❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás
■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛
Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠
▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
❆♣❧✐❝❛çõ❡s ♣❛r❛ ♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦
▼❛t❡♠át✐❝❛
◆♦r♠❛♥❞♦ ❙✐❧✈❛ ❏✉♥✐♦r
●♦✐â♥✐❛
TERMO DE CIÊNCIA E DE AUTORIZAÇÃO PARA DISPONIBILIZAR AS TESES E DISSERTAÇÕES ELETRÔNICAS (TEDE) NA BIBLIOTECA DIGITAL DA UFG
Na qualidade de titular dos direitos de autor, autorizo a Universidade Federal de Goiás (UFG) a disponibilizar, gratuitamente, por meio da Biblioteca Digital de Teses e Dissertações (BDTD/UFG), sem ressarcimento dos direitos autorais, de acordo com a Lei nº 9610/98, o do-cumento conforme permissões assinaladas abaixo, para fins de leitura, impressão e/ou down-load, a título de divulgação da produção científica brasileira, a partir desta data.
1. Identificação do material bibliográfico: Trabalho de Conclusão de Curso de Mestrado Profissional
2. Identificação da Tese ou Dissertação
Autor (a): Normando Silva Junior
E-mail: normado.prof@gmail.com
Seu e-mail pode ser disponibilizado na página? [X]Sim [ ] Não Vínculo empregatício do autor Mestrando Bolsista
Agência de fomento: Coord. Aperf. De Pessoal de Nível Superior Sigla: CAPES
País: Brasil UF: GO CNPJ: 00889834/0001-08
Título: Aplicações para o Princípio de Indução Matemática
Palavras-chave: Recorrência, Indução, Fibonacci, Teoria dos Números.
Título em outra língua: Applications for the principle of mathematical induction
Palavras-chave em outra língua: Recurrence, Induction, Fibonacci, Numbers Theory.
Área de concentração: Matemática do Ensino Básico
Data defesa:(dd/mm/aaaa) 26/09/2014
Programa de Pós-Graduação: Mestrado
Orientador (a): Professor Dr. Maxwell Lizete da Silva
E-mail: maxwelllizete@hotmail.com
Co-orientador (a):*
E-mail:
*Necessita do CPF quando não constar no SisPG
3. Informações de acesso ao documento:
Concorda com a liberação total do documento [X] SIM [ ] NÃO1
Havendo concordância com a disponibilização eletrônica, torna-se imprescindível o en-vio do(s) arquivo(s) em formato digital PDF ou DOC da tese ou dissertação.
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________________________________________ Data: ____ / ____ / _____ Assinatura do (a) autor (a)
1
◆♦r♠❛♥❞♦ ❙✐❧✈❛ ❏✉♥✐♦r
❆♣❧✐❝❛çõ❡s ♣❛r❛ ♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦
▼❛t❡♠át✐❝❛
❚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✳ ▼❛①✇❡❧❧ ▲✐③❡t❡
●♦✐â♥✐❛
Ficha catalográfica elaborada
automaticamente com os dados fornecidos pelo(a) autor(a).
Silva Júnior, Normando
Aplicações para o princípio de indução matemática [manuscrito] / Normando Silva Júnior. - 2014.
54 f.: il.
Orientador: Prof. Maxwell Lizete.
Dissertação (Mestrado) - Universidade Federal de Goiás, Instituto de Matemática e Estatística (IME) , Programa de Pós-Graduação em Matemática, Goiânia, 2014.
Bibliografia.
Inclui lista de figuras.
❚♦❞♦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✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✱ ❛♠✐❣♦s ❡ ❛ ❉❡✉s ♣❡❧❛ ♣r❡s❡♥ç❛ ❝♦♥st❛♥t❡ ❡ s✉♣♦rt❡ ❣r❛t✉✐t♦✳
