ALGORITMOS UTILIZADOS PARA AS QUATRO OPERAÇÕES ELEMENTARES

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛

Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠

▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧

❆❧❣♦r✐t♠♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛s ◗✉❛tr♦ ❖♣❡r❛çõ❡s

❊❧❡♠❡♥t❛r❡s

●r❛❝✐❡❧❧② ❞❛ ❙✐❧✈❛ ❙❛♥t❛♥❛

●♦✐â♥✐❛

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●r❛❝✐❡❧❧② ❞❛ ❙✐❧✈❛ ❙❛♥t❛♥❛

❆❧❣♦r✐t♠♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛s ◗✉❛tr♦

❖♣❡r❛çõ❡s ❊❧❡♠❡♥t❛r❡s

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼ár✐♦ ❏♦sé ❞❡ ❙♦✉③❛

●♦✐â♥✐❛

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Ficha de identificação da obra elaborada pelo autor, através do Programa de Geração Automática do Sistema de Bibliotecas da UFG.

CDU 51 Santana, Gracielly da Silva

Algoritmos utilizados para as Quatro Operações Elementares [manuscrito] / Gracielly da Silva Santana. - 2016.

iv, 54 f.: il.

Orientador: Prof. Dr. Mário José de Souza.

Dissertação (Mestrado) - Universidade Federal de Goiás, Instituto de Matemática e Estatística (IME), Programa de Pós-Graduação em Matemática, Goiânia, 2016.

Bibliografia.

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❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳ ➱ ♣r♦✐❜✐❞❛ ❛ r❡♣r♦❞✉çã♦ t♦t❛❧ ♦✉ ♣❛r❝✐❛❧ ❞❡st❡ tr❛❜❛❧❤♦ s❡♠ ❛ ❛✉t♦r✐③❛çã♦ ❞❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ❞♦ ❛✉t♦r ❡ ❞♦ ♦r✐❡♥t❛❞♦r✳

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❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡ à ❉❡✉s✱ ♣♦r t❡r ♣r♦♣♦r❝✐♦♥❛❞♦ ❛s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❛✈❛♥ç❛r ♠❛✐s ❡ss❡ ❞❡❣r❛✉✳

❆♦ ♠❡✉ ❡s♣♦s♦✱ ❘❛✐ss♦♥ ❆❧✈❡s ❞❛ ❙✐❧✈❛✱ ♣♦r t❡r ❛❝r❡❞✐t❛❞♦✱ ❞❡s❞❡ ♦ ♠♦♠❡♥t♦ ❞❛ ✐♥s❝r✐çã♦ ♥♦ ❡①❛♠❡ ❞❡ ❛❝❡ss♦ à ❝♦♥❝❧✉sã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ q✉❡ ❡ss❡ s♦♥❤♦ s❡r✐❛ ♣♦ssí✈❡❧✳ ❆♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ❚✐♦ ❏❛✐r♦ ❇❛st♦s✱ ♣♦r ❛❝r❡❞✐t❛r❡♠ q✉❡ ❡✉ s❡r✐❛ ❝❛♣❛③ ❞❡ ❛❧❝❛♥ç❛r ♠❛✐s ❡ss❡ ♦❜❥❡t✐✈♦ ❡ ♣♦r t❡r❡♠ ❢❡✐t♦✱ ❝❛❞❛ ✉♠ ❞❡♥tr♦ ❞❡ s✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✱ ❛❧❣♦ ♣❛r❛ ♠❡ ❛❥✉❞❛r ❛ ❝♦♥❝r❡t✐③❛r ❡ss❡ s♦♥❤♦✳

❆♦s ♠❡✉s ♣❛✐s ♣♦r t❡r❡♠ ❢❡✐t♦ ❡✉ ❛❝r❡❞✐t❛r q✉❡ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ é ❛ ♠❛✐♦r r✐q✉❡③❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♠❡✉ P❛✐✱ ❆♥tô♥✐♦ ❇❡♥t♦✱ q✉❡ ♣r♦♣♦r❝✐♦♥❛♥❞♦ ❛♦s ✜❧❤♦s ❝♦♥❞✐çõ❡s ♣❛r❛ ❛♥❣❛r✐❛r ❝♦♥❤❡❝✐♠❡♥t♦✱ ❝♦♥❝❧✉✐✉ ♠✉✐t♦ ❜❡♠ s✉❛ ♠✐ssã♦ ❥✉♥t♦ ❞❡ ♥ós✳

❆♦ ♠❡✉ Pr♦❢❡ss♦r ❖r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ▼ár✐♦ ❏♦sé ❞❡ ❙♦✉③❛✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❝♦♠♣r❡❡♥sã♦ ❡ s✉❣❡stõ❡s✳

❆ ❈❆P❊❙ ♣❡❧❛ ❜♦❧s❛ ❞❡ ❡st✉❞♦s ❝♦♥❝❡❞✐❞❛✳ ❆ t♦❞♦s ✈♦❝ês ♠❡✉ ♠✉✐t♦ ♦❜r✐❣❛❞❛✳

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❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s q✉❛tr♦ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛✲ r❡s ♥♦ ❈♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ◆❛t✉r❛✐s✳ ❱❡r✐✜❝❛r❡♠♦s ❛❧❣✉♥s ❛s♣❡❝t♦s ♣❡rt✐♥❡♥t❡s ❛♦ ❙✐st❡♠❛ ❞❡ ◆✉♠❡r❛çã♦ ❉❡❝✐♠❛❧✱ ❜❡♠ ❝♦♠♦ ❛ ❡①♣❛♥sã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❡ss❡ s✐st❡♠❛✳ ❆♣r♦✈❡✐t❛r❡♠♦s ♣❛r❛ ♠♦str❛r ✉♠ ♣♦✉❝♦ ❞❛ ✉t✐❧✐③❛çã♦ ❞♦ ▼❛t❡r✐❛❧ ❉♦✉r❛❞♦ q✉❡ é ✉♠ r❡❝✉rs♦ ♣❡❞❛❣ó❣✐❝♦ ♠✉✐t♦ út✐❧✱ q✉❛♥❞♦ s❡ tr❛t❛ ❞❡ ❝♦♠♣r❡❡♥❞❡r ♦ ❙✐st❡♠❛ ❞❡ ◆✉♠❡r❛✲ çã♦ ❉❡❝✐♠❛❧ ❡ ❛té ♠❡s♠♦ ♣❛r❛ ❡❢❡t✉❛r♠♦s ✉♠❛ ❞❛s ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s✳ ❆ ♣❛rt✐r ❞❛í ♠♦str❛r❡♠♦s ❛❧❣✉♥s ❛❧❣♦r✐t♠♦s q✉❡ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞♦s ♣❛r❛ r❡s♦❧✈❡r♠♦s ❝❛❞❛ ✉♠❛ ❞❛s q✉❛tr♦ ♦♣❡r❛çõ❡s✿ ❛❞✐çã♦✱ s✉❜tr❛çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❞✐✈✐sã♦✳

P❛❧❛✈r❛s✲❝❤❛✈❡

◆ú♠❡r♦s ◆❛t✉r❛✐s✳ ❙✐st❡♠❛ ❞❡ ◆✉♠❡r❛çã♦ ❉❡❝✐♠❛❧✳ ❆❧❣♦r✐t♠♦✳

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❆❜str❛❝t

❚❤✐s ✇♦r❦ ✐s ♠❡❛♥t t♦ st✉❞② s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❢♦✉r ❡❧❡♠❡♥t❛r② ♦♣❡r❛t✐♦♥s ♦♥ t❤❡ s❡t ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs✳ ❙♦♠❡ r❡❧❡✈❛♥t ❛s♣❡❝ts t♦ ❉❡❝✐♠❛❧ ◆✉♠❜❡r✐♥❣ ❙②st❡♠ ✇✐❧❧ ❜❡ ✈❡r✐✜❡❞ ❛s ✇❡❧❧ ❛s t❤❡ ❡①♣❛♥s✐♦♥ ♦❢ ❛ ♥✉♠❜❡r ♦♥ t❤❛t s②st❡♠✳ ■t ✇✐❧❧ ❛❧s♦ ❞❡♠♦♥str❛t❡ t❤❡ ✉s❛❣❡ ♦❢ t❤❡ ✧●♦❧❞❡♥ ❇❡❛❞s✧✱ ❛ ✈❡r② ✉s❡❢✉❧ ♣❡❞❛❣♦❣✐❝❛❧ r❡s♦✉r❝❡ ✇❤❡♥ ✐t ❝♦♠❡s t♦ ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ ❉❡❝✐♠❛❧ ◆✉♠❜❡r✐♥❣ ❙②st❡♠ ❛♥❞ ❡✈❡♥ ✐ts ✉s❛❣❡ t♦ s♦❧✈❡ ♦♥❡ ♦❢ t❤❡ ❡❧❡♠❡♥t❛r② ♦♣❡r❛t✐♦♥s✳ ❚❤❡r❡❢♦r❡✱ s♦♠❡ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠s t❤❛t ❝❛♥ ❜❡ ✉s❡❞ t♦ s♦❧✈❡ ❡❛❝❤ ♦❢ t❤❡ ❢♦✉r ♦♣❡r❛t✐♦♥s✿ ❛❞❞✐t✐♦♥✱ s✉❜tr❛❝t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥✳

❑❡②✇♦r❞s

◆❛t✉r❛❧ ◆✉♠❜❡rs✳ ❉❡❝✐♠❛❧ ◆✉♠❜❡r✐♥❣ ❙②st❡♠✳ ❆❧❣♦r✐t❤♠✳

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❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ ❖s ◆ú♠❡r♦s ◆❛t✉r❛✐s ✸

✶✳✶ ❆❞✐çã♦ ❡ ▼✉❧t✐♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❙✉❜tr❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❆①✐♦♠❛ ❞❡ ■♥❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹ ❉✐✈✐sã♦ ♥♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹✳✷ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✺ ❙✐st❡♠❛ ❞❡ ◆✉♠❡r❛çã♦ ❉❡❝✐♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✷ ▼❛t❡r✐❛❧ ❉♦✉r❛❞♦ ✶✾

✸ ❆❧❣♦r✐t♠♦s ♣❛r❛ ❛s q✉❛tr♦ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s ✷✶ ✸✳✶ ❆❧❣♦r✐t♠♦ ❞❛ ❛❞✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✶✳✶ ❆❧❣♦r✐t♠♦ ❞❛ ❛❞✐çã♦ ✉s❛♥❞♦ ❡①♣❛♥sã♦ ♥❛ ❜❛s❡ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✶✳✷ ▼ét♦❞♦ ❞❛s s♦♠❛s ♣❛r❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✶✳✸ ▼ét♦❞♦ ❞❛ ❣❡❧♦s✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✶✳✹ ❆❧❣♦r✐t♠♦ ❝♦♥✈❡❝✐♦♥❛❧ ❞❛ ❛❞✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✷ ❆❧❣♦r✐t♠♦ ❞❛ s✉❜tr❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✷✳✶ ❆❧❣♦r✐t♠♦ ❝♦♥✈❡❝✐♦♥❛❧ ❞❛ s✉❜tr❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✷✳✷ ❆❧❣♦r✐t♠♦ ❞❡ ✐❣✉❛❧❞❛❞❡ ❞❡ ❛❞✐çõ❡s ♦✉ ❛❧❣♦r✐t♠♦ ❞❡ ❝♦♠♣❡♥s❛çã♦ ✸✸ ✸✳✷✳✸ ❆❧❣♦r✐t♠♦ ❞❡ ❞✐❢❡r❡♥ç❛s ♣❛r❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✷✳✹ ❆❧❣♦r✐t♠♦ ❆❞❞✐♥❣ ✉♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷✳✺ ❆❧❣♦r✐t♠♦ ❞❛ s✉❜tr❛çã♦ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❞✐r❡✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✸ ❆❧❣♦r✐t♠♦ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

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✸✳✸✳✶ ❆❧❣♦r✐t♠♦ ❞❛ ❞❡❝♦♠♣♦s✐çã♦ ♣❛r❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✸✳✷ ❆❧❣♦r✐t♠♦ ✉s✉❛❧ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✸✳✸ ❆❧❣♦r✐t♠♦ ❞❡ ❣❡❧♦s✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✹ ❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✹✳✶ ❆❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❞❛ ❞✐✈✐sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✹✳✷ ❆❧❣♦r✐t♠♦ ❞❡ ❞✐✈✐sõ❡s s✉❝❡ss✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✹✳✸ ❆❧❣♦r✐t♠♦ ❞❡ ❞❡❝♦♠♣♦s✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✺✶

