, a₂, ...]✳

P❛r❛ ✐❧✉str❛r✱ ✈❛♠♦s ❡①♣r❡ss❛r √13❝♦♠♦ ✉♠❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ s✐♠♣❧❡s✳ ◆♦t❡ q✉❡ ⌊√13_⌋=_⌊3,605551..._⌋= 3✳

▲♦❣♦✱ a₀ = 3✳ ▼❛s✱

✶✳✹✳ ◆Ú▼❊❘❖❙ ■❘❘❆❈■❖◆❆■❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

√

13 = 3 + (√13₋3)

= 3 + 1 1 √

13₋3

= 3 + √ 1 13 + 3

4

▲♦❣♦✱ a₁ = 1✱ ♣♦✐s

$√

13 + 3 4

%

= 1✳ ▼❛s✱

√ 13 + 3

4 = 1 + √

13 + 3 4 −1

!

= 1 + √

13₋1 4

= 1 + 1 4 √

13₋1

= 1 + 1 4(√13 + 1)

12

= 1 + _√ 1 13 + 1

3

▲♦❣♦✱ a2 = 1✱ ♣♦✐s

$√

13 + 1 3

%

= 1✳

❈♦♥t✐♥✉❛♥❞♦ ♦ ♣r♦❝❡ss♦✱ ❡♥❝♦♥tr❛♠♦s

a3 = 1, a4 = 1, a5 = 6, a6 = 1, a7 = 1, a8 = 1, a9 = 1, a10 = 6, a11 = 1, ...

▲♦❣♦✱

✶✳✹✳ ◆Ú▼❊❘❖❙ ■❘❘❆❈■❖◆❆■❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

√

13 = 3 + 1 1 + 1

1 + 1 1 + 1

1 + 1 6✳✳✳

= [3; 1,1,1,1,6, ...]

P♦r ✜♠✱ ✈❛♠♦s ❡①♣r❡ss❛r π ❝♦♠♦ ✉♠❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ s✐♠♣❧❡s✳

◆♦t❡ q✉❡ ⌊π_⌋=_⌊3,1415926535897932..._⌋= 3✳

▲♦❣♦✱ a₀ = 3✳

▼❛s✱

π = 3 + 0,1415926535897932...

= 3 + 1 1

0,1415926535897932...

= 3 + 1

7,06251331041...

▲♦❣♦✱ a1 = 7✳

▼❛s✱

7,06251331041... = 7 + 0,06251331041...

= 7 + 1 1

0,06251331041...

= 7 + 1

15,9965932606...

▲♦❣♦✱ a₂ = 15✳

❘❡♣❡t✐♥❞♦ ♦ ♣r♦❝❡ss♦✱ ❡♥❝♦♥tr❛♠♦s✱

a₃ = 1, a₄ = 292, a₅ = 1, a₆ = 1, a₇ = 1, a₈ = 2...

✶✳✺✳ ❈❖◆❱❊❘●❊◆❚❊❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

▲♦❣♦✱ π= [3; 7,15,1,292,1,1,1,2...]✳

◆❡♠ t♦❞♦ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ t❡♠ ❡①♣❛♥sã♦ ❡♠ ❢r❛çã♦ ❝♦♥tí♥✉❛ s✐♠♣❧❡s✳ ◆♦t❡✱ ♥♦s ❡①❡♠♣❧♦s ❛❜❛✐①♦✱ ❛ ♣r❡s❡♥ç❛ ❞❡ ❛❧❣✉♥s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ❝♦♠ ❡①♣❛♥sã♦ ❡♠ ❢r❛çã♦ ❝♦♥tí♥✉❛ ♥ã♦ s✐♠♣❧❡s✳

e= 2 + 1 1 + 1

2 + 1 1 + 1

1 + 1 4✳✳✳

e₋1 = 1 + 2 2 + 3

3 + 4 4 + 5

5 + 6 6 +✳✳✳

(Euler)

4

π= 1 +

12

2 + 3 2

2 + 5 2

2 + 7 2

2 + 9 2

2 + ✳✳✳

(Brouncker)

✶✳✺ ❈♦♥✈❡r❣❡♥t❡s

❙❡❥❛ p

q ✉♠❛ ❢r❛çã♦ r❛❝✐♦♥❛❧✱ ❝✉❥❛ ❡①♣❛♥sã♦ ❡♠ ❢r❛çã♦ ❝♦♥tí♥✉❛ s✐♠♣❧❡s ✜♥✐t❛ é

❞❛❞❛ ♣♦r

p

q =a0 +

1

a1+

1

a2+✳✳✳+

1

an

= [a₀;a₁, a₂, ..., an]. ✭✶✳✻✮

✶✳✺✳ ❈❖◆❱❊❘●❊◆❚❊❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ ❛s ❢r❛çõ❡s ❝♦♥tí♥✉❛s

c0 =a0

c₁ =a₀+ 1

a₁

c1 =a0+

1

a₁+ 1

a2

✳✳✳

♦❜t✐❞❛s ❞❡ ✶✳✻✱ ❝♦♥s✐❞❡r❛♥❞♦✲s❡✱ s✉❝❡ss✐✈❛♠❡♥t❡✱ ♦ ♣r✐♠❡✐r♦ t❡r♠♦ ❞❛ ❡①♣❛♥sã♦✱ ♦ ♣r✐♠❡✐r♦ ❡ s❡❣✉♥❞♦ t❡r♠♦s ❞❛ ❡①♣❛♥sã♦ ❡✱ ❛ss✐♠✱ s✉❝❡ss✐✈❛♠❡♥t❡✱ ❛té ♦ ✭♥✰✶✮✲és✐♠♦ t❡r♠♦✳

❉❡✜♥✐çã♦ ✶✳✸ ❈❤❛♠❛♠♦s ❝♦♥✈❡r❣❡♥t❡ ❞❡ ♦r❞❡♠ ✐ ❞❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ ✶✳✻✱ ♦ ♥ú♠❡r♦

ci =a0+

1

a1+✳✳✳ +

1

ai

, 0_≤i_≤n.