❆❣r❛❞❡ç♦ à ❈❆P❊❙ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦✳ ❆ ❜♦❧s❛ ❞❡ ♠❡str❛❞♦ ♦♣♦rt✉♥✐③♦✉ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ♠✐♥✐♠✐③❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❛✉❧❛s ❞✉r❛♥t❡ ♦s ❝ré❞✐t♦s ❝✉rs❛❞♦s✱ ❡ q✉❡ ♥❡st❡ ♣❡rí♦❞♦✱ ✈✐❛❜✐❧✐③♦✉ ♦s ❡st✉❞♦s tã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ ❢♦r♠❛çã♦ só❧✐❞❛ q✉❡ ♦❜t✐✈❡ ♥❡st❡s ❞♦✐s ❛♥♦s ❥✉♥t♦ à ❯❋●✳
❆♦ ❣r❛♥❞❡ ♠❡str❡ ❡ ♦r✐❡♥t❛❞♦r ♣r♦❢❡ss♦r ▼❛①✇❡❧❧ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ✐♥✜♥✐t❛ ❡ t♦❧❡râ♥❝✐❛ ✐♥❝♦♠❡♥s✉rá✈❡❧ ❛♣r❡s❡♥t❛❞❛ ❞✉r❛♥t❡ t♦❞♦ ♦ ♣r♦❝❡ss♦ ❞❡ ❝r✐❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ♣r♦❝✉r♦✉✲s❡ ❛♣r❡s❡♥t❛r s✐st❡♠❛t✐❝❛♠❡♥t❡ ❝♦♥❤❡❝✐♠❡♥t♦s ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❞❡ ♠❛♥❡✐r❛ ❝❧❛r❛ ❡ ❛❝❡ssí✈❡❧ ❛ ✉♠ ♣ú❜❧✐❝♦ ♠❛✐s ❛❜r❛♥❣❡♥t❡ ❞♦ q✉❡ ♦ ✉s✉❛❧ ❡♠ tr❛❜❛❧❤♦s ❛❝❛❞ê♠✐❝♦s ❞❡♥tr♦ ❞❛ ♠❛t❡♠át✐❝❛✳ ❖ ♣r✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✱
P IM✱❢♦✐ s❡♠♣r❡ ♦ ♣❛♥♦ ❞❡ ❢✉♥❞♦ ❡ ❛❜♦r❞❛❞♦ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❞❡♠♦♥str❛çõ❡s✱❞❡
♠❛♥❡✐r❛ ❛ ♣❡r♠❡❛r q✉❛s❡ t♦❞♦s r❡s✉❧t❛❞♦s✱❡ ❡♠ ❝❛❞❛ s❡çã♦✱❛♦ ♠❡♥♦s ✉♠ ❡①❡♠♣❧♦ ♥✉✲ ♠ér✐❝♦ ❢♦✐ ❞❛❞♦ ❢❛❝✐❧✐t❛♥❞♦ ❛ss✐♠ ♦ ♣r♦❡♠✐♥❡♥t❡ ❧❡✐t♦r q✉❡ ❡st❡❥❛ ✐♥✐❝✐❛♥❞♦ s❡✉s ❡st✉❞♦s ❡♠ ♠❛t❡♠át✐❝❛ ❡ ✐♥t❡♥❝✐♦♥❛♥❞♦ s❡♠♣r❡ ♥♦ ♠í♥✐♠♦ ✐♥st✐❣❛r ♦ s❡♥t✐♠❡♥t♦ ✐♥✈❡st✐❣❛t✐✈♦ ❡♠ t♦❞♦s ❧❡✐t♦r❡s✳ ❆s ❢r❛çõ❡s ❝♦♥tí♥✉❛s✱❛ss✉♥t♦ ♥ã♦ tã♦ ❡①♣❧♦r❛❞♦ ♠❛s ❡①tr❡♠❛♠❡♥t❡ r✐❝♦ ❡♠ ❛♣❧✐❝❛çõ❡s ♥❛ ❢ís✐❝❛ ❡ ❝á❧❝✉❧♦✱t❛♠❜é♠ ♠♦str♦✉✲s❡ ❢❛♠✐❧✐❛r ❝♦♠ ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳ ❊♠ s❡q✉ê♥❝✐❛✱❢♦r❛♠ ❛♣r❡s❡♥t❛❞♦s ❞♦✐s ♣r♦❜❧❡♠❛s ❝❧áss✐❝♦s ❞❡ ❝❛rát❡r ❧ú✲ ❞✐❝♦s✱❚♦rr❡ ❞❡ ❍❛♥♦✐ ❡ ♦ Pr♦❜❧❡♠❛ ❞❛ ▼♦❡❞❛ ❋❛❧s❛✱q✉❡ ❛ ♣❛rt✐r ❞❡ ❡①❡♠♣❧♦s s✐♠♣❧❡s ❝♦♥s❡❣✉✐✉✲s❡ ❧♦❣♦ ❡♠ s❡q✉ê♥❝✐❛ ❣❡♥❡r❛❧✐③❛r ❞❡♠♦♥str❛♥❞♦ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ♦s ♣r♦✲ ❜❧❡♠❛s ♣❛r❛ q✉❛❧q✉❡r ♥ú♠❡r♦ ♥❛t✉r❛❧✳ P♦r ✜♠✱❛s ❘❡❝♦rrê♥❝✐❛s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ❡ Pr♦❣r❡ssã♦ ❆r✐t♠ét✐❝❛ ❞❡ ❖r❞❡♠ ❙✉♣❡r✐♦r ❢♦r❛♠ ❡①♣♦st❛s ❡ s✉❛s r❡❧❛çõ❡s í♥t✐♠❛s ❝♦♠ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱❡ ❡ss❡s ♥ú♠❡r♦s ❡♥tã♦ ❛❝❛❜❛r❛♠ s❡ t♦r♥❛♥❞♦ ♠♦t✐✈❛❞♦r❡s ♣❛r❛ t♦❞♦ ♦ tr❛❜❛❧❤♦✱q✉❡ ♣r❡③❛ ♣♦r r❡s✉❧t❛❞♦s ❞❡♠♦♥strá✈❡✐s ❛tr❛✈és ❞♦
P IM ♦✉ q✉❡ t❡♥❤❛ r❡❧❛çã♦ ❝♦♠ ❡ss❛ s❡q✉ê♥❝✐❛✱s❡♥❞♦ q✉❡ s❡♠♣r❡ s❡ ♣r♦❝✉r♦✉ ❢♦rt❛❧❡✲
❝❡r ❛ ❛❞♠✐r❛çã♦ ♣❡❧♦s s❡✉s ❞✐á❧♦❣♦s ❝♦♠ r❛♠♦s tã♦ ❛♣❛r❡♥t❡♠❡♥t❡ ❡st❛♥q✉❡s ❡ q✉❡ sã♦ ❢❛♠✐❧✐❛r❡s s❡♥ã♦ ♣❡❧♦ s✉r❣✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ♥❡ss❡s tó♣✐❝♦s✳
P❛❧❛✈r❛s✲❝❤❛✈❡
❘❡❝♦rrê♥❝✐❛✱■♥❞✉çã♦✱❋✐❜♦♥❛❝❝✐✱❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✳
❆❜str❛❝t
❚❤✐s ♣❛♣❡r s♦✉❣❤t t♦ s②st❡♠❛t✐❝❛❧❧② ♣r❡s❡♥t ❦♥♦✇❧❡❞❣❡ ♦♥ t❤❡ ♥✉♠❜❡r t❤❡♦r② ✐♥ ❛ ❝❧❡❛r ❛♥❞ ❛❝❝❡ss✐❜❧❡ ✇❛② t♦ ❛ ❜r♦❛❞❡r ❝♦♠♠✉♥✐t② t❤❛♥ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❛❝❛❞❡♠✐❝s✳ t❤❡
♣r✐♥❝✐♣❧❡ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ✐♥❞✉❝t✐♦♥ P M I✱ ✇❛s t❤❡ ❜❛❝❦❣r♦✉♥❞ ❛♥❞ t❤❡ ♠❛✐♥ t♦♦❧ t♦ ❛❧❧
❞❡♠♦♥str❛t✐♦♥s s♦ t♦ ♣❡r♠❡❛t❡ ❛❧♠♦st ❛❧❧ t❤❡ r❡s✉❧ts ❛♥❞✱ ✐♥ ❡❛❝❤ s❡❝t✐♦♥✱ ❛t ❧❡❛st ♦♥❡ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡ ✇❛s ❣✐✈❡♥✱ t❤✉s ♠❛❦✐♥❣ ✐t ❡❛s✐❡r t♦ r❡❛❞❡rs ❜❡❣✐♥♥✐♥❣ t❤❡✐r st✉❞✐❡s ✐♥ ♠❛t❤ ❛♥❞ ❛❧✇❛②s s❡❡❦✐♥❣ t♦ ❛t ❧❡❛st ❡♥❝♦✉r❛❣❡ t❤❡ ✐♥✈❡st✐❣❛t✐✈❡ ❢❡❡❧✐♥❣ ♦♥ ❛❧❧ r❡❛❞❡rs✳ ❚❤❡ ❝♦♥t✐♥✉♦✉s ❢r❛❝t✐♦♥s✱ s✉❜❥❡❝t ❝♦♠♠♦♥❧② ♦✈❡r❧♦♦❦❡❞✱ ②❡t ✇✐t❤ ✈❛st ❛♣♣❧✐❝❛t✐♦♥s ✐♥ P❤②s✐❝s ❛♥❞ ❈❛❧❝✉❧✉s✱ ♣r♦✈❡❞ t♦ ❜❡ ❢❛♠✐❧✐❛r ✇✐t❤ t❤❡ ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡rs✳ ❙❡q✉❡♥t✐❛❧❧②✱ t✇♦ ❝❧❛ss✐❝ ❣❛♠❡ ♣r♦❜❧❡♠s ✇❡r❡ ♣r❡s❡♥t❡❞✱ ❚❤❡ ❍❛♥♦✐ ❚♦✇❡r ❛♥❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❢❛❧s❡ ❝♦✐♥✱ ✇❤✐❝❤ ❝♦✉❧❞✱ ❢r♦♠ s✐♠♣❧❡ ❡①❛♠♣❧❡s ❜❡ ❣❡♥❡r❛❧✐③❡❞ ❞❡♠♦♥str❛t✐♥❣ s♦❧✉t✐♦♥s ❢♦r t❤❡ ♣r♦❜❧❡♠s ❛t ❛♥② ❣✐✈❡♥ ♥❛t✉r❛❧ ♥✉♠❜❡r✳ ❋✐♥❛❧❧②✱ t❤❡ ❧✐♥❡❛r r❡❝✉rr❡♥❝❡s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛♥❞ t❤❡ ❤✐❣❤❡r ♦r❞❡r ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ✇❡r❡ s❤♦✇♥ t♦ ❤❛✈❡ ❞❡❡♣ ❝♦♥♥❡❝t✐♦♥s ✇✐t❤ t❤❡ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡✱ s♦ t❤❡♥✱ t❤❡s❡ ♥✉♠❜❡rs ❜❡❝❛♠❡ t❤❡ ♠❛✐♥ ♠♦t✐✈❛t♦rs ❢♦r