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✸

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■♥tr♦❞✉çã♦

▼✉✐t♦s sã♦ ♦s ♥♦ss♦s ❝♦♥✢✐t♦s q✉❛♥❞♦ ♥♦s ❞❡♣❛r❛♠♦s ❝♦♠ ❛ r♦t✐♥❛ ❞❡ ✉♠ ❛♠❜✐❡♥t❡ ❡s❝♦❧❛r✳ ❱✐✈❡♥❝✐❛♠♦s s✐t✉❛çõ❡s q✉❡ ♥♦s ✐♥❝♦♠♦❞❛♠ ❡ ♥♦s ❢❛③❡♠ ❜✉s❝❛r ❛❧t❡r♥❛t✐✈❛s ♣❛r❛ ♠✐♥✐♠✐③❛r ❡ss❡ ✐♥❝ô♠♦❞♦✳ ❯♠❛ ❞❡❧❛s é ♦ ❢❛t♦ ❞❡ q✉❡ ♥❛s sér✐❡s ✐♥✐❝✐❛✐s ♦s ♣r♦✲ ❢❡ss♦r❡s q✉❡ ✐rã♦ ✏❛❧❢❛❜❡t✐③❛r ♠❛t❡♠❛t✐❝❛♠❡♥t❡✑ ♦s ❛❧✉♥♦s ❡♠ s✉❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ sã♦ ♣❡❞❛❣♦❣♦s✱ ♦✉ s❡❥❛✱ ♥ã♦ ♠❛t❡♠át✐❝♦s✳ ❊♠ s✉❛ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛ ♠❛t❡♠át✐❝❛ ❢♦✐ ✉♠❛ ❞✐s❝✐♣❧✐♥❛ ❡st✉❞❛❞❛ ❡♠ ♣♦✉❝♦s s❡♠❡str❡s✳ ❈♦♠ ✐ss♦✱ ❡❧❡ ✐rá ❡♥s✐♥❛r ♦s ❝♦♥t❡ú❞♦s às ❝r✐✲ ❛♥ç❛s ❝♦♥t❛♥❞♦ ❝♦♠ ♦ ♣♦✉❝♦ q✉❡ ❛♣r❡♥❞❡✉ ♥❛ ❣r❛❞✉❛çã♦ ❛❧✐❛❞❛ ❛♦ ♣ré✲❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ ❡❧❡ ❛❞q✉✐r✐✉ ❞✉r❛♥t❡ ❛ ✈✐❞❛✳ ◆ã♦ é ❞✐❢í❝✐❧ t❛♠❜é♠ ❡♥❝♦♥tr❛r♠♦s✱ ❡♠ ✉♠❛ ❡s✲ ❝♦❧❛✱ ✉♠❛ s❛❧❛ ❞❡ ❛✉❧❛ q✉❡ t❡♥❤❛ ❛❧✉♥♦s ❡♠ ❞✐❢❡r❡♥t❡s ♠♦♠❡♥t♦s ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♠ ❡s❝♦❧❛s ♣ú❜❧✐❝❛s✳ ◆❡ss❡s ❛♠❜✐❡♥t❡s ♦s ❛❧✉♥♦s sã♦✱ ❡♠ s✉❛ ♠❛✐♦r✐❛✱ ❛❣r✉♣❛❞♦s ♣♦r ✐❞❛❞❡ ❡ ♥ã♦ ♣❡❧♦ ❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ ♦ ❛❧✉♥♦ ✈❡♥❤❛ ❛ ♣♦ss✉✐r✳ ❈♦♠ ✐ss♦✱ ♥ós ♣r♦❢❡ss♦r❡s✱ ❧✐❞❛♠♦s ❝♦♠ ✉♠❛ s❛❧❛ ♠✉✐t♦ ❤❡t❡r♦❣ê♥❡❛✱ ❡♠ q✉❡ t❡♠♦s ❛❧✉♥♦s q✉❡ ❥á s❡ ❛♣♦❞❡r❛r❛♠ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♣❛r❛ àq✉❡❧❛ sér✐❡ ❡♠ q✉❡stã♦✱ ♦✉tr♦s ♥❡♠ t❛♥t♦ ❡ ❛q✉❡❧❡s q✉❡ s❡ ❡♥❝♦♥tr❛♠ ❡♠ ✉♠ ♥í✈❡❧ ♠✉✐t♦ ✐♥❢❡r✐♦r ❛♦ ❡s♣❡r❛❞♦✳

❈♦♠ r❡❧❛çã♦ à ❞✐s❝✐♣❧✐♥❛ ❞❡ ▼❛t❡♠át✐❝❛ ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ♣♦✐s é ♥❡❧❛ q✉❡ ✐r❡♠♦s ❢♦❝❛r ♥♦ss♦ tr❛❜❛❧❤♦✱ t❡♠♦s ❡st✉❞❛♥t❡s ❡♠ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ sér✐❡ ❞❛ ú❧t✐♠❛ ❡t❛♣❛ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ♣r❡st❡s ❛ ✐♥❣r❡ss❛r ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ q✉❡ ♥ã♦ ❞♦♠✐♥❛♠ ♥❡♠ ❛ ♦♣❡r❛çã♦ ❞❡ ❛❞✐çã♦✱ q✉✐çá ❛ ❞✐✈✐sã♦✳ ❈♦♠ ✐ss♦ t❡♠♦s ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡♥tr♦ ❞❛ ♥♦ss❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ s❛❧❛ ❞❡ ❛✉❧❛ r❡❛❧✐③❛r♠♦s ♠♦♠❡♥t♦s ❞❡ r❡❢♦rç♦ ❝♦♠ ♦s ❛❧✉♥♦s q✉❡ ❛♣r❡✲ s❡♥t❛♠ ❢❛❧❤❛s ♥♦ ❝♦♥t❡ú❞♦✳ ◆♦r♠❛❧♠❡♥t❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡st✉❞❛♥t❡s q✉❡ ♣♦ss✉❡♠ ❧✐♠✐t❛çõ❡s é ❣r❛♥❞❡✱ ♥❡❝❡ss✐t❛♥❞♦ ❞♦ ❛✉①í❧✐♦ ❞❡ t♦❞♦ ♦ ❣r✉♣♦ ❞❡ ♣r♦❢❡ss♦r❡s✱ ♣❛r❛ s❛✲ ♥❛r ♦ ♠á①✐♠♦ ❞❡ ❞✐✜❝✉❧❞❛❞❡s✳ P❡r❝❡❜❡♠♦s q✉❡ ♦ ♠í♥✐♠♦ q✉❡ ♦ ❛❧✉♥♦ ❞❡✈❡ ❞♦♠✐♥❛r✱ ♥❡st❛ ❡t❛♣❛ ❞♦ ❡♥s✐♥♦✱ sã♦ ❛s q✉❛tr♦ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s ❡ té❝♥✐❝❛s ♣❛r❛ ❝♦♥s❡❣✉✐r ❡❢❡t✉á✲❧❛s✳ ❚❛✐s té❝♥✐❝❛s t❛♠❜é♠ sã♦ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❛❧❣♦r✐t♠♦s✱ q✉❡ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❞✐❝✐♦♥ár✐♦ ❆✉ré❧✐♦✱ é ❞❡✜♥✐❞❛ ❝♦♠♦ ✏♦ ♣r♦❝❡ss♦ ❞❡ ❝á❧❝✉❧♦✱ ♦✉ ❞❡ r❡s♦❧✉çã♦ ❞❡ ✉♠ ❣r✉♣♦ ❞❡ ♣r♦❜❧❡♠❛s s❡♠❡❧❤❛♥t❡s✱ ❡♠ q✉❡ s❡ ❡st✐♣✉❧❛♠✱ ❝♦♠ ❣❡♥❡r❛❧✐❞❛❞❡ ❡ s❡♠ r❡str✐✲

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çõ❡s✱ r❡❣r❛s ❢♦r♠❛✐s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ r❡s✉❧t❛❞♦✱ ♦✉ ❞❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✑✳

❈♦♠♦ ♦s ♣r♦❢❡ss♦r❡s q✉❡ tr❛❜❛❧❤❛rã♦ ❝♦♠ ❡st❡s ❛❧✉♥♦s q✉❡ ❛♣r❡s❡♥t❛♠ ❞✐✜❝✉❧❞❛❞❡s ❝♦♠ r❡❧❛çã♦ à ♠❛t❡♠át✐❝❛ ❜ás✐❝❛ ♥ã♦ s❡rã♦ ❛♣❡♥❛s ♦s ❞❡ ♠❛t❡♠át✐❝❛✱ ♣♦✐s ♦ ♥ú♠❡r♦ ❞❡ ❛❧✉♥♦s é ❣r❛♥❞❡✱ s✉r❣❡ ♥❡st❡ ♠♦♠❡♥t♦ ✉♠ ♣r♦❜❧❡♠❛✳ ◆❡♠ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s s❡ s❡♥t❡♠ ❝♦♥❢♦rtá✈❡✐s ♣❛r❛ tr❛❜❛❧❤❛r ❝♦♠ ❛s q✉❛tr♦ ♦♣❡r❛çõ❡s✳ ❆❧❣✉♥s ♣♦rq✉❡ ❥á ♣♦ss✉❡♠ ❛q✉❡❧❡ ❡st✐❣♠❛ ❞❡ q✉❡ ♠❛t❡♠át✐❝❛ é ❞✐❢í❝✐❧ ❞❡ s❡r ❡♥s✐♥❛❞❛ ❡ ♦✉tr♦s ♣♦rq✉❡✱ ❞❡❞✐❝❛♥❞♦✲ s❡ ❛♣❡♥❛s à ❞✐s❝✐♣❧✐♥❛ ♣❛r❛ q✉❛❧ ❡❧❡ ❢♦✐ ❡❢❡t✐✈❛❞♦✱ ♠✉✐t♦ t❡♠♣♦ s❡ ♣❛ss♦✉ ❞❡s❞❡ ♦ ú❧t✐♠♦ ❝♦♥t❛t♦ ❝♦♠ t❛✐s ❝♦♥t❡ú❞♦s ❞❛ ♠❛t❡♠át✐❝❛✱ ♥ã♦ r❡❝♦r❞❛♥❞♦ ❞♦s ❛❧❣♦r✐t♠♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ s❡ ❡♥s✐♥❛r ❛s q✉❛tr♦ ♦♣❡r❛çõ❡s✳

❉✐❛♥t❡ ❛ ✐♥s❡❣✉r❛♥ç❛✱ ♦s ♣r♦❢❡ss♦r❡s ❞❛s ♦✉tr❛s ❞✐s❝✐♣❧✐♥❛s ♣r❡❝✐s❛♠ r❡❝♦r❞❛r ❛ ♠❡❧❤♦r ❢♦r♠❛ ♣❛r❛ s❡ ❡♥s✐♥❛r ♦ ❝♦♥t❡ú❞♦ ❡s❝♦❧❤✐❞♦ ♣❛r❛ ♦ s❡✉ ❣r✉♣♦ ❞❡ ❛❧✉♥♦s✳ ➱ ♥❡st❡ ♠♦♠❡♥t♦ q✉❡ ❡♥tr❛ ♦ ♥♦ss♦ tr❛❜❛❧❤♦ ❡♠ q✉❡stã♦✱ ✏❆❧❣♦r✐t♠♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛s q✉❛tr♦ ♦♣❡r❛çõ❡s✑✳ ❘❡❝♦r❞❛r ❝♦♠ ♦s ❝♦❧❡❣❛s ❛s ❢♦r♠❛s ❞❡ s❡ ❡♥s✐♥❛r ❛ ❛❞✐çã♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❣❛r❛♥t❡ ❛ ❡❧❡s s❡❣✉r❛♥ç❛ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ✉♠ ♠❡❧❤♦r ❛♣r❡♥❞✐③❛❞♦ ♣♦r ♣❛rt❡ ❞♦ ❛❧✉♥♦✳ ❊ ❝♦♠♦ s❡ tr❛t❛ ❞❡ ❡♥s✐♥❛r ✉♠ ❛❧✉♥♦ q✉❡ ❡stá ❛tr❛s❛❞♦ ❡♠ s❡✉ ❝♦♥t❡ú❞♦✱ ♥ã♦ ♣♦❞❡♠♦s ❝♦rr❡r ♦ r✐s❝♦ ❞❡ ❡♥s✐♥❛r ❡rr❛❞♦ ♦✉ ♠❡s♠♦ r❡♣❛ss❛r ❞ú✈✐❞❛s q✉❡ sã♦ ♥♦ss❛s✱ ❛♦s ❛❧✉♥♦s✳