◆♦t❡ q✉❡ n✲és✐♠♦ t❡r♠♦ ❝♦♥✈❡r❣❡♥t❡ ❞❡ ✭✶✳✻✮✱ cn✱ é ❛ ♣ró♣r✐❛ ❢r❛çã♦ ❝♦♥tí♥✉❛✳

❱❛♠♦s✱ ❛❣♦r❛✱ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ❢♦r♠❛ ♣rát✐❝❛ ♣❛r❛ ❝❛❧❝✉❧❛r ❡ss❡s ❝♦♥✈❡r❣❡♥t❡s✳ ▼❛s ❛♥t❡s✱ ✈❛♠♦s ❞❡✜♥✐r ♦s ♥ú♠❡r♦s pi ❡qi✳

❉❡✜♥✐çã♦ ✶✳✹ ❖s ♥ú♠❡r♦s pi ❡ qi✱ t❛✐s q✉❡ ci =

pi

qi

sã♦ ❝❤❛♠❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♥✉♠❡r❛❞♦r ❡ ❞❡♥♦♠✐♥❛❞♦r ❞♦ ✐✲és✐♠♦ ❝♦♥✈❡r❣❡♥t❡✳

❉❡✜♥✐❞♦s ♦s ♥ú♠❡r♦spi ❡qi ❡ ❞❛❞❛ ❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ ✶✳✻✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

c0 =

p₀ q₀

= a₀

= a0 1

✶✳✺✳ ❈❖◆❱❊❘●❊◆❚❊❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

c1 =

p₁ q₁

= a₀+ 1

a₁

= a1a0+ 1

a₁

c₂ = p2

q2

= a0+ 1

a1+

1

a2

= a₀+ 1

a₂a₁+ 1

a₂

= a0+

a₂

a2a1+ 1

= a0(a2a1+ 1) +a2

a₂a₁+ 1

= a2a1a0+a0+a2

a₂a₁+ 1

= a2(a1a0+ 1) +a0

a2a1+ 1

= a2p1+p0

a₂q₁+q₀

Pr♦❝❡❞❡♥❞♦ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ❡♥❝♦♥tr❛♠♦s✱

c₃ = p3

q3

= a3p2 +p1

a₃q₂ +q₁

❚❛✐s r❡s✉❧t❛❞♦s s✉❣❡r❡♠ ✉♠❛ ❡①♣r❡ssã♦ s✐♠♣❧❡s ♣❛r❛ ❡♥❝♦♥tr❛r ♦ ♥✉♠❡r❛❞♦rpi❡ ♦ ❞❡♥♦♠✐♥❛❞♦rqi ❞♦i✲és✐♠♦ ❝♦♥✈❡r❣❡♥t❡ci✳ ❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r ✈❛❧✐❞❛ ❡ss❛ ❡①♣r❡ssã♦✳

✶✳✺✳ ❈❖◆❱❊❘●❊◆❚❊❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

❚❡♦r❡♠❛ ✶✳✸ ❖ ♥✉♠❡r❛❞♦r pi ❡ ♦ ❞❡♥♦♠✐♥❛❞♦r qi ❞♦ ✐✲és✐♠♦ ❝♦♥✈❡r❣❡♥t❡ ci ❞❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ [a₀;a₁, a₂, ..., an] s❛t✐s❢❛③❡♠ às ❡q✉❛çõ❡s

pi =aipi−1+pi−2

qi =aiqi−1+qi−2

, i= 2,3, ..., n, ✭✶✳✼✮

❝♦♠ ❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s

p₀ =a₀

q0 = 1

e

p₁ =a₁a₀+ 1

q1 =a1 ✳

❉❡♠♦♥str❛çã♦✿ P❛r❛ ❡st❛ ❞❡♠♦♥str❛çã♦ ✈❛♠♦s ✉s❛r ✐♥❞✉çã♦✳

❖s ❝á❧❝✉❧♦s q✉❡ ✜③❡♠♦s ♣❛r❛ c₀ ❡ c₁ ♠♦str❛♠ q✉❡ ❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ❡stã♦

s❛t✐s❢❡✐t❛s✳

❖ ❝á❧❝✉❧♦ ❢❡✐t♦ ♣❛r❛c2 ♠♦str❛ q✉❡ ❛s ❡q✉❛çõ❡s ✶✳✼ sã♦ ✈❡r❞❛❞❡✐r❛s ♣❛r❛i= 2✳ ❙✉♣♦♥❤❛♠♦s✱ ❡♥tã♦✱ q✉❡ ❛s ❡q✉❛çõ❡s ✶✳✼ sã♦ ✈❡r❞❛❞❡✐r❛s ♣❛r❛k✱ ❝♦♠ 3_≤k < n✳