❛❧❧ t❤❡ ♣❛♣❡r✱ t❤❛t ♣r✐③❡s ❢♦r ❞❡♠♦♥str❛❜❧❡ r❡s✉❧ts t❤r♦✉❣❤ P M I ♦r r❡❧❛t❡❞ t♦ t❤✐s
s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs✱ ❛♥❞ ❛❧✇❛②s s♦✉❣❤t t♦ str❡♥❣t❤❡♥ t❤❡ ❛❞♠✐r❛t✐♦♥ ♦❢ t❤❡ ❞✐❛❧♦❣✉❡s ✇✐t❤ ❜r❛♥❝❤❡s ❛♣♣❛r❡♥t❧② s♦ ✜①❡❞ t❤❛t ❛r❡ t❤❡ ❢❛♠✐❧✐❛r t❤r♦✉❣❤ t❤❡ ❛♣♣❡❛r❛♥❝❡ ♦❢ t❤❡ ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡rs ✐♥ t❤❡s❡ t♦♣✐❝s✳
❑❡②✇♦r❞s
❘❡❝✉rr❡♥❝❡✱ ■♥❞✉❝t✐♦♥✱ ❋✐❜♦♥❛❝❝✐✱ ◆✉♠❜❡rs ❚❤❡♦r②✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✸✳✶ ❚♦rr❡ ❞❡ ❍❛♥♦✐✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✷ ❇❛❧❛♥ç❛ ❞❡ ❞♦✐s ♣r❛t♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
❙✉♠ár✐♦
✶ ❖ ❝♦♠❡ç♦ ❞❡ ✉♠❛ ♣❡q✉❡♥❛ ❡r❛ ✶✸
✷ ■♥❞✉çã♦ ❡ ❋✐❜♦♥❛❝❝✐ ✶✺
✷✳✶ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸❋r❛çõ❡s ❈♦♥tí♥✉❛s ❡ ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸ ❯s♦ ❞♦ P IM ❞❡ ♠❛♥❡✐r❛ ❧ú❞✐❝❛ ✸✹
✸✳✶ ❚♦rr❡ ❞❡ ❍❛♥♦✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✹ ✸✳✷ Pr♦❜❧❡♠❛ ❞❛s ▼♦❡❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✼
✹ ❙❡q✉ê♥❝✐❛s ❡ ❘❡❝♦rrê♥❝✐❛ ▲✐♥❡❛r ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ✹✶
✹✳✶ Pr♦❣r❡ssã♦ ❆r✐t♠ét✐❝❛ ❞❡ ❖r❞❡♠ ❙✉♣❡r✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✷ ❘❡❝♦rrê♥❝✐❛ ▲✐♥❡❛r ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✺ ❯♠❛ ❆✉❧❛ ❙♦❜r❡ ■♥❞✉çã♦ ✹✽
✻ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✺✶
❈❛♣ít✉❧♦ ✶
❖ ❝♦♠❡ç♦ ❞❡ ✉♠❛ ♣❡q✉❡♥❛ ❡r❛
❈♦♠♦ s✉r❣✐r❛♠ ❛s ✐❞❡✐❛s
◗✉❛♥❞♦ ❡♠ ✷✵✶✶ ✐♥✐❝✐❛✈❛ ♦ ♣r❡♣❛r❛tór✐♦ ♣❛r❛ r❡❛❧✐③❛çã♦ ❞♦ ❡①❛♠❡ ❞❡ ❛❝❡ss♦ ❛♦ ♣r♦✲ ❣r❛♠❛ ❞❡ ♠❡str❛❞♦ ♣r♦✜ss✐♦♥❛❧ ❡♠ ♠❛t❡♠át✐❝❛ ✲ P❘❖❋▼❆❚ ✲ ❡r❛ ✐♠♣♦ssí✈❡❧ ✈✐s❧✉♠✲ ❜r❛r ♦s ❝✉♥❤♦s q✉❡ s❡ ❢♦r♠❛r✐❛♠ ♣❡❧♦s ❡st✉❞♦s ❡ ❛❜❞✐❝❛çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ❝♦♥❝r❡t✐③á✲ ❧♦s✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛s ❞✐s❝✐♣❧✐♥❛s ❞❡ ▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛ ❥á ♥♦ ♣r✐♠❡✐r♦ s❡♠❡str❡ ❞❡ ❝✉rs♦ ❡ ❆r✐t♠ét✐❝❛ ♥♦ q✉❡ s❡ s❡❣✉✐r✐❛✱ s❡r✐❛♠ ✈✐t❛✐s ♣❛r❛ ❡ss❡ tr❛❜❛❧❤♦✱ q✉❛♥❞♦ ❡♠ t❡♠♣♦✱ ❡r❛ ♣♦ssí✈❡❧ ❛♥❛❧✐s❛r ❛s ❛✈❛❧✐❛çõ❡s ❛♥t❡r✐♦r❡s ❞❡st❛s ❞✐s❝✐♣❧✐♥❛s ♥♦ ♣♦rt❛❧ ♥❛❝✐♦♥❛❧ ❞♦ ♣r♦❣r❛♠❛✱ ❡ ❛q✉❡❧❛ ❝✉r✐♦s✐❞❛❞❡ ♥❛t✉r❛❧ s✉r❣✐♥❞♦ ♣❡❧❛ ❛❞♠✐r❛çã♦ s❡♠♣r❡ ♣r❡s❡♥t❡ ♥♦ q✉❡ t❛♥❣❡ tó♣✐❝♦s ❞❡ ♠❛t❡♠át✐❝❛✳
❊♠ ✉♠❛ t❡♥t❛t✐✈❛ ❞❡ ❢❛③❡r r❡❧❛çã♦ ❡♥tr❡ ♦s ❡st✉❞♦s r❡❛❧✐③❛❞♦s ♥♦ ❝✉rs♦ ❞❡ ♠❡str❛❞♦ ♣r♦✜ss✐♦♥❛❧ ✲ P❘❖❋▼❆❚ ✲ ❡ ♥♦s ❝♦♥t❡ú❞♦s ♠✐♥✐str❛❞♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❝♦♥t✐❞♦s ♥♦s P❈◆ ✭P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛❧✮ ♥❛s❝❡✉ ❛ ✐❞❡✐❛ ❞❡ ❞✐s❝♦rr❡r s♦❜r❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ ♣♦✐s ❛❧é♠ ❞❡ s❡r ❝♦♥t❡♠♣❧❛❞❛ ♥♦ ❡♥s✐♥♦ ❞❡ r❡❝♦rrê♥❝✐❛ ❞✉r❛♥t❡ ♦s ❡st✉❞♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ tr❛t❛✲s❡ ❞❡ ✉♠ ❛ss✉♥t♦ ✐♥t❡r❡ss❛♥tíss✐♠♦ ♣❡❧❛s ❝❛r❛❝t❡ríst✐❝❛s ✐♥trí♥✲ s❡❝❛s ❡ ❞❡ ✐♥❝rí✈❡❧ r❡❧❛çã♦ ❝♦♠ ♦✉tr♦s ❝❛♠♣♦s ❞❛ ♠❛t❡♠át✐❝❛✱ ✐♥❝❧✉✐♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ ✭P■▼✮ ❛❧é♠ ❞❛ r❛③ã♦ á✉r❡❛✳