❈♦♠ ✈ár✐❛s té❝♥✐❝❛s ❛♦ s❡✉ ❞✐s♣♦r ❡ ❛♥❛❧✐s❛♥❞♦ ❛ r❡❛❧✐❞❛❞❡ ❞❡ s❡✉s ❛❧✉♥♦s✱ ❜❛st❛r✐❛ ❡s❝♦❧❤❡r q✉❛❧ ❣❛r❛♥t✐r✐❛ ♠❡❧❤♦r ♦ ❛♣r❡♥❞✐③❛❞♦✳ P❛r❛ ❝❛❞❛ ♦♣❡r❛çã♦✱ ❡♥❝♦♥tr❛♠♦s ❞✐✈❡r✲ s❛s ❢♦r♠❛s ❞❡ t❡♥t❛r ❢❛③❡r ❝♦♠ q✉❡ ♦ ❛❧✉♥♦ ❡❧❛❜♦r❡ ✉♠ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦ ♣❛r❛ r❡s♦❧✈ê✲❧❛✳ ◆♦ ❝♦♥t❡①t♦ ❞❡ q✉❡ ❡st❛♠♦s tr❛❜❛❧❤❛♥❞♦ ❝♦♠ ❛❧✉♥♦s ❛ ♥í✈❡❧ ❞❡ r❡❢♦rç♦ t❡♠♦s q✉❡ ❝♦♥s✐❞❡r❛r ♦ ♣ré✲❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ ♦ ♠❡s♠♦ ❝❛rr❡❣❛ ♣❛r❛ ♣♦❞❡r♠♦s ❛♣r♦✈❡✐t❛r s❡✉ ❝♦✲ ♥❤❡❝✐♠❡♥t♦ ✐♥❢♦r♠❛❧ ❛❝❡r❝❛ ❞♦ ❛ss✉♥t♦ ❡♠ q✉❡stã♦✳ ❙♦♠❛♥❞♦ t✉❞♦ ✐st♦✱ ❛s ❝❤❛♥❝❡s ❞❡ s✉❝❡ss♦ ♥♦ r❡❢♦rç♦ ❛✉♠❡♥t❛♠✳

Pr✐♠❡✐r❛♠❡♥t❡✱ ♠♦str❛r❡♠♦s ❛s q✉❛tr♦ ♦♣❡r❛çõ❡s✱ ❞❡♥tr♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❢♦r♠❛❧ ❡ t❡ór✐❝❛✳ ❆♣r❡s❡♥t❛r❡♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s✱ ♣r♦♣♦s✐✲ çõ❡s✱ ❝♦r♦❧ár✐♦s ❡ t❡♦r❡♠❛s q✉❡ ❝♦♥s♦❧✐❞❛♠ ❡st❛ t❡♦r✐❛✱ ❢❡❝❤❛♥❞♦ ❡ss❡ ❝❛♣ít✉❧♦ ❝♦♠ ✉♠ ♣❡q✉❡♥♦ ❤✐stór✐❝♦ ❞♦ s✐st❡♠❛ ❞❡ ♥✉♠❡r❛çã♦ ❞❡❝✐♠❛❧✳

❈♦♥t✐♥✉❛r❡♠♦s ♥♦ss♦ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♥❞♦ ✉♠ ❜r❡✈❡ ❝❛♣ít✉❧♦ s♦❜r❡ ♦ s✉r❣✐♠❡♥t♦ ❞♦ ▼❛t❡r✐❛❧ ❉♦✉r❛❞♦ ❡ s✉❛ ♣r♦♣♦st❛ ❞❡ ✉t✐❧✐③❛çã♦✱ ❜❡♠ ❝♦♠♦ s❡✉s ❝♦♠♣♦♥❡♥t❡s✳

P♦r ✜♠✱ ✐r❡♠♦s ❛♣r❡s❡♥t❛r s✐t✉❛çõ❡s ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈❛♠ ❛s q✉❛tr♦ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s ❡ s✉❛s r❡s♦❧✉çõ❡s s❡❥❛ ♣♦r ✉♠❛ ♠❛♥❡✐r❛ ❧ú❞✐❝❛✱ ♣♦r ✈❡③❡s ❝♦♠ ❛ ✉t✐❧✐③❛çã♦ ❞♦ ▼❛t❡r✐❛❧ ❉♦✉r❛❞♦✱ ♦✉ ✉t✐❧✐③❛♥❞♦ ❛❧❣✉♠ ❛❧❣♦r✐t♠♦ ✭té❝♥✐❝❛✮ ❡s♣❡❝í✜❝❛✳

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❈❛♣ít✉❧♦ ✶

❖s ◆ú♠❡r♦s ◆❛t✉r❛✐s

❖s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❢♦r♠❛♠ ✉♠ ❞♦s ❝♦♥❝❡✐t♦s ♠❛✐s ❛♥t✐❣♦s ❝♦♥❝❡❜✐❞♦s ♣❡❧♦ s❡r ❤✉♠❛♥♦✳ ❊♥tr❡t❛♥t♦✱ ❛ s✉❛ ❡✈♦❧✉çã♦ ❞❡ ✉♠❛ ♥♦çã♦ ✐♥t✉✐t✐✈❛ ♣❛r❛ ✉♠ ❝♦♥❝❡✐t♦ ♠❛✐s ❡❧❛❜♦r❛❞♦ ❢♦✐ ♠✉✐t♦ ❧❡♥t❛✳ ❙ó ♥♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ✶✾✱ q✉❛♥❞♦ ♦s ❢✉♥❞❛♠❡♥t♦s ❞❡ t♦❞❛ ❛ ♠❛t❡♠át✐❝❛ ❢♦r❛♠ q✉❡st✐♦♥❛❞♦s ❡ ✐♥t❡♥s❛♠❡♥t❡ r❡♣❡♥s❛❞♦s✱ é q✉❡ ❛ ♥♦çã♦ ❞❡ ♥ú✲ ♠❡r♦ ♣❛ss♦✉ ❛ s❡r ❜❛s❡❛❞❛ ❡♠ ❝♦♥❝❡✐t♦s ❞❛ t❡♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s✱ ❝♦♥s✐❞❡r❛❞♦s ♠❛✐s ♣r✐♠✐t✐✈♦s✳

◆❡st❡ ❝❛♣ít✉❧♦ ♥ã♦ ♥♦s ❛t❡r❡♠♦s à q✉❡stã♦ ❤✐stór✐❝❛ ❡ s✐♠ ♠♦str❛r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❝✐r❝✉♥❞❛♠ t❛❧ ❝♦♥❥✉♥t♦✳

P❛rt✐r❡♠♦s ❞♦ ♣r✐♥❝í♣✐♦ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s s❡rá✿

N❂④✵✱✶✱✷✱✸✱✳✳✳⑥

❡ q✉❡ ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♥tr❡ ❛ ❡ ❜ s❡rã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦ra+b

❡a.b✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❝♦♥❝❡✐t♦s ♣r❡❧✐♠✐♥❛r❡s✱ ♥❡❝❡ssár✐♦s✱ ♣❛r❛ ♦ ❞❡s❡♥✲ ✈♦❧✈✐♠❡♥t♦ ❞❡ss❡ tr❛❜❛❧❤♦✳ ❖❜s❡r✈❛♠♦s q✉❡ ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s✱ ♥♦ ❞❡❝♦rr❡r ❞❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ♦♠✐t✐❞❛s✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s ❛ r❡s♣❡✐t♦ ❞♦ ❛ss✉♥t♦ ❛q✉✐ ❞❡s❡♥✈♦❧✈✐❞♦ ✐♥❞✐❝❛♠♦s ❬✺❪ ❡ ❬✻❪✳

✶✳✶ ❆❞✐çã♦ ❡ ▼✉❧t✐♣❧✐❝❛çã♦

✶✮ ❆ ❛❞✐çã♦ ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ sã♦ ❜❡♠ ❞❡✜♥✐❞❛s✿

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∀ a, b, a′, bNa=a′ ❡ b=b=a+b=a+b′ ❡ a.b=a.b′✳

✷✮ ❆ ❛❞✐çã♦ ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ sã♦ ❝♦♠✉t❛t✐✈❛s✿

∀ a, b∈N✱a+b =b+a ❡ a.b=b.a✳

✸✮ ❆ ❛❞✐çã♦ ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ sã♦ ❛ss♦❝✐❛t✐✈❛s✿

∀ a, b, c∈N✱ (a+b) +c=a+ (b+c) ❡ (a.b).c=a.(b.c)✳

✹✮ ❆ ❛❞✐çã♦ ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦ss✉❡♠ ❡❧❡♠❡♥t♦s ♥❡✉tr♦s✿

∀ a∈N✱ a+ 0 =a ❡ a.1 = a✳

✺✮ ❆ ♠✉❧t✐♣❧✐❝❛çã♦ é ❞✐str✐❜✉t✐✈❛ ❝♦♠ r❡❧❛çã♦ à ❛❞✐çã♦✿

∀ a, b, c∈N✱ a.(b+c) = a.b+a.c✳

❆ Pr♦♣r✐❡❞❛❞❡ ✶✮ é q✉❡ ♣❡r♠✐t❡ s♦♠❛r✱ ❛ ❛♠❜♦s ♦s ❧❛❞♦s ❞❡ ✉♠❛ ✐❣✉❛❧❞❛❞❡✱ ✉♠ ❞❛❞♦ ♥ú♠❡r♦✱ ♦✉ ♠✉❧t✐♣❧✐❝❛r ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣♦r ✉♠ ♠❡s♠♦ ♥ú♠❡r♦✳

❆❧❣✉♠❛s ✈❡③❡s tr❛❜❛❧❤❛r❡♠♦s ❝♦♠ ♦✉tr♦s ❝♦♥❥✉♥t♦s✱ ❞✐❢❡r❡♥t❡s ❞♦s ♥❛t✉r❛✐s✱ ♠✉✲ ♥✐❞♦s ❞❡ ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ q✉❡ ♣♦ss✉❡♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✶✮ ❛ ✺✮ ❛❝✐♠❛✳ ◆❡st❡ ❝❛s♦✱ ❞✐r❡♠♦s q✉❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ t❛✐s ❝♦♥❥✉♥t♦s✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛s ❞✉❛s ♦♣❡r❛çõ❡s✱ ❡stã♦ s✉❥❡✐t♦s às ❧❡✐s ❜ás✐❝❛s ❞❛ ❛r✐t♠ét✐❝❛✳ P♦r ❡①❡♠♣❧♦✱ s❛❜❡♠♦s q✉❡ ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s r❡❧❛t✐✈♦s✱ ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ ♦s ♥ú♠❡r♦s r❡❛✐s ❡ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡stã♦ s✉❥❡✐t♦s às ❧❡✐s ❜ás✐❝❛s ❞❛ ❛r✐t♠ét✐❝❛✳ ❆❧❡rt❛♠♦s ♦ ❧❡✐t♦r q✉❛♥t♦ ❛♦ ❢❛t♦ ❞❡ q✉❡ ❡st❡s ♥ú♠❡r♦s só s❡rã♦ ✉t✐❧✐③❛❞♦s ♥♦s ❡①❡♠♣❧♦s ❡ ♣r♦❜❧❡♠❛s❀ ♥✉♥❝❛✱ ♣♦ré♠✱ ❡♠ ❧✉❣❛r ❡ss❡♥❝✐❛❧ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛✳

❯s❛r❡♠♦s ❛ ♥♦t❛çã♦

N❂④✵✱✶✱✷✱✸✱✳✳✳⑥

❱❛♠♦s ❛❞♠✐t✐r✱ t❛♠❜é♠✱ q✉❡ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣♦ss✉❛♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❛ s❡❣✉✐r✿ ✻✮ ■♥t❡❣r✐❞❛❞❡✿ ❉❛❞♦sa, b∈N∗ t❡♠✲s❡ q✉❡a.bN∗✳

❊q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ♣❡❧❛ ❢♦r♠✉❧❛çã♦ ❝♦♥tr❛♣♦s✐t✐✈❛✿

∀ a, b∈N✱a.b= 0⇐⇒a= 0 ♦✉b= 0✳

✼✮ ❚r✐❝♦t♦♠✐❛✿ ❉❛❞♦s a, b∈ N✱ ✉♠❛✱ ❡ ❛♣❡♥❛s ✉♠❛✱ ❞❛s s❡❣✉✐♥t❡s ♣♦ss✐❜✐❧✐❞❛❞❡s é

✈❡r✐✜❝❛❞❛✿

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✐✮ a=b

✐✐✮∃ c∈N∗ t❛❧ q✉❡ b=a+c

✐✐✐✮∃ c∈N∗ t❛❧ q✉❡ a=b+c

❉✐r❡♠♦s q✉❡ ❛ é ♠❡♥♦r q✉❡ ❜✱ s✐♠❜♦❧✐③❛❞♦ ♣♦r ❛❁❜✱ t♦❞❛ ✈❡③ q✉❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ✭✐✐✮ ❛❝✐♠❛ é ✈❡r✐✜❝❛❞❛✳