■ss♦ s✐❣♥✐✜❝❛ q✉❡

ck =

pk

qk

= akpk−1+pk−2

akqk−1+qk−2 ✭✶✳✽✮

❙❛❜❡♠♦s q✉❡

ck =a0+

1

a₁+ 1

a2+✳✳✳+

1

ak₋1+

1

ak ❡ q✉❡

ck+1 =a0+

1

a₁+ 1

a₂+✳✳✳+ 1

ak−1 +

1

ak+

1

ak+1

◆♦t❡ q✉❡ ♣♦❞❡♠♦s ♦❜t❡r ck+1 ❛ ♣❛rt✐r ❞❡ ck s✐♠♣❧❡s♠❡♥t❡ ♣❡❧❛ s✉❜st✐t✉✐çã♦ ❞❡

ak♣♦rak+_ak1₊₁✳ ▼❛s ✐st♦ q✉❡r ❞✐③❡r q✉❡ s❡ ♣✉❞❡r♠♦s ♠♦str❛r q✉❡ ♦s ♥ú♠❡r♦spk−1✱

pk−2✱ qk−1 ❡ qk−2 ❞❡♣❡♥❞❡♠ s♦♠❡♥t❡ ❞♦s ♥ú♠❡r♦s a0✱ a1✱✳✳✳✱ ak−1✱ ♣♦❞❡r❡♠♦s ✉s❛r ✶✳✽ ♣❛r❛ ♦❜t❡r ck+1✱ ♣♦✐s ❡st❛♠♦s s✉♣♦♥❞♦✱ ❝♦♠♦ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ ❛ ✈❛❧✐❞❛❞❡ ❞❡ ✶✳✽✱ ♣❛r❛ t♦❞♦ k✱ ❝♦♠3_≤k < n✳ ❉❡ ❢❛t♦✱ ❝♦♠♦

✶✳✻✳ ❋❘❆➬Õ❊❙ P❊❘■Ó❉■❈❆❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

pk−1

qk−1

= ak−1pk−2+pk−3

ak−1qk−2+qk−3✱

s❡❣✉❡ q✉❡ ♦s ♥ú♠❡r♦s pk₋1 ❡qk₋1 ❞❡♣❡♥❞❡♠ s♦♠❡♥t❡ ❞♦ ♥ú♠❡r♦ak₋1 ❡ ❞♦s ♥ú♠❡r♦s

pk−2✱pk−3✱qk−2 ❡ qk−3✱ ♦s q✉❛✐s✱ ♣♦r s✉❛ ✈❡③✱ ❞❡♣❡♥❞❡♠ ❞❡ s❡✉s ♣r❡❝❡❞❡♥t❡s✳ ❉❡ss❛ ❢♦r♠❛✱ pk−1✱pk−1✱qk−1 ❡ qk−2 ❞❡♣❡♥❞❡♠ ❛♣❡♥❛s ❞♦s ♥ú♠❡r♦s a0✱ a1✱✳✳✳✱ ak−1✱ s❡♥❞♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ ak✳ ▲♦❣♦✱ ❡❧❡s ♥ã♦ sã♦ ❛❧t❡r❛❞♦s ❝♦♠ ❛ s✉❜st✐t✉✐çã♦ ❞❡ ak ♣♦r

ak+ _ak1₊₁✳ P♦❞❡♠♦s✱ ♣♦rt❛♥t♦✱ ✉t✐❧✐③❛r ❛ ❡①♣r❡ssã♦ ✶✳✽ ♣❛r❛ ♦❜t❡r ck+1✱ ❜❛st❛♥❞♦✱ ♣❛r❛ ✐ss♦✱ s✉❜st✐t✉✐r ak ♣♦r ak+_ak1₊₁✱ ♦✉ s❡❥❛✱

ck+1 =

pk+1

qk+1

=

(ak+ 1

ak+1

)pk−1+pk−2

(ak+ 1

ak+1

)qk₋1+qk₋2

= (ak+1ak+ 1)pk−1+ai+1pi−2 (ak+1ak+ 1)qk−1+ai+1qi−2

= ak+1akpk−1+pk−1+ak+1pk−2

ak+1akqk−1 +qk−1+ak+1qk−2

= ak+1(akpk−1+pk−2) +pk−1

ak+1(akqk−1+qk−2) +qk−1

= ak+1pk+pk−1

ak+1qk+qk−1

❖s ❝♦♥✈❡r❣❡♥t❡s ❞❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ s✐♠♣❧❡s ✐♥✜♥✐t❛

x= [a₀;a₁, a₂, a₃, ...]

sã♦ ❝❛❧❝✉❧❛❞♦s ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♥♦ ❝❛s♦ ❞❛s ❢r❛çõ❡s ❝♦♥tí♥✉❛s s✐♠♣❧❡s ✜♥✐t❛s✳

✶✳✻ ❋r❛çõ❡s ♣❡r✐ó❞✐❝❛s

❈❡rt❛s ❢r❛çõ❡s ❝♦♥tí♥✉❛s✱ ❝♦♠♦ √

2 = [1; 2,2,2...] = [1; ¯2]

sã♦ ♣❡r✐ó❞✐❝❛s ❛ ♣❛rt✐r ❞❡ ✉♠ ❝❡rt♦ t❡r♠♦ ❡ ♦✉tr❛s ❝♦♠♦

✶✳✻✳ ❋❘❆➬Õ❊❙ P❊❘■Ó❉■❈❆❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

√

2 + 1 = [2; 2,2,2, ...] = [¯2]

sã♦ ♣❡r✐ó❞✐❝❛s ❞❡s❞❡ ♦ ✐♥í❝✐♦✳

❊st❛s ú❧t✐♠❛s sã♦ ❝❤❛♠❛❞❛s ❞❡ ❢r❛çõ❡s ❝♦♥tí♥✉❛s ♣✉r❛♠❡♥t❡ ♣❡r✐ó❞✐❝❛s✳

◆❛ s❡çã♦ ✶✳✹✱ ❥á ♠❡♥❝✐♦♥❛♠♦s s♦❜r❡ ♦s ✐rr❛❝✐♦♥❛✐s q✉❛❞rát✐❝♦s✳ ❱❛♠♦s✱ ❛❣♦r❛✱ ❞❡✜♥✐r ❡ss❡s ❝♦♥❝❡✐t♦s✳