◆❛ ♣r✐♠❡✐r❛ s❡çã♦ ❞♦ ❝❛♣ít✉❧♦ ✷✱ q✉❛♥❞♦ ❢❛❧❛✲s❡ ❞♦ P■▼✱ ✉t✐❧✐③❛❞♦ ♣❛r❛ ♣r♦✈❛r ❛❧❣✉♥s t❡♦r❡♠❛s ❛❝❡r❝❛ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ ❞❡✜♥❡✲s❡ ♦ ❛①✐♦♠❛ ❞❡ ✐♥❞✉çã♦ ❡ ❡♥tã♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✱ ❡st❡ ❡♥tã♦ s❡rá ✐♥str✉♠❡♥t♦ ✈❛❧✐♦s♦ ♣❛r❛ ❞❡♠♦♥str❛çõ❡s ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✳
❏á ♥♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡✱ ❛♣r❡s❡♥t❛✲s❡ ❡①❡♠♣❧♦s ♣rát✐❝♦s ❞♦ ✉s♦ ❞♦ P■▼✱ t❛❧ ❝♦♠♦
♦ ♣r♦❜❧❡♠❛ ❜❛ ❚♦rr❡ ❜❡ ❍❛♥♦✐✱ ✐♥❝❧✉s✐✈❡ s✉❛ ♣❛rt❡ ❤✐stór✐❝❛✱ ❡ t❛♠❜é♠ ♦ ♣r♦❜❧❡♠❛ ❜❛ ▼♦❡❜❛ ❋❛❧s❛✱ q✉❛♥❜♦ t❡♠✲s❡ ✉♠ ♣r♦❜❧❡♠❛ ❜❡ tr❛♥s♣♦rt❛r ❜✐s❝♦s ❡♠ três ❤❛st❡s ❜❡ ✉♠❛ ♠❛♥❡✐r❛ ❜❡♠ ♣❛rt✐❝✉❧❛r ❝♦♠ r❡❣r❛s ❝❧❛r❛s✱ ❡ ♥❛ ♠♦❡❜❛✱ ❝♦♥s❡❣✉✐r ❡♥❝♦♥tr❛r ✉♠❛ ♠♦❡❜❛ ❝♦♠ ♣❡s♦ ❜✐st✐♥t♦ ❡♠ ♠❡✐♦ ❜❡ t❛♥t❛s ♦✉tr❛s r❡❛❧✐③❛♥❜♦ ❛♣❡♥❛s ♣❡s❛❣❡♥s ❡♠ ✉♠❛ ❜❛❧❛♥ç❛ ❝♦♥st✐t✉í❜❛ ❛♣❡♥❛s ♣♦r ❜♦✐s ♣r❛t♦s✳❊ss❡s ♣r♦❜❧❡♠❛s✱ ❢♦r❛♠ ✐♥✐❝✐❛❧♠❡♥t❡ ❢♦r♠✉❧❛❜♦s ♣❛r❛ ✉♠ ♥ú♠❡r♦ ❜❡♠ ♣❛rt✐❝✉❧❛r ❜❡ ❜✐s❝♦s ❡ ♠♦❡❜❛s✱ ❡♠ s❡❣✉✐❜❛ ❣❡♥❡r❛❧✐③❛❜♦ ♣❛r❛ q✉❛❧q✉❡r q✉❛♥t✐❜❛❜❡✱ s❡♥❜♦ q✉❡ ♣❛r❛ ✐ss♦✱ ❝❛♠✐♥❤❛♠♦s ❛tr❛✈és ❜♦ Pr✐♥❝í♣✐♦ ❜❛ ❇♦❛ ❖r❜❡♥❛çã♦ q✉❛♥❜♦ ♦♣♦rt✉♥❛♠❡♥t❡ tr❛③✲s❡ ❛ ❜❡♠♦♥str❛çã♦✳
◆♦ q✉❛rt♦ ❝❛♣ít✉❧♦✱ sã♦ ❛♣r❡s❡♥t❛❜❛s ❛s r❡❝♦rrê♥❝✐❛s ❧✐♥❡❛r❡s ❜❡ ♦r❜❡♠ s✉♣❡r✐♦r✱ ❡ ❛♣r♦✈❡✐t❛♥❜♦ ❛s r❡❝♦rrê♥❝✐❛s✱ ✉s❛✲s❡ ♦ P■▼ ♣❛r❛ ❜❡♠♦str❛r ❛s ❢ór♠✉❧❛s ❜❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛✲❣❡♦♠étr✐❝❛✳▲♦❣♦ ❡♠ s❡❣✉✐❜❛✱ ❢❛③✲s❡ ✉♠ ❡❧♦ ❝♦♠ ❛s r❡❝♦rrê♥❝✐❛s ❜❡ s❡❣✉♥❜❛ ♦r❜❡♠ ❡ ♦s ♥ú♠❡r♦s ❜❛ s❡q✉ê♥❝✐❛ ❜❡ ❋✐❜♦♥❛❝❝✐ q✉❛♥❜♦ é ❛♣r❡s❡♥t❛❜♦ ✉♠❛ ❢ór♠✉❧❛ ❡①♣❧í❝✐t❛ ♣❛r❛ ♦s ♥ú♠❡r♦s ❜❡ss❛ s❡q✉ê♥❝✐❛✳
❏á ♥♦ ✜♥❛❧✱ ❛♣r❡s❡♥t❛✲s❡ ✉♠ ♣❧❛♥♦ ❜❡ ❛✉❧❛ ♣❛r❛ ❛♣❧✐❝❛çã♦ ❜❡ ✐♥❜✉çã♦ ❡♠ ❢ór♠✉❧❛s ♠❛t❡♠át✐❝❛s ❛♣r❡s❡♥t❛❜❛s ♥♦ ❡♥s✐♥♦ ♠é❜✐♦✳
❈❛♣ít✉❧♦ ✷
■♥❞✉çã♦ ❡ ❋✐❜♦♥❛❝❝✐
✷✳✶ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛
◆❛ ♠♦t✐✈❛çã♦ ❞❡ ❢♦r♠❛❧✐③❛r ♦s ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s✱ ❡♠ ❡s♣❡❝✐❛❧ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ N = {0,1,2,3,· · · }✱ ❢❡③✲s❡ ♥❡❝❡ssár✐♦ ❛ ❝♦♥str✉çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✱ q✉❡ ❡♥tã♦✱ ❢♦✐ ❛❞♦t❛❞♦ ❝♦♠♦ ♦ ❛①✐♦♠❛✿
Axioma1✿ ❆①✐♦♠❛ ❞❡ ✐♥❞✉çã♦✿ ❙❡❥❛ ❙ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ N t❛❧ q✉❡
i)0∈S.
ii)S é ❢❡❝❤❛❞♦ ♥❛ ♦♣❡r❛çã♦ ❞❡ s♦♠❛r ✶ ❛ s❡✉s ❡❧❡♠❡♥t♦s✱ ♦✉ s❡❥❛✱ ∀n, n∈S ⇒n+1∈S
❡♥tã♦✱ S=N.
❆ ♥♦ss❛ ❢❡rr❛♠❡♥t❛ ♣r✐♥❝✐♣❛❧ s❡rá ♦ ♣r✐♠❡✐r♦ P■▼ q✉❡ ❢♦r♠✉❧❛✲s❡ ❝♦♠♦ ✉♠ t❡♦r❡♠❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ ❝♦♠ s✉❛ ❞❡♠♦♥str❛çã♦ ❜❛s❡❛❞❛ ❡♠ ❬✼❪✳
❚❡♦r❡♠❛ ✷✳✶✳✶ ✭Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✮✳ ❙❡❥❛ ❛ ∈ N ❡ s❡❥❛ p(n) ✉♠❛
s❡♥t❡♥ç❛ ❛❜❡rt❛ ❡♠ n ✶✳ ❙✉♣♦♥❤❛ q✉❡
✐✮ ♣✭❛✮ é ✈❡r❞❛❞❡✱ ❡ q✉❡
ii) ♣❛r❛ t♦❞♦ n ≥ ❛, p(n) ✐♠♣❧✐❝❛ p(n+ 1) é ✈❡r❞❛❞❡✱ ❡♥tã♦✱ p(n) é ✈❡r❞❛❞❡ ♣❛r❛ t♦❞♦ n ≥❛✳
❉❡♠♦♥str❛çã♦ ✶✳ ❙❡❥❛ W = {n ∈ N❀ ✈❛❧❡ p(n)⑥ ✱ ♦✉ s❡❥❛✱ W é ♦ s✉❜❝♦♥❥✉♥t♦ ❞♦s
❡❧❡♠❡♥t♦s ❞❡ N ♣❛r❛ ♦s q✉❛✐s p(n) é ✈❡r❞❛❞❡✳
✶❋r❛s❡ ❞❡ ❝✉♥❤♦ ♠❛t❡♠át✐❝♦ ♦♥❞❡ ✜❣✉r❛ ❛ ❧❡tr❛ n ❝♦♠♦ ♣❛❧❛✈r❛ ❡ q✉❡ s❡ t♦r♥❛ ✉♠❛ s❡♥t❡♥ç❛
✈❡r❞❛❞❡✐r❛ ♦✉ ❢❛❧s❛ q✉❛♥❞♦ né s✉❜st✐t✉í❞♦ ♣♦r ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❡s♣❡❝í✜❝♦✳
P❡❧♦ ❢❛t♦ ❞❡ p(a) s❡r ✈❡r❞❛❞❡✐r❛ ❛ss❡❣✉r❛ q✉❡ a ♣❡rt❡♥❝❡ ❛ W✳ ❆ ✈❛❧✐❞❛❞❡ ❞❡ p(n)
✐♠♣❧✐❝❛ ♥❛ ✈❛❧✐❞❛❞❡ ❞❡ p(n+ 1)✱ s❡ ♥ ♣❡rt❡♥❝❡ ❛ ❲✱ ❡♥tã♦ n+ 1 t❛♠❜é♠ ♣❡rt❡♥❝❡ ❛
❲✳ P❡❧♦ ❆①✐♦♠❛✶✱ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♣❡rt❡♥❝❡ ❛ W✱ q✉❡r ❞✐③❡r✱ p(n) é ✈❡r❞❛❞❡✐r❛
♣❛r❛ t♦❞♦ n.