❈♦♠ ❡st❛ ❞✐✜♥✐çã♦✱ t❡♠♦s q✉❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ✭✐✐✐✮ ❛❝✐♠❛ ❡q✉✐✈❛❧❡ ❛ ❛✜r♠❛r q✉❡ ❜❁❛✳ ❆ss✐♠✱ ❛ tr✐❝♦t♦♠✐❛ ♥♦s ❞✐③ q✉❡✱ ❞❛❞♦s a, b ∈ N ✉♠❛✱ ❡ s♦♠❡♥t❡ ✉♠❛✱ ❞❛s s❡❣✉✐♥t❡s

❝♦♥❞✐çõ❡s é ✈❡r✐✜❝❛❞❛✿

✐✮ a=b

✐✐✮a < b

✐✐✐✮ b < a

❯t✐❧✐③❛r❡♠♦s ❛ ♥♦t❛çã♦ ❜❃❛✱ q✉❡ s❡ ❧ê ❜ é ♠❛✐♦r ❞♦ q✉❡ ❛✱ ♣❛r❛ r❡♣r❡s❡♥t❛r ❛❁❜✳ ❉❡❝♦rr❡ ❞❛s ❞❡✜♥✐çõ❡s q✉❡ ✵❁❛✱ ♣❛r❛ t♦❞♦a∈N∗✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦aN∗ t❡♠♦s

q✉❡ ✵✰❛❂❛ ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ✵❁❛✳

❚❡♠♦s t❛♠❜é♠ q✉❡ a+b= 0✱ ❡♥tã♦ a=b = 0✳ ❉❡ ❢❛t♦✱ s❡ a6= 0 t❡rí❛♠♦s ❜❁✵✱ ♦

q✉❡ é ❛❜s✉r❞♦✱ ❧♦❣♦ ❛❂✵✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♠♦str❛✲s❡ q✉❡ ❜❂✵✳ P♦rt❛♥t♦✱ s❡ a ∈ N∗ ♦✉

b∈N∗✱ ❡♥tã♦ a+bN∗✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✳ P❛r❛ t♦❞♦ a∈N✱ a.0 = 0 ✳

❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡

a.0 =a(0 + 0) =a.0 +a.0✳

❙❡ a.0 6= 0✱ t❡rí❛♠♦s q✉❡ a.0 ∈ N∗ ❡✱ ♣♦rt❛♥t♦✱ s❡❣✉✐r✐❛✱ ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ q✉❡

a.0❃ a.0✱ ♦ q✉❡ é ❛❜s✉r❞♦✳ ▲♦❣♦ a.0 = 0✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✷✳ ❆ r❡❧❛çã♦ ✏♠❡♥♦r ❞♦ q✉❡✑ é tr❛♥s✐t✐✈❛

∀ a, b, c∈N✱ a❁b ❡ b❁c =⇒ a❁c✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❞♦a❁b ❡b❁c✱ t❡♠♦s q✉❡ ❡①✐st❡♠ d, f ∈N∗ t❛✐s q✉❡ b=a+d

c=b+f✳ ▲♦❣♦✱ ✉s❛♥❞♦ ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❛ ❛❞✐çã♦✱ t❡♠♦s q✉❡ c=b+f = (a+d) +f =a+ (d+f)✱

❝♦♠ d+f ∈N∗✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ac

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Pr♦♣♦s✐çã♦ ✶✳✶✳✸✳ ❆ ❛❞✐çã♦ é ❝♦♠♣❛tí✈❡❧ ❡ ❝❛♥❝❡❧❛t✐✈❛ ❝♦♠ r❡s♣❡✐t♦ à r❡❧❛çã♦ ✏♠❡♥♦r ❞♦ q✉❡✑

∀ a, b, c∈N✱ a❁b ⇐⇒ a+c❁b+c✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ a❁b✳ ▲♦❣♦✱ ❡①✐st❡ d ∈ N∗ t❛❧ q✉❡ b = a+d✳ ❙♦♠❛♥❞♦

c ❛ ❛♠❜♦s ♦s ❧❛❞♦s ❞❡st❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡✱ ♣❡❧❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❛

❛❞✐çã♦✱ t❡♠♦s

b+c=c+b=c+ (a+d) = (c+a) +d= (a+c) +d✱

♦ q✉❡ ♠♦str❛ q✉❡a+c❁b+c✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡a+c❁b+c✳ P❡❧❛ tr✐❝♦t♦♠✐❛✱ t❡♠♦s três ♣♦ss✐❜✐❧✐❞❛❞❡s✿

✭✐✮a=b✳ ■st♦ ❛❝❛rr❡t❛r✐❛ a+c=b+c✱ ♣♦rt❛♥t♦ ❢❛❧s♦✳ ✭✐✐✮ b❁a✳ ■ss♦ ❛❝❛rr❡t❛r✐❛✱ ♣❡❧❛

♣r✐♠❡✐r❛ ♣❛rt❡ ❞❛ ❞❡♠♦♥str❛çã♦✱ q✉❡b+c❁a+c❀ t❛♠❜é♠ é ❢❛❧s♦✳ ✭✐✐✐✮ a❁b✳ ❊st❛ é ❛

ú♥✐❝❛ ♣♦ss✐❜✐❧✐❞❛❞❡ q✉❡ r❡st❛✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✹✳ ❆ ♠✉❧t✐♣❧✐❝❛çã♦ é ❝♦♠♣❛tí✈❡❧ ❡ ❝❛♥❝❡❧❛t✐✈❛ ❝♦♠ r❡s♣❡✐t♦ à r❡❧❛çã♦ ✏♠❡♥♦r ❞♦ q✉❡✑✱ ♦✉ s❡❥❛✱

∀ a, b∈N✱ ❡ c∈N∗✱ ab ⇐⇒ a.cb.c

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ a❁b✳ ▲♦❣♦✱ ❡①✐st❡ d ∈ N∗✱ t❛❧ q✉❡ b = a+d✳ ▼✉❧t✐♣❧✐✲

❝❛♥❞♦ ♣♦rc❛ ❛♠❜♦s ♦s ❧❛❞♦s ❞❡ss❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡✱ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❝♦♠✉t❛t✐✈❛ ❡

❞✐str✐❜✉t✐✈❛ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❞❡❝♦rr❡

b.c=c.b=c.(a+d) =c.a+c.d=a.c+c.d✱

♦ q✉❡ ♠♦str❛ q✉❡a.c❁b.c✱ ♣♦✐s✱ ♣❡❧❛ ✐♥t❡❣r✐❞❛❞❡✱ c.d∈N∗✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ a.c❁b.c✳ P❡❧❛ tr✐❝♦t♦♠✐❛✱ t❡♠♦s três ♣♦ss✐❜✐❧✐❞❛❞❡s✿

✭✐✮ a = b✳ ■ss♦ ❛❝❛rr❡t❛r✐❛ a.c = b.c✱ ♣♦rt❛♥t♦ ❢❛❧s♦✳ ✭✐✐✮ b❁a✳ ■ss♦ ❛❝❛rr❡t❛r✐❛✱ ♣❡❧❛

♣r✐♠❡✐r❛ ♣❛rt❡ ❞❛ ❞❡♠♦♥str❛çã♦✱ q✉❡b.c❁a.c❀ t❛♠❜é♠ é ❢❛❧s♦✳ ✭✐✐✐✮a❁b✳ ❊st❛ é ❛ ú♥✐❝❛

♣♦ss✐❜✐❧✐❞❛❞❡ q✉❡ r❡st❛✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✺✳ ❆ ❛❞✐çã♦ é ❝♦♠♣❛tí✈❡❧ ❡ ❝❛♥❝❡❧❛t✐✈❛ ❝♦♠ r❡s♣❡✐t♦ à ✐❣✉❛❧❞❛❞❡

∀ a, b, c∈N✱ a❂b ⇐⇒ a+c❂b+c✳

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❉❡♠♦♥str❛çã♦✳ ❆ ✐♠♣❧✐❝❛çã♦ a❂b =⇒ a+c❂b+c é ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❢❛t♦ ❞❛ ❛❞✐çã♦

s❡r ❜❡♠ ❞❡✜♥✐❞❛ ✭Pr♦♣r✐❡❞❛❞❡ ✶✮✳

❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ a+c=b+c✳ ❚❡♠♦s três ♣♦ss✐❜✐❧✐❞❛❞❡s✿

✭✐✮a❁b✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✸✱ t❡♠♦s q✉❡a+c❁b+c✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳

✭✐✐✮ b❁a✳ P❡❧♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ ❛❝✐♠❛✱ b+c❁a+c✱ ♦ q✉❡ t❛♠❜é♠ é ✉♠ ❛❜s✉r❞♦✳

✭✐✐✐✮ a=b✳ ❊st❛ é ❛ ú♥✐❝❛ ❛❧t❡r♥❛t✐✈❛ ✈á❧✐❞❛✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✻✳ ❆ ♠✉❧t✐♣❧✐❝❛çã♦ é ❝♦♠♣❛tí✈❡❧ ❡ ❝❛♥❝❡❧❛t✐✈❛ ❝♦♠ r❡s♣❡✐t♦ à ✐❣✉❛❧❞❛❞❡

∀ a, b∈N✱ ∀ c∈N∗✱ ab ⇐⇒ a.cb.c

❉❡♠♦♥str❛çã♦✳ ❆ ✐♠♣❧✐❝❛çã♦ a❂b =⇒ a.c❂b.c é ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞♦ ❢❛t♦ ❞❛

♠✉❧t✐♣❧✐❝❛çã♦ s❡r ❜❡♠ ❞❡✜♥✐❞❛ ✭Pr♦♣r✐❡❞❛❞❡ ✶✮✳

❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ a.c=b.c✳ ❚❡♠♦s três ♣♦ss✐❜✐❧✐❞❛❞❡s✿

✭✐✮a❁b✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✹✱ t❡♠♦s q✉❡a.c❁b.c✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳

✭✐✐✮ b❁a✳ P❡❧♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ ❛❝✐♠❛✱ b.c❁a.c✱ ♦ q✉❡ t❛♠❜é♠ é ✉♠ ❛❜s✉r❞♦✳

✭✐✐✐✮ a=b✳ ❊st❛ é ❛ ú♥✐❝❛ ❛❧t❡r♥❛t✐✈❛ ✈á❧✐❞❛✳

◆♦t❡ q✉❡ ❛ r❡❧❛çã♦ < ♥ã♦ é ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✱ ♣♦✐s ♥ã♦ é r❡✢❡①✐✈❛✱ ♥❡♠ ❛♥✲

t✐s✐♠étr✐❝❛✳ P♦❞❡♠♦s✱ ❡♥tr❡t❛♥t♦✱ ❛tr❛✈és ❞❡❧❛✱ ♦❜t❡r ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✳ ❆ ✐❞❡✐❛ ✐♥t✉✐t✐✈❛ q✉❡ tr❛③❡♠♦s ❞❡s❞❡ ❛ ❡s❝♦❧❛✱ ❞❡ q✉❡ ✵ é ♠❡♥♦r q✉❡ ✶✱ q✉❡ é ♠❡♥♦r q✉❡ ✷ ❡ ❛s✲ s✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✱ ✈❡♠ ❞❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ q✉❡ ❡①✐st❡ ♥♦s ♥❛t✉r❛✐s✱ q✉❡ ♥♦s ♣❡r♠✐t❡ ❝♦♠♣❛r❛r ♦s ♥ú♠❡r♦s ❞❡st❡ ❝♦♥❥✉♥t♦✱ ❢♦r♠❛❧✐③❛♥❞♦ ❛ ✐❞❡✐❛ ✐♥t✉✐t✐✈❛✳ ❉❡s❝r❡✈❡♠♦s ❛ s❡❣✉✐r ❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✳

❉✐r❡♠♦s q✉❡ a é ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❞♦ q✉❡ b✱ ♦✉ q✉❡ b é ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❞♦ q✉❡ a✱