❉❡✜♥✐çã♦ ✶✳✺ ❯♠❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ s✐♠♣❧❡s é ❞❡♥♦♠✐♥❛❞❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ ♣❡r✐ó✲ ❞✐❝❛ s❡ ❛ s❡q✉ê♥❝✐❛ ❞♦s ✈❛❧♦r❡s ai ❛♣r❡s❡♥t❛ r❡♣❡t✐çã♦ ✭♣❡rí♦❞♦✮✱ q✉❡ ❞❡♥♦t❛♠♦s ♣♦r

[a₀;a₁, a₂, ..., ak−1, ak, ak+1, ..., ak+n−1]✱

♦♥❞❡ ak+n =ak ❡ ♦s ✈❛❧♦r❡s ak✱ ak+1✱✳✳✳✱ak+n−1 ❢♦r♠❛♠ ♦ ♣❡rí♦❞♦ q✉❡ s❡ r❡♣❡t❡✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ ❢r❛çã♦ ❝♦♥tí♥✉❛

[a0;a1, a2, ..., an−1]

é ❝❤❛♠❛❞❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ ♣✉r❛♠❡♥t❡ ♣❡r✐ó❞✐❝❛✳

❉❡✜♥✐çã♦ ✶✳✻ ❈❤❛♠❛♠♦s ✐rr❛❝✐♦♥❛❧ q✉❛❞rát✐❝♦ ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ x q✉❡ é r❛✐③

❞❛ ❡q✉❛çã♦ q✉❛❞rát✐❝❛ ax2+bx+c= 0✱ ♦♥❞❡ a, b, c sã♦ ✐♥t❡✐r♦s ❡ b2₋4ac >0 ♥ã♦ é ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✳

❍á ❞♦✐s r❡s✉❧t❛❞♦s ❢✉♥❞❛♠❡♥t❛✐s s♦❜r❡ ❢r❛çõ❡s ❝♦♥tí♥✉❛s ♣❡r✐ó❞✐❝❛s ❡ ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s q✉❛❞rát✐❝♦s✱ ♦s t❡♦r❡♠❛s ❞❡ ❊✉❧❡r ❡ ▲❛❣r❛♥❣❡✳

❚❡♦r❡♠❛ ✶✳✹ ✭❊✉❧❡r✮ ❙❡ x é ✉♠❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ ♣❡r✐ó❞✐❝❛✱ ❡♥tã♦x é ✉♠ ✐rr❛❝✐✲

♦♥❛❧ q✉❛❞rát✐❝♦✳

❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ s❡❥❛ x = [a₀;a₁, a₂, ..., ak₋1, ak, ak+1, ..., ak+n₋1] ✉♠❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ ♣❡r✐ó❞✐❝❛✳

❱❛♠♦s t♦♠❛rx= [a0;a1, ..., ak−1, xk]✱ ♦♥❞❡ xk = [ak;ak+1, ...]✳ ❆ss✐♠

xk = [ak;ak+1, ..., ak+n−1, xk]

❡✱ ❡♥tã♦✱

xk=

xkp′ +p′′

xkq′ +q′′✱ ♦✉ s❡❥❛✱

q′x2k+ (q′′−p′)xk−p′′= 0, ✭✶✳✾✮

♦♥❞❡ p′′

q′′ ❡ p′

q′ sã♦ ♦s ❞♦✐s ú❧t✐♠♦s ❝♦♥✈❡r❣❡♥t❡s ❞❡[ak, ak+1, ..., ak+n−1]✳ ▼❛s✱

✶✳✻✳ ❋❘❆➬Õ❊❙ P❊❘■Ó❉■❈❆❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

x= xkpk−1+pk−2

xkqk−1+qk−2✳ ▲♦❣♦✱

xk =

pk−2−qk−2x

qk−1x−pk−1✳

❙✉❜st✐t✉✐♥❞♦✲s❡ ♦ ✈❛❧♦r ❞❡xk ❞❛❞♦ ❛❝✐♠❛ ❡♠ ✶✳✾ ❡ s✐♠♣❧✐✜❝❛♥❞♦✱ ♦❜t❡♠♦s

ax2 _{+bx+c= 0✱}

♦♥❞❡

a=q′q2_k−2−(q′′₋q′)qk−2qk−1−p′′q2k−1✱

b= 2(p′′_p

k−1qk−1−q′pk−2qk−2) + (q′′−p′)(pk−2qk−1−qk−2pk−1)✱

c=q′_p2

k₋2−(q′′−p′)pk−2pk−1−p′′pk−1

❡✱ ❛ss✐♠✱ a, b, c sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❈♦♠♦x é ✐rr❛❝✐♦♥❛❧✱ ❡♥tã♦ b2_{−4ac > 0✳} P♦rt❛♥t♦✱ x é ✉♠ ✐rr❛❝✐♦♥❛❧ q✉❛❞rát✐❝♦✳

❚❡♦r❡♠❛ ✶✳✺ ✭▲❛❣r❛♥❣❡✮ ❆ ❢r❛çã♦ ❝♦♥tí♥✉❛ q✉❡ r❡♣r❡s❡♥t❛ ✉♠ ✐rr❛❝✐♦♥❛❧ q✉❛❞rá✲ t✐❝♦ x é ♣❡r✐ó❞✐❝❛✳

❉❡♠♦♥str❛çã♦✿ ❙❛❜❡♠♦s q✉❡ ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ q✉❛❞rát✐❝♦ s❛t✐s❢❛③ ❛ ✉♠❛ ❡q✉❛çã♦ q✉❛❞rát✐❝❛ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ✐♥t❡✐r♦s✱ q✉❡ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦

ax2+bx+c= 0 ✭✶✳✶✵✮

❙❡x= [a0;a1, a2, ...ak, ..]✱ t♦♠❛♥❞♦✲s❡xk = [ak;ak+1, ...]✱ ❡♥tã♦x= [a0;a1, ..., ak−1, xk]✳