❯♠❛ ♦✉tr❛ ❞❡♠♦♥str❛çã♦ ❞❡♠❛s✐❛❞❛♠❡♥t❡ s✐♠♣❧✐st❛ ♣♦❞❡ s❡r ❛ss✐♠ ❢♦r♠✉❧❛❞❛✿
❉❡♠♦♥str❛çã♦ ✷✳ ❙✉♣♦♥❞♦ ♣♦r s✐♠♣❧✐❝✐❞❛❞❡ q✉❡ a = 0 ❡ ♣♦r ❛❜s✉r❞♦ q✉❡ W = N✱
t❡♠✲s❡ n0 ∈ N ❡ ∈/ W✱ ❧♦❣♦✱ P(n0) ♥ã♦ é ✈á❧✐❞♦✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ♦ s❡✉ ❛♥t❡❝❡ss♦r
t❛♠❜é♠ ♥ã♦ ♦ é✱ ♦✉ s❡❥❛✱ Pn0−1 ♥ã♦ é ✈á❧✐❞♦✱ ❡ r❡♣❡t✐♥❞♦ ❡ss❡ ♣r♦❝❡ss♦ t♦❞♦ n0+ 1
✈❡③❡s é s✉✜❝✐❡♥t❡ ♣❛r❛ ❝❤❡❣❛r ❛ P(0) t❛♠❜é♠ ✐♥✈❛❧✐❞♦✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦ ❥á q✉❡
P(a) é ✈á❧✐❞♦ ♣♦r ❤✐♣ót❡s❡✱ ❧♦❣♦✱ W =N.
❆❧é♠ ❞♦ ♣r✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ ❝❧áss✐❝♦✱ ❛❝✐♠❛ ❞❡♠♦♥str❛❞♦✱ ❡①✐st❡ ✉♠❛ s❡❣✉♥❞❛ ✈❡rsã♦ ✉s❛❞❛ ♣❛r❛ ♣r♦✈❛r ✐❞❡♥t✐❞❛❞❡s q✉❛♥❞♦ é ♣r❡❝✐s♦ ❝♦♥✜r♠❛r ❛ ✈❛❧✐❞❛❞❡
❞❛ ♣r♦♣♦s✐çã♦ ♣❛r❛ ✈❛❧♦r❡s ♠❡♥♦r❡s ❞♦ q✉❡ n✱ ❣❡r❛❧♠❡♥t❡ ♥❛ ♣❛ss❛❣❡♠ ❞❛ ❤✐♣ót❡s❡ ❞❡
✐♥❞✉çã♦ s✉♣♦♥❞♦ ✈á❧✐❞❛ ♣❛r❛ n ❡ ♣r♦✈❛r ❛ ✐♠♣❧✐❝❛çã♦ ❞❡ n+ 1✳
P❛r❛ ✐ss♦✱ ❝♦♠♦ ❤á ♦ r❡q✉✐s✐t♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✱ s❡❣✉❡ s✉❛ ❢♦r♠✉❧❛çã♦ ❡ s✉❛ ❞❡♠♦♥str❛çã♦ ❧♦❣♦ ❡♠ s❡❣✉✐❞❛ ✉s❛♥❞♦ ❛r❣✉♠❡♥t♦s ❞❡ ❬✹❪✳
❚❡♦r❡♠❛ ✷✳✶✳✷ ✭Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✮✳ ❚♦❞♦ s✉❜❝♦♥❥✉♥t♦ ❆ ⊂ N ♥ã♦ ✈❛③✐♦
♣♦ss✉✐ ✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦✱ ✐st♦ é✱ ✉♠ ❡❧❡♠❡♥t♦ t t❛❧ q✉❡ t≤ n ♣❛r❛ t♦❞♦ n∈A✳
❉❡♠♦♥str❛çã♦✳ ◆❡st❛ ❞❡♠♦♥str❛çã♦ ✈❛♠♦s ❞❡♥♦t❛r In = {p ∈ N/0 ≤ p ≤ n} ♦ ❝♦♥✲
❥✉♥t♦ ❞♦s ♥❛t✉r❛✐s ♠❡♥♦r❡s ♦✉ ✐❣✉❛✐s ❛ n✳ ❙ ❡1∈A❡♥tã♦ s❡rá ♦ ♠❡♥♦r ❡❧❡♠❡♥t♦ ❞❡ A✳
P♦ré♠✱ s❡ 1 ∈ A ❡♥tã♦ ❝♦♥s✐❞❡r❛✲s❡ ♦ ❝♦♥❥✉♥t♦ X ❞♦s ♥❛t✉r❛✐s n ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡
❞❡ q✉❡ In ⊂Ac✱ ❯♠❛ ✈❡③ q✉❡ I1 ⊂N\A✱ ♥♦t❛✲s❡ q✉❡ 1∈X✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❥á q✉❡ A
é ♥ã♦ ✈❛③✐♦✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ X = N ♥❡❣❛♥❞♦ ❛ ❤✐♣ót❡s❡ ✭✐✐✮✱ ❞❡✈❡♥❞♦ ❡♥tã♦ ❡①✐st✐r ✉♠
n ∈ X t❛❧ q✉❡ n+ 1 ∈ X✳ ❊♥tã♦✱ In = 1,2,3,· · · , n ⊂ N−A ♠❛s n0 = n+ 1 ∈ A✳
P♦rt❛♥t♦ t❡♠✲s❡ q✉❡ n0 é ♦ ♠❡♥♦r ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ❆✳
❖♣♦rt✉♥❛♠❡♥t❡✱ s❡♥❞♦ ♣♦ssí✈❡❧ ❛❣♦r❛ ♣❡❧♦ ❢❛t♦ ❞♦ P✳❇✳✵✳ ❡①♣❧✐❝✐t❛❞♦ ❡ ❝♦♠♣r♦✈❛❞♦✱ ❢♦r♠✉❧❛✲s❡✿
❚❡♦r❡♠❛ ✷✳✶✳✸ ✭✷❛ ❢♦r♠❛ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✮✳ ❙❡❥❛ ♣✭♥✮ ✉♠❛ s❡♥✲
t❡♥ç❛ ❛❜❡rt❛ t❛❧ q✉❡
✐✮ ♣✭❛✮ é ✈❡r❞❛❞❡✱ ❡ q✉❡
✐✐✮ ❙❡ ♣❛r❛ ❝❛❞❛ ♥ ♥❛t✉r❛❧✱ ♣✭❛✮✱ ♣✭❛✰✶✮✱ · · ·✱ ♣✭♥✮ ✐♠♣❧✐❝❛ q✉❡ ♣✭♥✰✶✮ é ✈❡r❞❛❞❡✱
❡♥tã♦ ♣✭♥✮ é ✈❡r❞❛❞❡ ♣❛r❛ t♦❞♦ ♥ ≥ ❛✳
❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦ ✭❛✰N✮ ❂ (a, a + 1, a + 2,· · ·)✳ ❈♦♥str✉✐♥❞♦ ♦
❝♦♥❥✉♥t♦ S ={m ∈(a+N)t❛❧ q✉❡ ✈❛❧❡ p(m)}.❊ss❡ é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s m q✉❡
✈❛❧❡♠ p(m)s❡♥❞♦ ♦s t❡r♠♦s ❞♦s ♥❛t✉r❛✐s ♠❛✐s a q
■♥t❡♥❝✐♦♥❛✲s❡ ♠♦str❛r q✉❡
W ={(a+N)\S}
é ✈❛③✐♦✳ ❙✉♣♦♥❞♦ ♣♦r r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦ q✉❡ ✈❛❧❡ ♦ ❝♦♥trár✐♦✱ ❡♥tã♦✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❛
❜♦❛ ♦r❞❡♥❛çã♦✱ W t❡r✐❛ ❞❡ ❢❛t♦ ✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦ ❝❤❛♠❛❞♦ ❛q✉✐ ❞❡t✱ ❡ ♣❡❧❛ ❤✐♣ót❡s❡
(i)♥♦s ❣❛r❛♥t❡ q✉❡ t ∈W é ✐♠❡❞✐❛t♦ ✐♥❢❡r✐r q✉❡ ❡①✐st❡ ✉♠ ♥❛t✉r❛❧ m ❞❡ ♠❛♥❡✐r❛ ❛ ♥♦s
❢❛③❡r t=a+m > a✳ ❊♥tã♦✱a, a+ 1, a+ 2,· · · , a−1∈W✱ ❡♥tã♦ a, a+ 1,· · · , a−1∈S✳
❆❣♦r❛✱ (ii) ♣❡r♠✐t❡ ❝♦♥❝❧✉✐r q✉❡ t = t−1 + 1 ∈ S✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ ❢❛t♦ ❞❡ t ∈ W
❣❛r❛♥t✐♥❞♦ W =∅.