❡s❝r❡✈❡♥❞♦a ≤b ♦✉b ≥a s❡a❁b ♦✉ a=b✳

◆♦t❡ q✉❡ a≤ b s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ c∈ N✱ t❛❧ q✉❡b = a+c✳ ❈♦♠ ✐st♦✱ é ❢á❝✐❧

✈❡r✐✜❝❛r q✉❡ ❡st❛ ♥♦✈❛ r❡❧❛çã♦ é ❡❢❡t✐✈❛♠❡♥t❡ ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✱ ♣♦✐s ♣♦ss✉✐ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✮ ➱ r❡✢❡①✐✈❛✿ ∀ a, a≤a✳

✷✮ ➱ ❛♥t✐✲s✐♠étr✐❝❛✿ ∀ a, b, a≤b ❡ b≤a=⇒a=b✳

✸✮ ➱ tr❛♥s✐t✐✈❛✿ ∀a, b, c, a≤b ❡b ≤c=⇒a≤c✳

✶✳✷ ❙✉❜tr❛çã♦

❉❛❞♦s ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s a ❡ b ❝♦♠ a ≤ b✱ s❛❜❡♠♦s q✉❡ ❡①✐st❡ ✉♠ ♥ú♠❡r♦

♥❛t✉r❛❧c t❛❧ q✉❡ b =a+c✳ ◆❡st❡ ❝❛s♦✱ ❞❡✜♥✐♠♦s ♦ ♥ú♠❡r♦ ❜ ♠❡♥♦s ❛✱ ❞❡♥♦t❛❞♦ ♣♦r

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b ✲ a✱ ❝♦♠♦ s❡♥❞♦ ♦ ♥ú♠❡r♦ c✳ ❊♠ sí♠❜♦❧♦s✿ b−a=c✳

❉✐③❡♠♦s q✉❡ cé ♦ r❡s✉❧t❛❞♦ ❞❛ s✉❜tr❛çã♦ ❞❡ a ❞❡ b✳

P♦rt❛♥t♦✱ t❡♠♦s ♣♦r ❞❡✜♥✐çã♦

c=b−a⇐⇒b =a+c✳

◆♦ ✉♥✐✈❡rs♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ♥❡♠ s❡♠♣r❡ ❡①✐st❡ ❛ s✉❜tr❛çã♦ ❞❡ ❞♦✐s ♥ú♠❡r♦s❀ só ❡①✐st❡b−a q✉❛♥❞♦ a≤b✳

◆♦t❡ q✉❡ a−a= 0 ♣❛r❛ t♦❞♦a∈N✱ ❡ q✉❡✱ ♣♦r ❞❡✜♥✐çã♦✱ (b−a) +a=b✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✶✳ ❙❡❥❛♠ a, b, c∈N✳ ❙❡ a≤b✱ ❡♥tã♦ c.(b−a) = c.b−c.a✳

❉❡♠♦♥str❛çã♦✳ ◆♦t❡ q✉❡✱ s❡ b ≥ a✱ ❡♥tã♦ c.b ≥ c.a✱ ♦ q✉❡ ♥♦s ❞✐③ q✉❡ c.b−c.a ❡stá

❜❡♠ ❞❡✜♥✐❞♦✳

❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡b−a=d✱ ❧♦❣♦b =a+d✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ♣♦rc❛♠❜♦s ♦s ♠❡♠❜r♦s

❞❡st❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡✱ ♦❜t❡♠♦sc.b=c.(a+d) =c.a+c.d✱ ♦ q✉❡ ✐♠♣❧✐❝❛ c.d=c.b−c.a✳

❙✉❜st✐t✉✐♥❞♦d ♣♦r b−a ♥❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ ♦❜t❡♠♦s c.(b−a) =c.b−c.a✳

✶✳✸ ❆①✐♦♠❛ ❞❡ ■♥❞✉çã♦

❆s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ ❞❡ s✉❛s ♦♣❡r❛çõ❡s q✉❡ ❞❡s❝r❡✈❡♠♦s ❛té ♦ ♠♦♠❡♥t♦ ♥ã♦ ❜❛st❛♠ ♣❛r❛ ❝❛r❛❝t❡r✐③á✲❧♦s✳ P♦r ❡①❡♠♣❧♦✱ ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ♥ã♦ ♥❡❣❛t✐✈♦s✱ ❛ss✐♠ ❝♦♠♦ ♦s ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ♥❡❣❛t✐✈♦s ♣♦ss✉❡♠ t♦❞❛s ❛ ♣r♦♣r✐❡❞❛❞❡s ❛❝✐♠❛✳ ◆♦ ❡♥t❛♥t♦✱ ❤á ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❛❞✐❝✐♦♥❛❧ q✉❡ só ♦s ♥❛t✉r❛✐s ♣♦ss✉❡♠✱ q✉❡ é ♦ ❆①✐♦♠❛ ❞❡ ■♥❞✉çã♦ q✉❡ ♣❛ss❛♠♦s ❛ ❞❡s❝r❡✈❡r✳

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❆①✐♦♠❛ ✶ ✭❆①✐♦♠❛ ❞❡ ■♥❞✉çã♦✮✳ ❙❡❥❛ ❙ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ N t❛❧ q✉❡

✐✮ 0∈❙✳

✐✐✮ ❙ é ❢❡❝❤❛❞♦ ❝♦♠ r❡s♣❡✐t♦ à ♦♣❡r❛çã♦ ❞❡ ✏s♦♠❛r ✶✑ ❛ s❡✉s ❡❧❡♠❡♥t♦s✱ ♦✉ s❡❥❛✱

∀ n ∈❙ =⇒ n+ 1 ∈❙✳

❊♥tã♦✱ ❙❂N✳

❙❡ ❆⊂N ❡ ❛ ∈N✱ ✉s❛r❡♠♦s ❛ s❡❣✉✐r ❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦✿

❛+❆❂{a+x;x∈A}✳

➱ ✐♠❡❞✐❛t♦ ✈❡r✐✜❝❛r q✉❡

❛+N❂{m∈N;m≥a}✳

❙❡❣✉❡✲s❡✱ ❞♦ ❆①✐♦♠❛ ❞❡ ■♥❞✉çã♦✱ ♦ s❡❣✉✐♥t❡ ✐♠♣♦rt❛♥t❡ ✐♥str✉♠❡♥t♦ ♣❛r❛ ♣r♦✈❛r t❡♦r❡♠❛s✿

❚❡♦r❡♠❛ ✶✳✸✳✶ ✭Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✮✳ ❙❡❥❛ a ∈ N ❡ s❡❥❛ ♣✭♥✮ ✉♠❛

s❡♥t❡♥ç❛ ❛❜❡rt❛ ❡♠ ♥✳ ❙✉♣♦♥❤❛ q✉❡ ✭✐✮ ♣✭❛✮ é ✈❡r❞❛❞❡✱ ❡ q✉❡

✭✐✐✮ ∀ n ≥a✱ p(n) =⇒p(n+ 1) é ✈❡r❞❛❞❡✱ ❡♥tã♦✱ ♣✭♥✮ é ✈❡r❞❛❞❡ ♣❛r❛ t♦❞♦ n ≥a✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ ❱ ❂{n ∈ N;p(n)}❀ ♦✉ s❡❥❛✱ ❱ é ♦ s✉❜❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡

N♣❛r❛ ♦s q✉❛✐s ♣✭♥✮ é ✈❡r❞❛❞❡✳

❈♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦

❙❂{m∈N;a+m∈❱}✱

q✉❡ ✈❡r✐✜❝❛ tr✐✈✐❛❧♠❡♥t❡a+❙⊂❱✳

❈♦♠♦ ♣❡❧❛ ❝♦♥❞✐çã♦ ✭✐✮✱ t❡♠♦s q✉❡ a+ 0 =a∈❱✱ s❡❣✉❡✲s❡ q✉❡ 0∈❙✳

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ m ∈ ❙✱ ❡♥tã♦ a+m ∈❱ ❡✱ ♣♦r ✭✐✐✮✱ t❡♠♦s q✉❡ a+m+ 1 ∈ ❱❀

❧♦❣♦ m+ 1∈❙✳ ❆ss✐♠✱ ♣❡❧♦ ❆①✐♦♠❛ ❞❡ ■♥❞✉çã♦✱ t❡♠♦s q✉❡ ❙=N✳ P♦rt❛♥t♦✱

{m∈N;m ≥a}=a+N⊂❱✱

♦ q✉❡ ♣r♦✈❛ ♦ r❡s✉❧t❛❞♦✳

❈♦r♦❧ár✐♦ ✶✳ ◆ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♥ t❛❧ q✉❡ 0< n <1✳

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♣✭♥✮✿ n >0 =⇒n ≥1

é ✈❡r❞❛❞❡ ♣❛r❛ t♦❞♦n∈N✳

❙❡♥❞♦ ✵❃✵ ❢❛❧s♦✱ s❡❣✉❡✲s❡ q✉❡ ♣✭✵✮✿ ✵❃✵ =⇒✵❃✶ ♥ã♦ é ✈❡r❞❛❞❡✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ♥♦t❡ q✉❡ ♣✭♥✰✶✮✿ n+1>0 =⇒n+1≥1é ✈❡r❞❛❞❡ ♣❛r❛ t♦❞♦n ∈N✳

❉❡ ❢❛t♦✱ n+ 1≥1 é ✈❡r❞❛❞❡ ♣❛r❛ t♦❞♦n ∈N✱ ♣♦✐s é ❡q✉✐✈❛❧❡♥t❡✱ ♣♦r ❝❛♥❝❡❧❛♠❡♥t♦✱ ❛

n≥0✱ ♦ q✉❡ é s❡♠♣r❡ ✈❡r❞❛❞❡✳

▲♦❣♦✱ s❡♥❞♦ ♣✭♥✰✶✮ ✈❡r❞❛❞❡ ♣❛r❛ t♦❞♦ ♥✱ s❡❣✉❡✲s❡ q✉❡ ♣✭♥✮ =⇒ ♣✭♥✰✶✮ é ✈❡r❞❛❞❡

♣❛r❛ t♦❞♦n∈N✳

P♦rt❛♥t♦✱ ♦ r❡s✉❧t❛❞♦ ❞❡❝♦rr❡ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✳

❈♦r♦❧ár✐♦ ✷✳ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♥ q✉❛❧q✉❡r✱ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♠ t❛❧ q✉❡ n < m < n+ 1✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ ❡①✐st❛ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♠ ❝♦♠n < m < n+ 1✳ ▲♦❣♦✱ ❡①✐st✐r✐❛ ✉♠ ♥ú♠❡r♦ k ∈ N∗ t❛❧ q✉❡ n+k = m < n + 1✱ q✉❡✱ ♣❡❧❛

Pr♦♣♦s✐çã♦ ✶✳✶✳✸✱ ✐♠♣❧✐❝❛r✐❛ q✉❡ ✵❁❦❁✶✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ t❡♥❞♦ ❡♠ ✈✐st❛ ♦ ❈♦r♦❧ár✐♦ ✶ ❛❝✐♠❛✳

❈♦r♦❧ár✐♦ ✸✳ ❙❡❥❛♠ a, b∈N✳ ❙❡ a.b= 1✱ ❡♥tã♦ a=b = 1✳

❉❡♠♦♥str❛çã♦✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ♥♦t❡ q✉❡ a6= 0 ❡ b6= 0✱ ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ a.b= 0✳

❆❣♦r❛✱ s❡ a 6= 1 ❡ b 6= 1✱ ❡♥tã♦✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✶✱ s❡❣✉❡✲s❡ q✉❡ a > 1 ❡ b > 1✳

▲♦❣♦✱ a.b > b > 1❀ ❝♦♥tr❛❞✐çã♦✳ P♦rt❛♥t♦✱ a = 1 ♦✉ b = 1✳ ◗✉❛❧q✉❡r ✉♠❛ ❞❡ss❛s

♣♦ss✐❜✐❧✐❞❛❞❡s ✐♠♣❧✐❝❛a=b= 1✳

✶✳✹ ❉✐✈✐sã♦ ♥♦s ◆❛t✉r❛✐s

❈♦♠♦ ❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♣♦r ♦✉tr♦ ♥❡♠ s❡♠♣r❡ é ♣♦ssí✈❡❧✱ ❡①♣r❡ss❛✲s❡ ❡st❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❛tr❛✈és ❞❛ r❡❧❛çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✳ ◗✉❛♥❞♦ ♥ã♦ ❡①✐st✐r ✉♠❛ r❡❧❛çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s✱ ✈❡r❡♠♦s q✉❡✱ ❛✐♥❞❛ ❛ss✐♠✱ s❡rá ♣♦ssí✈❡❧ ❡❢❡t✉❛r ✉♠❛ ❞✐✈✐sã♦ ❝❤❛♠❛❞❛ ❞❡ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✳ ❖ ❢❛t♦ ❞❡ s❡♠♣r❡ s❡r ♣♦ssí✈❡❧ ❡❢❡t✉❛r t❛❧ ❞✐✈✐sã♦ é r❡s♣♦♥sá✈❡❧ ♣♦r ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥❛t✉r❛✐s q✉❡ ❡①♣❧♦r❛r❡♠♦s ♥❡st❛ s❡çã♦✳