❆❧é♠ ❞✐ss♦✱

x= xkpk−1+pk−2

xkpk−1+pk−2✳ ❙✉❜st✐t✉✐♥❞♦✲s❡ ♦ ✈❛❧♦r ❛❝✐♠❛ ❡♠ ✶✳✶✵✱ ♦❜t❡♠♦s

Akx2k+Bkxk+Ck= 0, ✭✶✳✶✶✮

♦♥❞❡

Ak =ap2k−1+bpk−1qk−1+cq2k−1

Bk = 2apk−1pk−2+b(pk−1qk−2+pk−2qk−1) + 2cqk−1qk−2, ✭✶✳✶✷✮

Ck =ap2k−2+bpk−2qk−2+cq2k−2✳

❙❡Ak = 0✱ ✐st♦ é✱ ap2k−1+bpk−1qk−1+cq2k−1 = 0✱ ♦❜t❡♠♦s

✶✳✻✳ ❋❘❆➬Õ❊❙ P❊❘■Ó❉■❈❆❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

pk₋1 = −

bqk−1±

p

(b2_−4ac)q2 k₋1 2a =

= −b± √

b2_−4ac

2a qk−1✳

▲♦❣♦✱ −b± √

b2_−4ac

2a = pk−1

qk−1✳ ❊♥tã♦✱ ❛ ❡q✉❛çã♦ ✶✳✶✵ t❡♠ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧

pk−1

qk−1 ❝♦♠♦ r❛✐③✱ ♦ q✉❡ é ✐♠♣♦ssí✈❡❧✱ ♣♦✐s

xé ✐rr❛❝✐♦♥❛❧✳ P♦rt❛♥t♦✱Ak 6= 0❡ ❛ ❡q✉❛çã♦ q✉❛❞rát✐❝❛

Aky2+Bky+Ck = 0✱

t❡♠ xk ❝♦♠♦ ✉♠❛ ❞❡ s✉❛s r❛í③❡s✳ ❉❡ ✶✳✶✷✱ ♦❜t❡♠♦s

B2₋4AkCk = [2apk−1pk−2+b(pk−1qk−2+pk−2qk−1) + 2cqk−1qk−2]2− 4(ap2_k−1+bpk−1qk−1+cq2k₋1)(ap2k₋2+bpk−2qk−2+cq2k₋2) = (b2₋4ac)(pk−1qk−2−pk−2qk−1)2

= =b2₋4ac

▲♦❣♦✱

B2₋4AkCk=b2−4ac. ✭✶✳✶✸✮

➱ ♣♦ssí✈❡❧ ♣r♦✈❛r q✉❡ xqk−1 −pk−1 >− 1

qk✳ ▲♦❣♦✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦

δk−1✱ ❝♦♠ |δk−1|<1✱ t❛❧ q✉❡

pk−1 =xqk−1+

δk₋1

qk−1

P♦rt❛♥t♦✱ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛ ❡ ❞❡ ✶✳✶✵✱ ♦❜t❡♠♦s

Ak = a xqk−1+

δk−1

qk−1

!2

+bqk−1 xqk−1+

δk−1

qk−1

!

+cq2_k−1

= (ax2+bx+c)q2_k−1+ 2axδk−1+a

δ2 k−1

q2 k−1

+bδk−1

= 2axδk−1+a

δ2k−1

q2 k−1

+bδk−1

▲♦❣♦✱

✶✳✻✳ ❋❘❆➬Õ❊❙ P❊❘■Ó❉■❈❆❙ ❈❆P❮❚❯▲❖ ✶✳ ❋❘❆➬Õ❊❙ ❈❖◆❚❮◆❯❆❙

|Ak|<2|ax|+|a|+|b|✳

▼❛s✱ ❝♦♠♦Ck=Ak−1✱

|Ck|<2|ax|+|a|+|b|✳

❉❡ ✶✳✶✸✱ ♦❜t❡♠♦s

B2k ≤a|AkCk|+|b2−4ac|<4(2|ax|+|b|+|c|)2+|b2−4ac|✳

❖❜s❡r✈❡ q✉❡ ♦s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s ❞❡ Ak✱ Bk ❡ Ck sã♦ ♠❡♥♦r❡s ❞♦ q✉❡ ♥ú♠❡r♦s q✉❡ ♥ã♦ ❞❡♣❡♥❞❡♠ ❞❡ k✳ ❈♦♠♦ Ak✱ Bk ❡ Ck sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❡①✐st❡ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ tr✐♣❧❛s (Ak, Bk, Ck) ❞✐❢❡r❡♥t❡s ❡♥tr❡ s✐✳ ▲♦❣♦✱ ♣♦❞❡♠♦s ❡♥❝♦♥✲ tr❛r ✉♠❛ tr✐♣❧❛ (A, B, C) q✉❡ ♦❝♦rr❡ ♣❡❧♦ ♠❡♥♦s ✸ ✈❡③❡s✱ ❞✐❣❛♠♦s (Ak1, Bk1, Ck1)✱ (Ak2, Bk2, Ck2)❡ (Ak3, Bk3, Ck3)✳ P♦rt❛♥t♦✱ ❞❡ ✭✶✳✶✶✮✱ xk1✱xk2 ❡ xk3 sã♦ r❛í③❡s ❞❡

Ay2+By+C = 0✳

➱ ❝❧❛r♦ q✉❡ ♣❡❧♦ ♠❡♥♦s ❞✉❛s ❞❡❧❛s sã♦ ✐❣✉❛✐s✱ ♣♦r ❡①❡♠♣❧♦✱ xk1 ❡ xk2✳ ❊♥tã♦✱

ak2 =ak1✱ak2+1 =ak1+1✱ ak2+2 =ak1+2✱...