✷✳✷ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐
▲❡♦♥❛r❞♦ ❋✐❜♦♥❛❝❝✐ ✭✜❧❤♦ ❞❡ ❇♦♥❛❝❝✐♦✮✱ ♥❛t✉r❛❧ ❞❡ P✐s❛✱ r❡❣✐ã♦ ❝♦♠❡r❝✐❛❧ ♥♦ ❝❡♥tr♦ ❞❛ ■tá❧✐❛✱ ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ♠✉✐t♦ ✐♥✢✉❡♥t❡ ♥♦ s❡✉ t❡♠♣♦✱ ❡ ❛té ❤♦❥❡ r❡❝♦♥❤❡❝✐❞♦ ♣❡❧❛s s✉❛s ❝♦♥tr✐❜✉✐çõ❡s✱ ❞❡♥tr❡ ❡❧❛s ❛ r❡❝♦♠❡♥❞❛çã♦ ❞♦ ✉s♦ ❞♦s ❛❧❣❛r✐s♠♦s ✐♥❞♦✲❛rá❜✐❝♦s ❡ ♦ ✉s♦ ❞♦ sí♠❜♦❧♦ ✵ ♣❛r❛ ♦ ♥ú♠❡r♦ ③❡r♦ ✭❝❤❛♠❛❞♦ ❞❡ ③❡♣❤✐r✉♠ ❡♠ ár❛❜❡✮✳
❉❡♥tr❡ s❡✉s ♣r♦❜❧❡♠❛s ❝♦♥t✐❞♦s ❡♠ s❡✉s tr❛❜❛❧❤♦s✱ t❛❧✈❡③ ♦ ♠❛✐s ❢❛♠♦s♦ s❡ tr❛t❛ ❞♦ ♣r♦❜❧❡♠❛ ❞♦s ❝♦❡❧❤♦s✱ ♦✉ ❝♦♠♦ ✜❝❛r✐❛ ❝♦♥❤❡❝✐❞♦✱ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ q✉❡ ♥♦ ♦r✐❣✐♥❛❧✱ ▲✐❜❡r ❆❜❛❝❝✐ ❞❡ ✶✷✵✷ r♦t✉❧❛✲s❡ ❛ss✐♠✿ ◗✉♦t ♣❛r✐❛ ❝♦♥✐❝✉❧♦r✉♠ ✐♥ ✉♥♦ ❛♥♥♦ ❡① ✉♥♦ ♣❛r✐♦ ❣❡r♠✐♥❡♥t✉r✳
❚r❛❞✉③✐♥❞♦ ♣❛r❛ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❝♦♥t❡♠♣♦râ♥❡❛ ♣♦❞❡r✐❛ ❧✐✈r❡♠❡♥t❡ s❡r ❛ss✐♠✿ ✉♠ ❝❛s❛❧ ❞❡ ❝♦❡❧❤♦s r❡❝é♠ ♥❛s❝✐❞♦s ❢♦✐ s❡♣❛r❛❞♦✳ ❙✉♣♦♥❞♦ q✉❡ ❛ ❝❛❞❛ ♠ês ✉♠ ❝❛s❛❧ ❞❡ ❝♦❡❧❤♦s ♣r♦❞✉③ ♦✉tr♦ ❝❛s❛❧ ❡ q✉❡ ❝❛❞❛ ✉♠ ❞❡st❡s ❝♦♠❡ç❛ ❛ ♣r♦❝r✐❛r ❞♦✐s ♠❡s❡s ❞❡♣♦✐s ❞♦ s❡✉ ♥❛s❝✐♠❡♥t♦✱ ❞❡t❡r♠✐♥❛r q✉❛♥t♦s ❝❛s❛✐s ❤❛✈❡rã♦ ♣❛ss❛❞♦ ✉♠ ❛♥♦✳
❯♠❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✱ ♣♦❞❡r✐❛ s❡r ❛tr❛✈és ❞❛ ❝♦♥str✉çã♦ ❞❛ s❡q✉ê❝♥✐❛ ♦♥❞❡ ❝❛❞❛ ♥ú♠❡r♦ r❡♣r❡s❡♥t❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛s❛✐s✿ ✶✱✶✱✷✱✸✱✺✱✽✱✶✸✱✷✶✱· · · un✱· · ·✱ ♦♥❞❡ ❝❛❞❛
t❡r♠♦✱ ♣♦❞❡ s❡r ❡①♣r✐♠✐❞♦ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ r❡❝♦rrê♥❝✐❛ un=un−1+un−2✳ ❊st❛ ❢ór♠✉❧❛
t❡♠ ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s ❡ ❝✉r✐♦s✐❞❛❞❡s✱ q✉❡✱ ♣r❡♣♦t❡♥t❡♠❡♥t❡ ♣♦❞❡rí❛♠♦s ❞✐③❡r q✉❡ ▲❡♦♥❛r❞♦ ♥ã♦ ❤❛✈✐❛ ❝♦♠♦ ♠❡♥s✉r❛r ❡♠ s✉❛ ❝r✐❛çã♦✱ ❛❧❣✉♠❛s ❞❡st❛s ❜❡❧❛s ♣r♦♣r✐❡❞❛❞❡s s❡rã♦ ❢♦r♠✉❧❛❞❛s ❡ ❞❡♠♦♥str❛❞❛s ❛q✉✐ ❢❛③❡♥❞♦ ✉s♦ ❞♦ P■▼✳
P❛r❛ ✈✐s✉❛❧✐③❛r ❛ s❡q✉ê♥❝✐❛✱ ✈❛♠♦s ❝♦♥tr✉✐r ✉♠❛ t❛❜❡❧❛ ❝♦♥t❡♥❞♦ ♦ ♠ês ❡♠ q✉❡stã♦✱ ♥ú♠❡r♦ ❞❡ ❝❛s❛✐s ❞❡ ❝♦❡❧❤♦s r❡❝é♠✲♥❛s❝✐❞♦s✱ ♦ ♥ú♠❡r♦ ❞❡ ❝❛s❛✐s ❞❡ ❝♦❡❧❤♦s ❞♦ ♠ês ❛♥t❡r✐♦r ❡ ♦ t♦t❛❧ ❞❡ ❝❛s❛✐s ❞❡ ❝♦❡❧❤♦s✳
❚❛❜❡❧❛ ✷✳✶✿ ◆ú♠❡r♦ ❞❡ ❝❛s❛✐s ❞❡ ❝♦❡❧❤♦s ♠ês ❛ ♠ês ▼ês ◆ú♠❡r♦ ❞❡ ❝❛s❛✐s ◆ú♠❡r♦ ❞❡ ❝❛s❛✐s ❚♦t❛❧ ❞❡ ❝❛s❛✐s
r❡❝é♠✲♥❛s❝✐❞♦s ❞♦ ♠ês ❛♥t❡r✐♦r ❞❡ ❝♦❡❧❤♦s
✶♦ ✶ ✵ ✶
✷◦ ✵ ✶ ✶
✸◦ ✶ ✶ ✷
✹◦ ✶ ✷ ✸
✺◦ ✷ ✸ ✺
✻◦ ✸ ✺ ✽
✼◦ ✺ ✽ ✶✸
✽◦ ✽ ✶✸ ✷✶
✾◦ ✶✸ ✷✶ ✸✹
✶✵◦ ✷✶ ✸✹ ✺✺
✶✶◦ ✸✹ ✺✺ ✽✾
✶✷◦ ✺✺ ✽✾ ✶✹✹
❋✐❝❛ ❢á❝✐❧ ❡♥t❡♥❞❡r q✉❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦❡❧❤♦s ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♠ês é ❞❡ ❢❛t♦ ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❝❛s❛✐s ❞♦ ♠ês ❛♥t❡r✐♦r ❛✉♠❡♥t❛❞♦ ❞♦ ♠ês ❛♥t❡r✐♦r ❛♦ ❛♥t❡r✐♦r✳
✷✳✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐
❉❡♥♦t❛♥❞♦ ♣♦r un ✉♠ ♥ú♠❡r♦ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ ✈❛♠♦s ♣r♦✈❛r ❛❧❣✉♠❛s
♣r♦♣r✐❡❞❛❞❡s r❡❧❛❝✐♦♥❛♥❞♦ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ✭▼❉❈✮ ❝♦♠ ♦s í♥❞✐❝❡s ❞♦s ♥ú♠❡r♦s
❞❡ ❋✐❜♦♥❛❝❝✐ ❡ ♦s ♣ró♣r✐♦s ♥ú♠❡r♦s✳ P❛r❛ ✐ss♦✱ s❡rá ✉s❛❞❛ ❛ ♥♦t❛çã♦ (un, um) ♣❛r❛
r❡♣r❡s❡♥t❛r ♦ ▼✳❉✳❈✳ ❡♥tr❡ ♦s ♥ú♠❡r♦s un ❡ um✳ q✉❡ ♥❡st❡ t❡①t♦✱ s❡rá ❛ss✉♠✐❞♦ q✉❡ ♦