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✶✳✹✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡

❉❛❞♦s ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❛ ❡ ❜ ❝♦♠ a 6= 0✱ ❞✐r❡♠♦s q✉❡ ❛ ❞✐✈✐❞❡ ❜✱ ❡s❝r❡✈❡♥❞♦

a | b✱ q✉❛♥❞♦ ❡①✐st✐r c ∈ N t❛❧ q✉❡ b = a.c✳ ◆❡st❡ ❝❛s♦✱ ❞✐r❡♠♦s t❛♠❜é♠ q✉❡ ❛ é ✉♠

❞✐✈✐s♦r ♦✉ ✉♠ ❢❛t♦r ❞❡ ❜ ♦✉✱ ❛✐♥❞❛✱ q✉❡ ❜ é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ❛✳ ❖❜s❡r✈❛r❡♠♦s q✉❡ ❛ ♥♦t❛çã♦ a | b ♥ã♦ r❡♣r❡s❡♥t❛ ♥❡♥❤✉♠❛ ♦♣❡r❛çã♦ ❡♠ N✱ ♥❡♠ ✉♠❛ ❢r❛çã♦✳ ❆ ♥❡❣❛çã♦

❞❡ss❛ s❡♥t❡♥ç❛ é r❡♣r❡s❡♥t❛❞❛ ♣♦r a∤b✳

Pr♦♣♦s✐çã♦ ✶✳✹✳✶✳ ❙❡❥❛♠ a, b∈N∗ ❡ cN✳ ❚❡♠✲s❡ q✉❡

✐✮ 1|c✱ a|a ❡ a|0✳

✐✐✮ s❡ a|b ❡ b|c✱ ❡♥tã♦ a |c✳

❉❡♠♦♥str❛çã♦✳ ✭✐✮ ■st♦ ❞❡❝♦rr❡ ❞❛s ✐❣✉❛❧❞❛❞❡s c= 1.c✱ a = a.1 ❡ a.0 = 0✳ ✭✐✐✮ a | b ❡ b| c✐♠♣❧✐❝❛ q✉❡ ❡①✐st❡♠ f, g ∈ N✱ t❛✐s q✉❡ b =a.f ❡ c=b.g✳ ❙✉❜st✐t✉✐♥❞♦ ♦ ✈❛❧♦r ❞❡

❜ ❞❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ♥❛ ♦✉tr❛✱ ♦❜t❡♠♦s

c=b.g= (a.f).g =a.(f.g)✱

♦ q✉❡ ♥♦s ♠♦str❛ q✉❡a |c✳

❖ ✐t❡♠ ✭✐✮ ❞❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛ ♥♦s ❞✐③ q✉❡ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ é ❞✐✈✐sí✈❡❧ ♣♦r ✶ ❡✱ s❡ ♥ã♦ ♥✉❧♦✱ ♣♦r s✐ ♠❡s♠♦✳

Pr♦♣♦s✐çã♦ ✶✳✹✳✷✳ ❙❡a, b, c, d∈N✱ ❝♦♠ a6= 0 ❡ c6= 0✱ ❡♥tã♦

a|b ❡ c|d =⇒ a.c|b.d✳

❉❡♠♦♥str❛çã♦✳ s❡ a | b ❡ c | d✱ ❡♥tã♦ ∃f, g ∈ N, b = a.f ❡ d = c.g✳ P♦rt❛♥t♦✱ b.d = (a.c)(f.g)✱ ❧♦❣♦✱ a.c|b.d✳

❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ a|b✱ ❡♥tã♦ a.c|b.c✱ ♣❛r❛ t♦❞♦ c∈N∗✳

Pr♦♣♦s✐çã♦ ✶✳✹✳✸✳ ❙❡❥❛♠ a, b, c∈N✱ ❝♦♠ a6= 0✱ t❛✐s q✉❡ a|(b+c)✳ ❊♥tã♦

a|b ⇐⇒ a|c✳

❉❡♠♦♥str❛çã♦✳ ❈♦♠♦a |(b+c)✱ ❡①✐st❡ f ∈N t❛❧ q✉❡ b+c=f.a✳

❆❣♦r❛✱ s❡a|b✱ t❡♠♦s q✉❡ ❡①✐st❡g ∈Nt❛❧ q✉❡b=a.g✳ ❏✉♥t❛♥❞♦ ❛s ❞✉❛s ✐❣✉❛❧❞❛❞❡s

❛❝✐♠❛✱ t❡♠♦s

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a.g+c=f.a=a.f✱

❞♦♥❞❡ s❡❣✉❡✲s❡ q✉❡ a.f > a.g✱ ❡✱ ❝♦♥s❡q✉❡t❡♠❡♥t❡✱ f > g✳ P♦rt❛♥t♦✱ ❞❛ ✐❣✉❛❧❞❛❞❡

❛❝✐♠❛ ❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✷✳✶✱ ♦❜t❡♠♦s

c=a.f −a.g =a.(f −g)✱

♦ q✉❡ ✐♠♣❧✐❝❛ q✉a|c✱ ❥á q✉❡f −g ∈N✳

❆ ♣r♦✈❛ ❞❛ ♦✉tr❛ ✐♠♣❧✐❝❛çã♦ é t♦t❛❧♠❡♥t❡ ❛♥á❧♦❣❛✳

Pr♦♣♦s✐çã♦ ✶✳✹✳✹✳ ❙❡❥❛♠ a, b, c∈N✱ ❝♦♠ a6= 0✱ ❡ b≥c✱ t❛✐s q✉❡ a|(b−c)✳ ❊♥tã♦

a|b ⇐⇒ a|c✳

Pr♦♣♦s✐çã♦ ✶✳✹✳✺✳ ❙❡ a, b, c ∈ N✱ ❝♦♠ a 6= 0✱ ❡ x, y ∈ N✱ sã♦ t❛✐s q✉❡ a | b ❡ a | c✱

❡♥tã♦a |(xb+yc)❀ ❡ s❡ xb≥yc✱ ❡♥tã♦ a |(xb−yc)✳

❉❡♠♦♥str❛çã♦✳ a | b ❡ a | c ✐♠♣❧✐❝❛♠ q✉❡ ❡①✐st❡♠ f, g ∈ N t❛✐s q✉❡ b = af ❡ c =ag✳

▲♦❣♦

xb±yc=x(af)±y(ag) = a(xf±yg)✱

♦ q✉❡ ♣r♦✈❛ ♦ r❡s✉❧t❛❞♦✱ ♣♦✐s✱ ♥❛s ❝♦♥❞✐çõ❡s ❞❛❞❛s✱xf ±yg∈N✳

Pr♦♣♦s✐çã♦ ✶✳✹✳✻✳ ❉❛❞♦s a, b∈N∗✱ t❡♠♦s q✉❡

a|b =⇒ a ≤b✳

❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ s❡a|b✱ ❡①✐st❡ c∈N∗ t❛❧ q✉❡b =ac✳ ❈♦♠♦ ❞♦ ❈♦r♦❧ár✐♦ ✶ ❞♦

❚❡♦r❡♠❛ ✶✳✸✳✶✱c≥1✱ s❡❣✉❡✲s❡ q✉❡ a≤ac=b✳

◆♦t❡ q✉❡ ❛ r❡❧❛çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡♠ N∗ é ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✱ ♣♦✐s

✐✮ é r❡✢❡①✐✈❛✿ ∀a∈N∗✱ a|a✳ ✭Pr♦♣♦s✐çã♦ ✶✳✹✳✶✳✶✭✐✮✮✱

✐✐✮ é tr❛♥s✐t✐✈❛✿ s❡ a|b ❡ b |c✱ ❡♥tã♦ a|c✳ ✭Pr♦♣♦s✐çã♦ ✶✳✹✳✶✳✶✭✐✐✮✮✱

✐✐✐✮ é ❛♥t✐✲s✐♠étr✐❝❛✿ s❡ a|b ❡ b |a✱ ❡♥tã♦ a=b✳ ✭Pr♦♣♦s✐çã♦ ✶✳✹✳✶✳✻✮

Pr♦♣♦s✐çã♦ ✶✳✹✳✼✳ ❙❡❥❛♠ a, b, n∈N✱ ❝♦♠ a > b >0✳ ❚❡♠♦s q✉❡ a−b ❞✐✈✐❞❡ anbn

❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ♣r♦✈❛r ✐st♦ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ ♥✳

➱ ♦❜✈✐♦ q✉❡ ❛ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❡ ♣❛r❛ n= 0✱ ♣♦✐sa−b ❞✐✈✐❞❡ a0b0 = 0

❙✉♣♦♥❤❛♠♦s✱ ❛❣♦r❛✱ q✉❡ a−b |anbn✳ ❊s❝r❡✈❛♠♦s

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an+1bn+1 =aanban+banbbn= (ab)an+b(anbn)

❈♦♠♦ a−b | a−b ❡✱ ♣♦r ❤✐♣ót❡s❡✱ a−b | anbn✱ ❞❡❝♦rr❡ ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ❡

❞❛ Pr♦♣♦s✐çã♦ ✶✳✹✳✶✳✺ q✉❡ a−b | an+1bn+1✳ ❊st❛❜❡❧❡❝❡♥❞♦ ♦ r❡s✉❧t❛❞♦ ♣❛r❛ t♦❞♦

n∈N✳

Pr♦♣♦s✐çã♦ ✶✳✹✳✽✳ ❙❡❥❛♠ a, b, n∈N✱ ❝♦♠ a+b6= 0✳ ❚❡♠♦s q✉❡ a+b ❞✐✈✐❞❡ a2n+1+

b2n+1

❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ♣r♦✈❛r ✐st♦ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ ♥✳

➱ ♦❜✈✐♦ q✉❡ ❛ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❡ ♣❛r❛ n= 0✱ ♣♦✐sa+b ❞✐✈✐❞❡ a1+b1 =a+b✳

❙✉♣♦♥❤❛♠♦s✱ ❛❣♦r❛✱ q✉❡ a+b|a2n+1+b2n+1✳ ❊s❝r❡✈❛♠♦s

a2(n+1)+1+b2(n+1)+1 =a2a2n+1b2a2n+1+b2a2n+1+b2b2n+1 = (a2−b2)a2n+1+b2a2n+1+b2n+1

❈♦♠♦a+b|a2−b2 ❡✱ ♣♦r ❤✐♣ót❡s❡✱a+b |a2n+1+b2n+1✱ ❞❡❝♦rr❡ ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛

❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✹✳✶✳✺ q✉❡a+b|a2(n+1)+1+b2(n+1)+1✳ ❊st❛❜❡❧❡❝❡♥❞♦ ♦ r❡s✉❧t❛❞♦ ♣❛r❛

t♦❞♦n∈N✳

Pr♦♣♦s✐çã♦ ✶✳✹✳✾✳ ❙❡❥❛♠a, b, n∈N✱ ❝♦♠a ≥b >0✳ ❚❡♠♦s q✉❡ a+b❞✐✈✐❞❡a2nb2n

❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ♣r♦✈❛r ✐st♦ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ ♥✳

❆ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❡ ♣❛r❛ n= 0✱ ♣♦✐sa+b ❞✐✈✐❞❡ a0b0 = 0

❙✉♣♦♥❤❛♠♦s✱ ❛❣♦r❛✱ q✉❡ a+b|a2nb2n✳ ❊s❝r❡✈❛♠♦s

a2(n+1)b2(n+1) =a2a2nb2a2n+b2a2nb2b2n= (a2b2)a2n+b2(a2nb2n)

❈♦♠♦a+b |a2−b2 ❡✱ ♣♦r ❤✐♣ót❡s❡✱a+b|a2nb2n✱ ❞❡❝♦rr❡ ❞❛s ✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛

❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✹✳✶✳✺ q✉❡ a+b | a2(n+1)b2(n+1)✳ ❊st❛❜❡❧❡❝❡♥❞♦ ♦ r❡s✉❧t❛❞♦ ♣❛r❛

t♦❞♦n∈N✳

✶✳✹✳✷ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛

❚❡♦r❡♠❛ ✶✳✹✳✶ ✭❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛✮✳ ❙❡❥❛♠ ❛ ❡ ❜ ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❝♦♠0< a < b✳ ❊①✐st❡♠ ❞♦✐s ú♥✐❝♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s q ❡ r t❛✐s q✉❡

b=a.q+r✱ ❝♦♠ r < a✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡b > a ❡ ❝♦♥s✐❞❡r❡✱ ❡♥q✉❛♥t♦ ✜③❡r s❡♥t✐❞♦✱ ♦s ♥ú♠❡r♦s

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b, b−a, b−2a, ..., b−n.a, ...