P♦rt❛♥t♦✱ ❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ é ♣❡r✐ó❞✐❝❛✳

❈❛♣ít✉❧♦ ✷

◆ú♠❡r♦s ▼❡tá❧✐❝♦s

✷✳✶ ■♥tr♦❞✉çã♦

❖ ♥ú♠❡r♦ ♠❡tá❧✐❝♦ ♠❛✐s ❝♦♥❤❡❝✐❞♦ é ♦ ♥ú♠❡r♦ ❞❡ ♦✉r♦✱φ = 1+₂√5✱ q✉❡ é ❛ r❛✐③ ♣♦s✐t✐✈❛ ❞❛ ❡q✉❛çã♦ x2₋x₋1 = 0✳

❖s ❡❧❡♠❡♥t♦s ❞❡st❛ ❢❛♠í❧✐❛ ❣♦③❛♠ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ♠❛t❡♠át✐❝❛s ❝♦♠✉♥s✱ ♥♦tá✈❡✐s ❡ ❞❡ ✐♥t❡r❡ss❡ ✐♥❞✉❜✐tá✈❡❧✳ ❊stã♦ ♣r❡s❡♥t❡s ♥❛ ♥❛t✉r❡③❛ ❡ s❡r✈✐r❛♠ ❞❡ ❜❛s❡ ♣❛r❛ ❝♦♥str✉çõ❡s q✉❡ ✈ã♦ ❞❡s❞❡ ❛ ❝✐✈✐❧✐③❛çã♦ r♦♠❛♥❛ à ❛t✉❛❧✐❞❛❞❡✳

✷✳✷ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s

❉❡✜♥✐çã♦ ✷✳✶ ❆ ❢❛♠í❧✐❛ ❞♦s ♥ú♠❡r♦s ♠❡tá❧✐❝♦s é ❢♦r♠❛❞❛ ♣❡❧❛s r❛í③❡s ♣♦s✐t✐✈❛s ❞❛s ❡q✉❛çõ❡s ❞❛ ❢♦r♠❛ x2 ₋px₋q = 0✱ ♦♥❞❡ p, q _∈N✳

❊ss❛ ❞❡✜♥✐çã♦ ❢♦✐ ❞❛❞❛ ❡♠ ✶✾✾✹✱ ♣❡❧❛ ❛r❣❡♥t✐♥❛ ❉r❛✳ ❱❡r❛ ❲✳ ❙♣✐♥❛❞❡❧✱ Pr♦❢❡s✲ s♦r❛ ❚✐t✉❧❛r ❊♠ér✐t❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇✉❡♥♦s ❆✐r❡s✳

❖ t❡♦r❡♠❛ ❛❜❛✐①♦ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ú♥✐❝❛ r❛✐③ ♣♦s✐t✐✈❛ ♣❛r❛ ❝❛❞❛ ❝♦♠❜✐♥❛çã♦ p, q✱ ❝♦♠ p, q _∈N✳

❚❡♦r❡♠❛ ✷✳✶ ◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ✈❛❧♦r❡s ❞❡ p, q _∈N✱ ❛ ❡q✉❛çã♦x2₋px₋q = 0 t❡♠ ❛♣❡♥❛s ✉♠❛ ú♥✐❝❛ r❛✐③ ♣♦s✐t✐✈❛✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ ❛s ❢✉♥çõ❡s f ❡g ❞❡✜♥✐❞❛s ♣♦r f(x) = x2 ❡g(x) = px+q✳

◆♦t❡ q✉❡ ❛ ❡q✉❛çã♦x2₋px₋q= 0 ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❝♦♠♦x2 =px+q✳ ❊♥tã♦

❛s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ x2 _{−px−q = 0 sã♦ ❛s ❛❜❝✐ss❛s ❞♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦} ❞♦s ❣rá✜❝♦s f ❡g✳

❖ ❣rá✜❝♦ ❞❡ f é ✉♠❛ ♣❛rá❜♦❧❛ ❝♦♠ ❝♦♥❝❛✈✐❞❛❞❡ ✈♦❧t❛❞❛ ♣❛r❛ ❝✐♠❛✱ ✈ért✐❝❡ ♥❛

♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❝❛rt❡s✐❛♥♦ ❡ ❡✐①♦ ❞❡ s✐♠❡tr✐❛ ✐❣✉❛❧ ❛♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s✳

✷✳✷✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❈❆P❮❚❯▲❖ ✷✳ ◆Ú▼❊❘❖❙ ▼❊❚➪▲■❈❖❙

❋✐❣✉r❛ ✷✳✶✿ ❚❡♦r❡♠❛ ✷✳✶

❖ ❣rá✜❝♦ ❞❡gé ✉♠❛ r❡t❛ ❝♦♠ ❞❡❝❧✐✈❡ ♣♦s✐t✐✈♦ ✐❣✉❛❧ ❛p✱ ♣♦✐sp_∈N✱ q✉❡ ✐♥t❡rs❡❝t❛

♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s ❡♠ q_∈N✳

◆❡❝❡ss❛r✐❛♠❡♥t❡✱ ❡①✐st❡ ❛♣❡♥❛s ✉♠ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❛ r❡t❛ ❝♦♠ ❛ ♣❛rá❜♦❧❛✱ ❝✉❥❛ ❛❜❝✐ss❛ é ♣♦s✐t✐✈❛✳