▼✳❉✳❈✳ ❡♥tr❡ q✉❛✐sq✉❡r ❞♦✐s ♥ú♠❡r♦s s❡♠♣r❡ ❡①✐st❡✱ s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❞❡♠♦♥str❛r t❛❧ ❢❛t♦✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♥❡❝❡ss✐❞❛❞❡✱ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s s❡rã♦ ♣r❡s❝r✐t❛s ♣♦r r❡s✉❧t❛✲ ❞♦s ❡ s✉❛s r❡s♣❡❝t✐✈❛s ❞❡♠♦♥str❛çõ❡s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦s ♥ú♠❡r♦s ❝♦♥s❡❝✉t✐✈♦s ❞❡ ❋✐❜♦♥❛❝❝✐ q✉❡ ❢❛rá✉s♦ ❞♦ ▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s q✉❡ s❡❣✉❡✳
▲❡♠❛ ✷✳✷✳✶ ✭▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s✮✳ ❙❡❥❛♠ a, b, n ∈N ❝♦♠ a < na < b✱ ❡♥tã♦
(a, b) = (a, b−na)
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛d = (a, b−na)✳ ❈♦♠♦aé ✉♠ ♠ú❧t✐♣❧♦ ❞❡d❡(b−na)é ✉♠ ♠ú❧t✐♣❧♦
❞❡ d✱ s❡❣✉❡ q✉❡ bé ♠ú❧t✐♣❧♦ ❞❡ d❡b =b−na+na✳ ▲♦❣♦✱ dé ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡
b✱ q✉❡r ❞✐③❡r✱ d(a, b)✳ ❙✉♣♦♥❞♦ x= (a, b), q✉❡r ❞✐③❡r✱ x(a, b−na)⇔xd. ▼❛s
❝♦♠♦ d = (a, b−na) ❡♥tã♦ d x✱ ❢❛③❡♥❞♦ ❡♥tã♦ ❝♦♠ q✉❡ x = d ❡ ♥ã♦ só ✉♠ ❞✐✈✐s♦r
❞❡ (a, b−na) ❝♦♠♦ t❛♠❜é♠ ♦ ♠❛✐♦r ❞❡❧❡s✳
▲❡♠❛ ✷✳✷✳✷✳ ❉♦✐s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❝♦♥s❡❝✉t✐✈♦s sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✳
❉❡♠♦♥str❛çã♦✳ ❋❛③❡♥❞♦ ✉s♦ ❞♦ P■▼ ✈❛♠♦s ♠♦str❛r q✉❡ ♦ ▼✳❉✳❈✳ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s ❝♦♥s❡❝✉t✐✈♦s é ✉♠✳ ❈♦♠♦ ♦s ❞♦✐s ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ sã♦ ♦ ✶ ❡ ♦ ♣ró♣r✐♦ ✶✱ (1,1) = 1✱ q✉❡r ❞✐③❡r✱
(u1, u2) = 1.
❡ ❝♦♠ ✐ss♦ ❥át❡♠♦s ✈á❧✐❞❛ ❛ ❛✜r♠❛çã♦ ♣❛r❛ n = 1
❙✉♣♦♥❞♦ ✈á❧✐❞♦ ❛ ❛✜r♠❛çã♦ ♣❛r❛ ❛❧❣✉♠ n✱ q✉❡r ❞✐③❡r✱ (un+1, un) = 1 ❛❧é♠ ❞♦ ❢❛t♦
❞❡ un+2 =un+1+un ❡♥tã♦ un =un+2−un+1
❚❡♠✲s❡ ❡♥tã♦ ♣❡❧♦ ▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s ❛❝✐♠❛✱ ♦♥❞❡ a=un+1 ❡ b=un+2 q✉❡
(un+2, un+1) = (un+2−un+1, un+1).
▲♦❣♦✱ (un+2−un+1, un+1) = (un, un+1) = 1 ✜❝❛♥❞♦ ✈❛❧✐❞❛❞♦ ♣❡❧♦ P.I.M. ♣❛r❛ t♦❞♦ n ♥❛t✉r❛❧✳
❊①❡♠♣❧♦ ✶✳ ❊s❝♦❧❤❡♥❞♦ u8 ❡ u9 t❡♠♦s q✉❡ 21 ❡ 34 sã♦ ❞❡ ❢❛t♦ ♣r✐♠♦s ❡♥tr❡ s✐✱q✉❡r
❞✐③❡r✱ (u8, u9) = (21,34) = 1
P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ♣ró①✐♠♦ ▲❡♠❛✱ s❡rá✉s❛❞❛ ❛ s❡❣✉✐♥t❡ ✐❞❡♥t✐❞❛❞❡✿
un+m=un−1um+unum+1 ✭✷✳✶✮
✈❡r✐✜❝á✈❡❧ ❡♠ ❬✶❪✳
▲❡♠❛ ✷✳✷✳✸✳ ❙❡❥❛♠ n, m∈N∗ t❛✐s q✉❡✱ m❡n sã♦ ♠ú❧t✐♣❧♦s ❡♥tr❡ s✐✱ ❡♥tã♦ ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ♥♦s r❡s♣❡❝t✐✈♦s í♥❞✐❝❡s t❛♠❜é♠ ♦ sã♦✳
❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ❞❡♠♦♥s♦r❛r ♣❡❧♦ P■▼ ♣❛r❛ ❦ q✉❡ ✉♠❛ ✈❡③ m♠ú❧♦✐♣❧♦ ❞❡n ✈❛❧❡
um ♠ú❧♦✐♣❧♦ ❞❡ un✳ P❛r❛ k = 1 ♦❡♠♦s q✉❡ 1 = k∗1❝♦♠ k = 1 ❡ u1 = 1∗u1✳ ❙✉♣♦♥❞♦
❛❣♦r❛ ✈á❧✐❞♦ ♣❛r❛ ❛❧❣✉♠ k✱ ✐s♦♦ é✱um =kun✱ q✉❡r ❞✐③❡r✱ um ♠ú❧♦✐♣❧♦ ❞❡ un✳ ❯♠❛ ✈❡③
q✉❡ ✭✷✳✶✮❣❛r❛♥♦❡
um(k+1) =umk+m =umk−1um+umkum+1 ✭✷✳✷✮
♦❡♠♦s q✉❡ umk−1um é ♠ú❧♦✐♣❧♦ ❞❡um ❡ ♣❡❧❛ ❤✐♣ó♦❡s❡ ❞❡ ✐♥❞✉çã♦✱ umkum+1 ♠ú❧♦✐♣❧♦ ❞❡
um✱ ✐♠♣❧✐❝❛♥❞♦ q✉❡ um(k+1) é ♠ú❧♦✐♣❧♦ ❞❡ um ✈❛❧✐❞❛♥❞♦ ❛ss✐♠ ♦ r❡s✉❧♦❛❞♦✳
❊①❡♠♣❧♦ ✷✳
❊s❝♦❧❤❡♥❞♦ ✹ ❡ ✶✷✱ ✈❛♠♦s ✈❡r✐✜❝❛r q✉❡ u12 é ♠ú❧♦✐♣❧♦ ❞❡ u4. ❉❡ ❢❛♦♦✱ u12 = 144 ❡
u4 = 3✳ ❋❛❝✐❧♠❡♥♦❡ ✈❡r✐✜❝❛✲s❡ q✉❡ ✶✹✹❂✸·✹✽ ❝♦♥❢♦r♠❡ q✉❡rí❛♠♦s ❞❡♠♦♥s♦r❛r✳