P❡❧❛ Pr♦♣r✐❡❞❛❞❡ ❞❛ ❇♦❛ ❖r❞❡♠✱ q✉❡ ❛✜r♠❛ q✉❡ t♦❞♦ s✉❜❝♦♥❥✉♥t♦ ❞♦s ♥❛t✉r❛✐s ♣♦ss✉✐ ✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦✱ ♦ ❝♦♥❥✉♥t♦ ❙ ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❛❝✐♠❛ t❡♠ ✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦r =b−q.a✳ ❱❛♠♦s ♣r♦✈❛r q✉❡ r t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ r❡q✉❡r✐❞❛✱ ♦✉ s❡❥❛✱

q✉❡r < a✳

❙❡ a | b✱ ❡♥tã♦ r = 0 ❡ ♥❛❞❛ ♠❛✐s t❡♠♦s q✉❡ ♣r♦✈❛r✳ ❙❡✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ a ∤ b✱

❡♥tã♦ r 6=a✱ ❡✱ ♣♦rt❛♥t♦✱ ❜❛st❛ ♠♦str❛r q✉❡ ♥ã♦ ♣♦❞❡ ♦❝♦rr❡r r > a✳ ❉❡ ❢❛t♦✱ s❡ ✐st♦

♦❝♦rr❡ss❡✱ ❡①✐st✐r✐❛ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ c < r t❛❧ q✉❡ r = c+a✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

s❡♥❞♦r =c+a=b−q.a✱ t❡rí❛♠♦s

c=b−(q+ 1).a∈ ❙✱ ❝♦♠ c < r✱

❝♦♥tr❛❞✐çã♦ ❝♦♠ ♦ ❢❛t♦ ❞❡ r s❡r ♦ ♠❡♥♦r ❡❧❡♠❡♥t♦ ❞❡ ❙✳

P♦rt❛♥t♦✱ t❡♠♦s q✉❡ b=a.q+r ❝♦♠ r < a✱ ♦ q✉❡ ♣r♦✈❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ q ❡ r✳

❆❣♦r❛✱ ✈❛♠♦s ♣r♦✈❛r ❛ ✉♥✐❝✐❞❛❞❡✳ ◆♦t❡ q✉❡✱ ❞❛❞♦s ❞♦✐s ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s ❞❡ ❙✱ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ♠❛✐♦r ❡ ♦ ♠❡♥♦r ❞❡ss❡s ❡❧❡♠❡♥t♦s✱ s❡♥❞♦ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ❛✱ é ♣❡❧♦ ♠❡♥♦s ❛✳ ▲♦❣♦✱ s❡r =b−a.q ❡ r′ = ba.q′✱ ❝♦♠r < r< a✱ t❡rí❛♠♦s rr a✱ ♦

q✉❡ ❛❝❛rr❡t❛r✐❛r′ r+a a✱ ❛❜s✉r❞♦✳ P♦rt❛♥t♦✱ r=r′✳

❉❛í s❡❣✉❡✲s❡ q✉❡ b−a.q =b−a.q′✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡a.q=a.q′ ❡✱ ♣♦rt❛♥t♦✱ q=q′✳

◆❛s ❝♦♥❞✐çõ❡s ❞♦ t❡♦r❡♠❛ ❛❝✐♠❛✱ ♦s ♥ú♠❡r♦s q ❡ r sã♦ ❝❤❛♠❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ q✉♦❝✐❡♥t❡ ❡ ❞❡ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❜ ❡ ❛✳ ❙❡ ❛ ❞✐✈✐❞❡ ❜✱ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❜ ♣♦r ❛ é ③❡r♦✳

❈♦r♦❧ár✐♦ ✹✳ ❉❛❞♦s ❞♦✐s ♥ú♠❡r♦s ❛ ❡ ❜ ❝♦♠ 1< a≤ b✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♥

t❛❧ q✉❡

na≤b <(n+ 1)a✳

❉❡♠♦♥str❛çã♦✳ P❡❧❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✱ t❡♠♦s q✉❡ ❡①✐st❡♠q, r ∈N ❝♦♠ r < a✱ ✉♥✐✈♦✲

❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦s✱ t❛✐s q✉❡b =a.q+r✳ ❇❛st❛ ❛❣♦r❛ t♦♠❛r n=q✳

✶✳✺ ❙✐st❡♠❛ ❞❡ ◆✉♠❡r❛çã♦ ❉❡❝✐♠❛❧

➱ ❝♦♠✉♠ ♣❡♥s❛r♠♦s q✉❡ ♦s ❤♦♠❡♥s t✐✈❡r❛♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝r✐❛r ♦s ♥ú♠❡r♦s ♣❛r❛ ❛ ❝♦♥t❛❣❡♠ ❞❡ s❡✉s ❜❡♥s✱ ♣♦ré♠✱ ❞❡s❝♦❜❡rt❛s ♥♦s r❡✈❡❧❛♠ q✉❡ ❛ ♥♦çã♦ ❞❡ q✉❛♥t✐❞❛❞❡

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❡①✐st❡ ❞❡s❞❡ à é♣♦❝❛ ❞♦s ❤♦♠❡♥s ❞❛s ❝❛✈❡r♥❛s✱ ♦s q✉❛✐s ♥ã♦ ♣❧❛♥t❛✈❛♠ ♥❡♠ ♣♦ss✉✐❛♠ ❜❡♥s✳

❙❡❣✉♥❞♦ ❬✹❪✱ ♦s ♣r✐♠❡✐r♦s s❡r❡s ❤✉♠❛♥♦s ✈✐✈✐❛♠ ❞❛ ❝♦❧❡t❛ ❞♦ q✉❡ ❥á ❡①✐st✐❛ ♥❛ ♥❛t✉r❡③❛ ❡ ❞❛ ❝❛ç❛✳ P♦st❡r✐♦r♠❡♥t❡✱ s❡✉s ❞❡s❝❡♥❞❡♥t❡s ❞❡s❡♥✈♦❧✈❡r❛♠ ❛ ❛❣r✐❝✉❧t✉r❛ ❡ té❝♥✐❝❛s ❞❡ ❞♦♠❡st✐❝❛çã♦ ❞❡ ❛♥✐♠❛✐s ❡ ♣❛ss❛r❛♠ ❛ ❝♦♠❡r❝✐❛❧✐③❛r ❛q✉✐❧♦ q✉❡ ❧❤❡s s♦❜r❛✈❛♠ ♦✉ ♥ã♦ t✐♥❤❛♠✳ ❈♦♠ ✐ss♦ ✜❝♦✉ ❝♦♠♣❧✐❝❛❞♦ r❡❣✐str❛r ♦ q✉❡ ♣♦ss✉í❛♠ ♦✉ ❞❡✈✐❛♠ ❛tr❛✈és ❞❡ ❝♦rt❡s ❡♠ ♦ss♦s✱ ♠❛❞❡✐r❛s✱ ♣❡❞r❛s ♦✉ r✐s❝♦s ❡♠ ♣❛r❡❞❡s✳ ❙✉r❣✐✉ ❛í ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝r✐❛r ♠♦❞♦s ❛❞❡q✉❛❞♦s ♣❛r❛ r❡❣✐str❛r q✉❛♥t✐❞❛❞❡s ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛q✉❡❧❛s ♠❛✐♦r❡s✳

❱ár✐❛s ❝✐✈✐❧✐③❛çõ❡s ❞❡s❡♥✈♦❧✈❡r❛♠ ❞✐❢❡r❡♥t❡s s✐st❡♠❛s ♣❛r❛ r❡♣r❡s❡♥t❛r ♦s ♥ú♠❡r♦s✳ ❯♠ ❞❡❧❡s✱ ♦ ❙✐st❡♠❛ ❞❡ ◆✉♠❡r❛çã♦ ❉❡❝✐♠❛❧ ✭❙◆❉✮ ♦ q✉❛❧ ✉t✐❧✐③❛♠♦s✱ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ♥❛ ➪s✐❛ ♥♦ ✈❛❧❡ ❞♦ r✐♦ ■♥❞♦ ♣❡❧♦s ✐♥❞✐❛♥♦s✳ ❖s ♣♦✈♦s ár❛❜❡s q✉❡ ❝♦♥str✉ír❛♠ ♦ ■♠♣ér✐♦ ■s❧â♠✐❝♦ ♦ ❛❞♦t❛r❛♠ ❧❡✈❛♥❞♦✲♦ ♣❛r❛ ❛ ❊✉r♦♣❛ s✉❜st✐t✉✐♥❞♦ ♦s q✉❡ ❛❧✐ ❡①✐st✐❛♠✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♦ ❙✐st❡♠❛ ❞❡ ◆✉♠❡r❛çã♦ ❘♦♠❛♥♦✳ P♦r ❡ss❡ ♠♦t✐✈♦ ❡❧❡ ✜❝♦✉ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ s✐st❡♠❛ ❞❡ ♥✉♠❡r❛çã♦ ✐♥❞♦✲❛rá❜✐❝♦✳

❖ s✐st❡♠❛ ❞❡ ♥✉♠❡r❛çã♦ ❞❡❝✐♠❛❧ ❛♣r❡s❡♥t❛✈❛ ✈ár✐❛s ✈❛♥t❛❣❡♥s ❡♠ r❡❧❛çã♦ ❛ ♦✉tr♦s s✐st❡♠❛s ❞❛ é♣♦❝❛✳ ❊❧❡ ❢♦✐ ❡str✉t✉r❛❞♦ ❛ ♣❛rt✐r ❞❛ ❜❛s❡ ❞❡③ ✭✶✵✮✳ ❉❡✈❡♠♦s t❡r ❡♠ ♠❡♥t❡ q✉❡ ❛ ❧❡✐t✉r❛✱ ❡s❝r✐t❛✱ ❝♦♠♣❛r❛çã♦✱ ❝♦♠♣♦s✐çã♦ ❡ t♦❞❛s ❛s ♦♣❡r❛çõ❡s sã♦ r❡❛❧✐③❛❞❛s ❛ ♣❛rt✐r ❞❡ ❛❣r✉♣❛♠❡♥t♦s ❞❡ ✶✵ ❡♠ ✶✵✱ ♦ q✉❡ ✜❝❛ ❝❧❛r♦ q✉❡ ♦ ❙◆❉ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ q✉❡ ❞❡✈❡ s❡r ❝♦♠♣r❡❡♥❞✐❞❛ ♣❛r❛ ♠❡❧❤♦r s❡r ✉t✐❧✐③❛❞❛✳