❆s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ x2_{−px−q = 0 sã♦}

x= p±

p

p2_{+ 4q}

2

❈♦♠♦p, q _∈N✱ ❡♥tã♦

p2_{+ 4q >0 ❡p}2_{+ 4q > p}2

❉❡ p2+ 4q >0✱ s❡❣✉❡ q✉❡ ❛ ❡q✉❛çã♦ s❡♠♣r❡ t❡♠ s♦❧✉çã♦✳

❉❡ p2_{+ 4q > p}2_{✱ s❡❣✉❡ q✉❡}

p

p2_{+ 4q >}p

p2✱ ♦✉ s❡❥❛✱ p

p2_{+ 4q > p}✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

x= p+

p

p2_{+ 4q}

2 é ✉♠❛ r❛✐③ ♣♦s✐t✐✈❛✳

✷✳✸✳ ▼❊❚➪▲■❈❖❙ ■◆❚❊■❘❖❙ ❈❆P❮❚❯▲❖ ✷✳ ◆Ú▼❊❘❖❙ ▼❊❚➪▲■❈❖❙

❡

x= p−

p

p2_{+ 4q}

2 é ✉♠❛ r❛✐③ ♥❡❣❛t✐✈❛✳

P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ q✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ ✈❛❧♦r ❞❡p, q _∈N✱ ❛ ❡q✉❛çã♦ x2₋px₋q= 0 t❡♠ ❛♣❡♥❛s ✉♠❛ r❛✐③ ♣♦s✐t✐✈❛✳

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ ❚❡♦r❡♠❛ ✷✳✶ ❣❛r❛♥t❡ q✉❡ ❛ ❡q✉❛çã♦

x2 ₋px₋q = 0 ❞á ♦r✐❣❡♠ ❛ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ ♠❡tá❧✐❝♦ p+

p

p2_{+ 4q}

2 ✱ q✉❡ r❡♣r❡✲ s❡♥t❛r❡♠♦s ♣♦r σp,q✳

❆❧❣✉♥s ❞❡ss❡s ♥ú♠❡r♦s ♠❡tá❧✐❝♦s tê♠ ♥♦♠❡ ❞❡ ♠❡t❛✐s✳ ❆❧é♠ ❞♦ ❢❛♠♦s♦ ◆ú♠❡r♦ ❞❡ ❖✉r♦✱ ❤á ♦ ◆ú♠❡r♦ ❞❡ Pr❛t❛✱ ❞❡ ❇r♦♥③❡✱ ❞❡ ❈♦❜r❡✱ ❞❡ ◆íq✉❡❧ ❡ ❞❡ P❧❛t✐♥❛✳

◆❛ t❛❜❡❧❛ ❛❜❛✐①♦ ❡♥❝♦♥tr❛♠✲s❡ ❞❡t❛❧❤❡s s♦❜r❡ ❝❛❞❛ ✉♠ ❞♦s ♥ú♠❡r♦s ❝✐t❛❞♦s✳

♣ q ❙❮▼❇❖▲❖ ◆❖▼❊ ❱❆▲❖❘

✶ ✶ φ ◆Ú▼❊❘❖ ❉❊ ❖❯❘❖ 1 +

√ 5 2

✷ ✶ σ₂,1 ◆Ú▼❊❘❖ ❉❊ P❘❆❚❆ 1 +

√ 2

✸ ✶ σ₃,1 ◆Ú▼❊❘❖ ❉❊ ❇❘❖◆❩❊

3 +√13 2

✶ ✷ σ₁,2 ◆Ú▼❊❘❖ ❉❊ ❈❖❇❘❊ ✷

✶ ✸ σ₁,3 ◆Ú▼❊❘❖ ❉❊ ◆❮◗❯❊▲

1 +√13 2

✷ ✷ σ₂,2 ◆Ú▼❊❘❖ ❉❊ P▲❆❚■◆❆ a+

√ 3

◆♦t❡ q✉❡ ❡①✐st❡♠ ♥ú♠❡r♦s ♠❡tá❧✐❝♦s ✐♥t❡✐r♦s✳ ◆❛ ♣ró①✐♠❛ s❡çã♦✱ ✈❛♠♦s ❛♥❛❧✐s❛r q✉❛♥❞♦ ❡❧❡s ❛♣❛r❡❝❡♠✳

✷✳✸ ▼❡tá❧✐❝♦s ✐♥t❡✐r♦s

P❛r❛ q✉❡ ✉♠ ♥ú♠❡r♦ ♠❡tá❧✐❝♦ σp,q =

p+√p2_+4q

2 s❡❥❛ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✱ p

2_{+ 4q}

❞❡✈❡ s❡r ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ ❡ p+pp2_{+ 4q} ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ✷✳

❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r ❣❛r❛♥t❡ q✉❡ ❛ s❡❣✉♥❞❛ ❝♦♥❞✐çã♦ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞❛ ♣r✐♠❡✐r❛✱ ♦✉ s❡❥❛✱ s❡ ❛ ♣r✐♠❡✐r❛ ❝♦♥❞✐çã♦ ❢♦r s❛t✐s❢❡✐t❛✱ ❛ s❡❣✉♥❞❛ t❛♠❜é♠ s❡rá✳ ❚❡♦r❡♠❛ ✷✳✷ ❙❡❥❛♠ p, q ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❙❡ p2+ 4q é ✉♠ ♥ú♠❡r♦ q✉❛❞r❛❞♦

♣❡r❢❡✐t♦✱ ❡♥tã♦ p+pp2_{+ 4q} é ♠ú❧t✐♣❧♦ ❞❡ 2✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛♠♦s q✉❡ p2 + 4q s❡❥❛ ✉♠ ♥ú♠❡r♦ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✳