◆❛ s✉❛ ❡str✉t✉r❛ ✈❛❧❡ r❡ss❛❧t❛r ❛ ✉t✐❧✐③❛çã♦ ❞❡ ❞❡③ sí♠❜♦❧♦s ✭✵✱ ✶✱ ✷✱ ✸✱ ✹✱ ✺✱ ✻✱ ✼✱ ✽ ❡ ✾✮✱ ♦s q✉❛✐s s❡rã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ q✉❛❧q✉❡r ♥ú♠❡r♦✳ ❉❡♥tr❡ ❡st❡s sí♠❜♦❧♦s ✈❛❧❡ r❡ss❛❧t❛r ♦ ③❡r♦ ✭✵✮ q✉❡ r❡♣r❡s❡♥t❛ ❛ ❛✉sê♥❝✐❛ ❞❡ ✈❛❧♦r✱ ♦ q✉❛❧ ♥ã♦ ❡①✐st✐❛ ❡♠ ♦✉tr♦s s✐st❡♠❛s ❞❛ é♣♦❝❛✳ ❖✉tr♦ ❛s♣❡❝t♦ ✐♠♣♦rt❛♥t❡ é q✉❡ ♦s sí♠❜♦❧♦s ❛❝✐♠❛ ❧✐st❛❞♦s ♣♦ss✉❡♠ ✈❛❧♦r❡s ❞✐st✐♥t♦s✱ s❡❣✉♥❞♦ ❛ s✉❛ ♣♦s✐çã♦ ♥♦ ♥ú♠❡r♦✱ ♦✉ s❡❥❛✱ ❡❧❡ é ♣♦s✐❝✐♦♥❛❧✳ ❯♠ ❡①❡♠♣❧♦ q✉❡ ♥♦s ❡s❝❧❛r❡❝❡ ❡ss❡ ♣♦♥t♦ ❞❛ ♣♦s✐çã♦ ❞♦ ❛❧❣❛r✐s♠♦ ♥♦ ♥ú♠❡r♦ sã♦ ♦s ♥ú♠❡r♦s ✶✷ ❡ ✷✶✳ ❆♣❡s❛r ❞❡st❡s ❞♦✐s ♥ú♠❡r♦s s❡r❡♠ ❢♦r♠❛❞♦s ♣❡❧♦s ❛❧❣❛r✐s♠♦s ✶ ❡ ✷✱ ❡st❡s✱ ❡♠ ❝❛❞❛ ✉♠ ❞♦s ♥ú♠❡r♦s ❝✐t❛❞♦s✱ ♦❝✉♣❛♠ ✉♠❛ ♣♦s✐çã♦ ❞✐❢❡r❡♥t❡✱ ♦r❛ ♥❛s ✉♥✐❞❛❞❡s✱ ♦r❛ ♥❛s ❞❡③❡♥❛s✳ ❖ sí♠❜♦❧♦ ✷✱ ♥♦ ♣r✐♠❡✐r♦ ✈❛❧❡ ❞✉❛s ✉♥✐❞❛❞❡s ❡ ♥♦ s❡❣✉♥❞♦ ❥á ✈❛❧❡rá ❞✉❛s ❞❡③❡♥❛s ♦✉ ✷✵ ✉♥✐❞❛❞❡s✳

Pr❡❝✐s❛♠♦s r❡ss❛❧t❛r q✉❡ ❛ ♣❛rt✐r ❞♦ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ✶✵ ❡♠ ✶✵ s✉r❣❡♠ ♥♦♠❡♥❝❧❛t✉✲ r❛s ❡ ❞❡✜♥✐çõ❡s ✐♠♣♦rt❛♥t❡s✱ ❝♦♠♦ ❛s ❝✐t❛❞❛s ♥♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r✳ ❆ ❞❡✜♥✐çã♦ ❞❡ q✉❡ ♦ ❣r✉♣♦ ❞❡ ✶✵ ✉♥✐❞❛❞❡s r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ❞❡③❡♥❛✱ ♦ ❣r✉♣♦ ❞❡ ✶✵ ❞❡③❡♥❛s é ♥♦♠❡❛❞♦ ❝♦♠♦ ❝❡♥t❡♥❛✱ ♦ ❣r✉♣♦ ❞❡ ✶✵ ❝❡♥t❡♥❛s ❡q✉✐✈❛❧❡ ❛ ✶ ✉♥✐❞❛❞❡ ❞❡ ♠✐❧❤❛r ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❚❛✐s ♥♦♠❡♥❝❧❛t✉r❛s ♥♦s r❡♠❡t❡ ❛♦ ❢❛t♦ ❞❡ q✉❡ ♥❛ ❝♦♥str✉çã♦ ❞♦ ♥ú♠❡r♦ ♥♦ ❙✐st❡♠❛ ❞❡

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◆✉♠❡r❛çã♦ ❉❡❝✐♠❛❧ q✉❛❧q✉❡r ✉♠ ❞♦s ❞❡③ sí♠❜♦❧♦s q✉❡ ❢♦r ✉t✐❧✐③❛❞♦ ♣❡rt❡♥❝❡rá ❛ ✉♠❛ ♦r❞❡♠ ❡ ❝❧❛ss❡✳ ❈♦♠❡ç❛♠♦s ❛ ❡♥✉♠❡r❛r ❛s ♦r❞❡♥s ❞❛ ❞✐r❡✐t❛ ♣❛r❛ ❛ ❡sq✉❡r❞❛ ❡ ❛ ❝❛❞❛ três ♦r❞❡♥s t❡♠♦s ✉♠❛ ❝❧❛ss❡✱ ❛ q✉❛❧ é s❡♣❛r❛❞❛ ✉♠❛ ❞❛ ♦✉tr❛ ♣♦r ✉♠ ♣♦♥t♦ ✭✳✮✳

❆s três ♦r❞❡♥s ❞❡ ❝❛❞❛ ❝❧❛ss❡ r❡❝❡❜❡♠ ♦ ♥♦♠❡ ❞❡ ✉♥✐❞❛❞❡s✱ ❞❡③❡♥❛s ❡ ❝❡♥t❡♥❛s s❡❣✉✐❞❛s ♣❡❧♦ ♥♦♠❡ ❞❛ ❝❧❛ss❡ q✉❡ ❡❧❛s ❡stã♦✱ t❛♠❜é♠ ♥♦♠❡❛❞❛s ❞❛ ❞✐r❡✐t❛ ♣❛r❛ ❛ ❡q✉❡r❞❛✳ ❱❛❧❡ ❧❡♠❜r❛r q✉❡ t❛♥t♦ ❛ ❧❡✐t✉r❛ ❝♦♠♦ ❛ ❡s❝r✐t❛ ❞♦ ♥ú♠❡r♦ s❡ ❢❛③ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✳

❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❛ ❝❧❛ss✐✜❝❛çã♦ ❞♦ ♥ú♠❡r♦ ❡♠ ♦r❞❡♥s ❡ ❝❧❛ss❡s✳ ❙❡❥❛ ♦ ♥ú♠❡r♦ ✹✳✸✼✷✳✶✵✹✱ t❡r❡♠♦s ❡♥tã♦ s❡t❡ ♦r❞❡♥s ❡ três ❝❧❛ss❡s✱ ❝♦♠♦ ♠♦str❛ ❛ t❛❜❡❧❛ ❛❜❛✐①♦✿

❖✉tr❛ ❝❛r❛❝t❡ríst✐❝❛ r❡❧❡✈❛♥t❡ ❞♦ ❙◆❉ é q✉❡ ❡❧❡ ♦❜❡❞❡❝❡ ❛♦s ♣r✐♥❝í♣✐♦s ❛❞✐t✐✈♦s ❡ ♠✉❧t✐♣❧✐❝❛t✐✈♦s✳ ❖ ♣r✐♥❝í♣✐♦ ❛❞✐t✐✈♦ s❡ ❞❡✈❡ ❛♦ ❢❛t♦ ❞❡ ♦❜t❡r♠♦s ♦ ✈❛❧♦r ❞❡ ✉♠ ♥ú♠❡r♦ ♣❡❧❛ ❛❞✐çã♦ ❞♦s ✈❛❧♦r❡s ♣♦s✐❝✐♦♥❛✐s ❞❡ ❝❛❞❛ ❛❧❣❛r✐s♠♦✳ ❯♠ ❡①❡♠♣❧♦ q✉❡ ♣♦❞❡♠♦s ❢♦r♥❡❝❡r é ♦ ♥ú♠❡r♦ ✶✹✼✱ ❡❧❡ ❡q✉✐✈❛❧❡ ❛ ✶✵✵✰✹✵✰✼✱ ♦✉ s❡❥❛✱ ♦ ❛❧❣❛r✐s♠♦ ✶ ❡stá ♥❛ ✸❛

♦r❞❡♠ ❞❛ ✶❛ ❝❧❛ss❡ ✭ ❝❡♥t❡♥❛ s✐♠♣❧❡s ✮✱ ✈❛❧❡ ✶✵✵❀ ♦ ✈❛❧♦r ✹ ❡stá ♥❛ ✷♦r❞❡♠ ❞❛ ✶❝❧❛ss❡

✭❞❡③❡♥❛ s✐♠♣❧❡s✮✱ ✈❛❧❡ ✹✵❀ ❡ ♦ ✼ ❡stá ♥❛ ✶❛ ♦r❞❡♠ ❞❛ ✶❝❧❛ss❡ ✭✉♥✐❞❛❞❡ s✐♠♣❧❡s✮✱ ✈❛❧❡

✼✱ ♦ q✉❡ ❛❞✐❝✐♦♥❛❞♦s r❡s✉❧t❛ ♥♦ ✈❛❧♦r ❞❡ ✶✹✼ q✉❡ é ♦ ♥ú♠❡r♦ ♣r♦♣♦st♦ ❝♦♠♦ ❡①❡♠♣❧♦✳ ❏á ♦ ♣r✐♥❝í♣✐♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ♥♦s ❣❛r❛♥t❡ q✉❡ ♦ ✈❛❧♦r ❞♦ ❛❧❣❛r✐s♠♦ é ♠✉❧t✐♣❧✐❝❛❞♦ ♣❡❧♦ ✈❛❧♦r ❞❛ ♣♦s✐çã♦ ♦❝✉♣❛❞❛✳ ❯t✐❧✐③❛r❡♠♦s ♥♦✈❛♠❡♥t❡ ♦ ✶✹✼ ❝♦♠♦ ❡①❡♠♣❧♦✳ ❚❡♠♦s q✉❡ ✶✹✼ ❂ ✶①✶✵✵ ✰ ✹①✶✵ ✰ ✼①✶✱ ❛❞♠✐t✐♥❞♦ ❛s ♠❡s♠❛s ❡①♣❧✐❝❛çõ❡s ✉s❛❞❛s ♣❛r❛ ♦ ♣r✐♥❝í♣✐♦ ❛❞✐t✐✈♦✳

❖s ♣r✐♥❝í♣✐♦s ❛❞✐t✐✈♦ ❡ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❣❡r❛♠ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞♦s ♥ú♠❡r♦s✳ ❱♦❧t❛♥❞♦ ❛♦ ♥ú♠❡r♦ ✶✹✼✱ t❡♠♦s q✉❡ ✉t✐❧✐③❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❡ ♣♦st❡r✐♦r♠❡♥t❡ ♦ ❛❞✐t✐✈♦ t❡r♠✐♥❛♠♦s ♣♦r ❞❡❝♦♠♣♦r ❡st❡ ♥ú♠❡r♦✱ ♦✉ s❡❥❛✱ ✶✹✼ ❂ ✶①✶✵✵ ✰ ✹①✶✵ ✰ ✼①✶ ❂ ✶✵✵ ✰ ✹✵ ✰ ✼✳ ❆ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ✏❞❡s♠❛♥❝❤❛r✑ ❡ss❡ ♥ú♠❡r♦ ❡♠ ♣❡❞❛ç♦s✳ P♦❞❡♠♦s ❞❡❝♦♠♣♦r ♦ ♥ú♠❡r♦ ❡♠ ♦r❞❡♥s ♦✉ ❡♠ ❝❧❛ss❡s✳ ❙❡ ✉t✐❧✐③❛r♠♦s ❛ ♣r✐♠❡✐r❛ ❡st❛r❡♠♦s ♠♦str❛♥❞♦ q✉❛♥t❛s ✉♥✐❞❛❞❡s t❡rá ❝❛❞❛ ♦r❞❡♠✱ ❥á ❛ s❡❣✉♥❞❛ ♥♦s ❞❛rá q✉❛♥t❛s ✉♥✐❞❛❞❡s t❡♠ ❡♠ ❝❛❞❛ ❝❧❛ss❡✳ P❡❣✉❡♠♦s ♦ ♥ú♠❡r♦ ✼✹✺✳✷✺✽ ✭ s❡t❡❝❡♥t♦s ❡ q✉❛r❡♥t❛ ❡ ❝✐♥❝♦ ♠✐❧✱ ❞✉③❡♥t♦s ❡ ❝✐♥q✉❡♥t❛ ❡ ♦✐t♦ ✮✳ ❋❛③❡♥❞♦ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❡♠ ♦r❞❡♥s t❡r❡♠♦s✿ ✼✹✺✳✷✺✽ ❂ ✼✵✵✳✵✵✵ ✰ ✹✵✳✵✵✵ ✰ ✺✳✵✵✵ ✰ ✷✵✵ ✰ ✺✵ ✰ ✽✱ ♥♦s ❢♦r♥❡❝❡♥❞♦✱ ♣♦rt❛♥t♦✱ q✉❛♥t❛s ✉♥✐❞❛❞❡s ♦ ♥ú♠❡r♦ t❡rá ❡♠ ❝❛❞❛ ✉♠❛ ❞❛s s✉❛s s❡✐s ♦r❞❡♥s✳ ❏á s❡ ✜③❡r♠♦s ♣♦r ❝❧❛ss❡s ✜❝❛rá✿ ✼✹✺✳✷✺✽ ❂ ✼✹✺✳✵✵✵ ✰ ✷✺✽✱ ♦ q✉❡ ♥♦s ❞á

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