❊♥tã♦✱ ❡①✐st❡ r_∈N✱ t❛❧ q✉❡ _r2 _=p2_{+ 4q}✳ ❉❛í✱

✷✳✸✳ ▼❊❚➪▲■❈❖❙ ■◆❚❊■❘❖❙ ❈❆P❮❚❯▲❖ ✷✳ ◆Ú▼❊❘❖❙ ▼❊❚➪▲■❈❖❙

r2 =p2+ 4q

r2_−p2 _{= 4q}

(r₋p)(r+p) = 4q

r+p= 4q

r₋p

r+p= 2_· 2q

r₋p

♦✉ s❡❥❛✱ r+p é ♠ú❧t✐♣❧♦ ❞❡ 2✳ ▼❛s r+p=p+pp2_{+ 4q✳} P♦rt❛♥t♦✱ p+pp2_{+ 4q} é ♠ú❧t✐♣❧♦ ❞❡ 2✳

❱❛♠♦s ❛❣♦r❛ ✈❡r✐✜❝❛r ♣❛r❛ q✉❛✐s ✈❛❧♦r❡s ❞❡ p ❡ q ❛ ❡①♣r❡ssã♦ p2 + 4q é ✉♠

q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✳ P❛r❛ ✐st♦✱ ✈❛♠♦s ✜①❛r ♦ ✈❛❧♦r ❞❡ p ❡♠ 1 ❡ ✈❛r✐❛r ♦ ✈❛❧♦r ❞❡q✳

p q p2_{+ 4q ❖❇❙❊❘❱❆➬➹❖}

✶ ✶ ✺

✶ ✷ ✾ é q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦

✶ ✸ ✶✸

✶ ✹ ✶✼

✶ ✺ ✷✶

✶ ✻ ✷✺ é q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦

✶ ✼ ✷✾

✶ ✽ ✸✸

✶ ✾ ✸✼

✶ ✶✵ ✹✶

✶ ✶✶ ✹✺

✶ ✶✷ ✹✾ é q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦

✶ ✶✸ ✺✸

✶ ✶✹ ✺✼

✶ ✶✺ ✻✶

✶ ✶✻ ✻✺

✶ ✶✼ ✻✾

✶ ✶✽ ✼✸

✶ ✶✾ ✼✼

✶ ✷✵ ✽✶ é q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦

◆♦t❡ q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ✈❛❧♦r❡s ❞❡q♣❛r❛ ♦s q✉❛✐sp2+4qé ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦

é

2,6,12,20, ...

✷✳✸✳ ▼❊❚➪▲■❈❖❙ ■◆❚❊■❘❖❙ ❈❆P❮❚❯▲❖ ✷✳ ◆Ú▼❊❘❖❙ ▼❊❚➪▲■❈❖❙

❡ q✉❡ ♦ t❡r♠♦ ❣❡r❛❧✱ qn✱ ❝♦♠ n_∈N✱ é

qn=n(n+ 1)

Pr♦❝❡❞❡♥❞♦ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ❡♥❝♦♥tr❛♠♦s✿

• ♣❛r❛ p= 2✱ qn=n(n+ 2)

• ♣❛r❛ p= 3✱ qn=n(n+ 3)

❚❛✐s r❡s✉❧t❛❞♦s s✉❣❡r❡♠ q✉❡ ♦ t❡r♠♦ ❣❡r❛❧ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ✈❛❧♦r❡s ❞❡ q ♣❛r❛ ♦s

q✉❛✐s p2 _{+ 4q é ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ é q}

n = n(n+p)✱ ♦✉ s❡❥❛✱ q✉❡ ♦s ♥ú♠❡r♦s ♠❡tá❧✐❝♦s σp,n(n+p) sã♦ ✐♥t❡✐r♦s✱ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ✈❛❧♦r❡s ❞❡ n, p ∈ N✳ ❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r ❣❛r❛♥t❡ ✐ss♦✳

❚❡♦r❡♠❛ ✷✳✸ ❚♦❞♦ ♥ú♠❡r♦ ♠❡tá❧✐❝♦ ❞❛ ❢♦r♠❛ σp,n(n+p) é ✐♥t❡✐r♦✱ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ n, p _∈N✳

❉❡♠♦♥str❛çã♦✿ ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❚❡♦r❡♠❛ ✷✳✷✱ ❜❛st❛ ♠♦str❛r q✉❡ p2 + 4q =

p2_{+ 4n(n+p) é ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✳} ❉❡ ❢❛t♦✱

p2+ 4n(n+p) = p2 + 4n2 + 4np

= 4n2+ 4np+p2

= 4(n2+np+p 2

4) = 22_·(n+p

2) 2

= 2_·n+ p 2

2

= (2n+p)2

❆té ❛❣♦r❛✱ ✈✐♠♦s q✉❡ ❤á ♥ú♠❡r♦s ♠❡tá❧✐❝♦s ✐rr❛❝✐♦♥❛✐s ❡ ✐♥t❡✐r♦s✳ ❯♠❛ ✐♥❞❛❣❛çã♦ s✉r❣❡ ♥❡ss❡ ♠♦♠❡♥t♦✿ ❡①✐st❡♠ ♥ú♠❡r♦s ♠❡tá❧✐❝♦s q✉❡ s❡❥❛♠ r❛❝✐♦♥❛✐s ♥ã♦ ✐♥t❡✐r♦s❄ ❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r ❣❛r❛♥t❡ q✉❡ ♥ã♦✳

❚❡♦r❡♠❛ ✷✳✹ ❚♦❞♦ ♥ú♠❡r♦ ♠❡tá❧✐❝♦ é ✉♠ ✐rr❛❝✐♦♥❛❧ q✉❛❞rát✐❝♦ ♦✉ ✉♠ ✐♥t❡✐r♦ ♠❛✐♦r q✉❡ 1✳