❚❡♦r❡♠❛ ✷✳✷✳✶✳ ❉❛❞❛ ✉♠❛ s❡q✉ê♥❝✐❛ (un)n t❛❧ q✉❡ ∀m≥n,(am, an) = (an, ar)✱ ♦♥❞❡ r é ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ♠ ♣♦r ♥✱ ❡♥tã♦ t❡♠✲s❡ q✉❡
(am, an) = a(m,n)
✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ r1, r2,· · · , rs, rs+1 = 0 ♦s r❡s♦♦s ♣❛r❝✐❛✐s ♥♦ ❆❧❣♦r✐♦♠♦ ❞❡ ❊✉❝❧✐✲
❞❡s✱ ❧♦❣♦✱ ♦❡♠♦s q✉❡ rs = (m, n)✳ P♦r♦❛♥♦♦✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ (an)✱
(am, an) = (an, ar1) =· · ·= (ars, ars+1) = (ars,0) =a(m,n)
❈♦♠ ❡ss❡s ♦rês ❧❡♠❛s ❡ ♦ ♦❡♦r❡♠❛ ❛❝✐♠❛ ♣♦❞❡♠♦s ♣r♦✈❛r ✉♠ ♦❡♦r❡♠❛ ❞❡ ❣r❛♥❞❡ r❡❧❡✈â♥❝✐❛✱ ✉♠❛ ✈❡③ q✉❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ♣♦❞❡ ♥ã♦ s❡r ♦ã♦ ❡✈✐❞❡♥♦❡ ❞❡ s❡ ✈✐s✉❛❧✐③❛r ♣♦ré♠✱ ❞❡ ♠♦❞♦ ♠✉✐♦♦ ❡❧❡❣❛♥♦❡ ♣♦❞❡♠♦s ❢♦r♠✉❧❛r ❛ss✐♠✿
❚❡♦r❡♠❛ ✷✳✷✳✷✳ ❖ ▼✳❉✳❈✳ ❞❡ ❞♦✐s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ un ❡ um é ✉♠ ♥ú♠❡r♦ ❞❡
❋✐❜♦♥❛❝❝✐ ❡♠ q✉❡ s❡✉ í♥❞✐❝❡ é ❞❛❞♦ ♣❡❧♦ ▼✳❉✳❈✳ ❡♥tr❡ ♦s í♥❞✐❝❡s ♥ ❡ ♠✱ q✉❡r ❞✐③❡r✱
(un, um) =u(n,m)
❉❡♠♦♥str❛çã♦✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ✈❛♠♦s s✉♣♦r q✉❡ m ≥ n✱ ❡♥tã♦✱ ♣♦❞❡♠♦s
❡s❝r❡✈❡r m=nq+r✱ ❡ ♣♦rt❛♥t♦✱ ✉s❛♥❞♦ ✭✷✳✶✮ t❡♠♦s
um =unq+r =unq−1+unqur+1.
▲♦❣♦✱ ❝♦♠♦ ♦ ▲❡♠❛ ✷✳✷✳✷ ♥♦s ❣❛r❛♥t❡ q✉❡ unq é ♠ú❧t✐♣❧♦ ❞❡ un, t❡♠✲s❡ t❛♠❜é♠ q✉❡
♦ ▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s ♣❡r♠✐t❡ ❛✜r♠❛r q✉❡
(un, um) = (unq−1ur+unqur+1, un) = (unq−1ur, un) ✭✷✳✸✮
❖ ▲❡♠❛ ✷✳✷✳✶ ❣❛r❛♥t❡ q✉❡ (unq−1, unq) = 1✱ ❡ ♣♦r ✐ss♦ (unq−1, un)✱ ❝♦♥s❡q✉❡♥t❡✲ ♠❡♥t❡ ❞❡ ✭✷✳✷✮ s❡❣✉❡✲s❡ q✉❡
(um, un) = (un, ur).
❯♠❛ ✈❡③ q✉❡ ❥á ❢♦✐ ♣r♦✈❛❞♦ ♦ ❚❡♦r❡♠❛ ✷✳✷✳✶✱ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❡
(un, um) =u(n,m)
❊①❡♠♣❧♦ ✸✳ ◆❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ ♦ ❞é❝✐♠♦ q✉✐♥t♦ ♥ú♠❡r♦ é ✻✶✵ ❡ ♦ ✈✐❣és✐♠♦ t❡r♠♦ é ♦ ✻✼✻✺✱ ♣❡❧♦ t❡♦r❡♠❛ ❛❝✐♠❛✱
(u15, u20) = (610,6765) =u(15,20).
▲♦❣♦✱ ❝♦♠♦ (15,20) = 5 ❡♥tã♦ (610,6765) =u5 = 5
❊①❡♠♣❧♦ ✹✳ ◆❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ ♦ ✈✐❣és✐♠♦ q✉❛rt♦ t❡r♠♦ é ♦ ♥ú♠❡r♦ ✹✻✸✻✽ ❡ ♦ ❞é❝✐♠♦ s❡①t♦ é ♦ ✾✽✼ ❞❡ ♠❛♥❡✐r❛ q✉❡
(u24, u16) = (46368,987) =u(24,16).
▲♦❣♦✱ ❝♦♠♦ (24,16) = 8 ❡♥tã♦ (46368,987) =u8 = 21
❖s ♣ró①✐♠♦s ♣r♦❜❧❡♠❛s sã♦ ❝❧áss✐❝♦s ❡ s❡ r❡❢❡r❡♠ ❛ s♦♠❛s ❞❡ t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ♦♥❞❡ tr❛r❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ✉♠❛ s♦❧✉çã♦ ♠❛✐s s✐♠♣❧❡s ❡ ❡♥tã♦ ❛♣r❡s❡♥t❛r❡♠♦s t♦❞❛s ❡❧❛s ❢❛③❡♥❞♦ ✉s♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✿
Pr♦❜❧❡♠❛ ✶✳ ❉ê ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡
❋✐❜♦♥❛❝❝✐✳
❙♦❧✉çã♦✿ ❚❡♠♦s q✉❡ ♣❡❧❛ ♥❛t✉r❡③❛ ❞❛ s❡q✉ê♥❝✐❛ t♦❞♦ t❡r♠♦ é ❛ ❛❞✐çã♦ ❞♦s ❞♦✐s ❛♥t❡❝❡❞❡♥t❡s s❡♥❞♦ q✉❡ ❛ s❡q✉ê♥❝✐❛ ❝♦♠❡ç❛ ♣❡❧♦s ❞♦✐s ♣r✐♠❡r✐♦s ♥✉♠❡r✐❝❛♠❡♥t❡ ✐❣✉❛✐s ❛ ✶✳ ■ss♦ q✉❡r ❞✐③❡r q✉❡
u1 = 1
u2 = 1
u3 =u1+u2 = 2
❊♥tã♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ss✐♠✿
u1 =u3−u2 = 1
q✉❡r ❞✐③❡r✱ ❝❛❞❛ t❡r♠♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❞✐❢❡r❡♥ç❛ ❞♦s ❞♦✐s s✉❝❡ss♦r❡s ❡♠ ♠ó❞✉❧♦✳
u1 =u3−u2
u2 =u4−u3
u3 =u5−u4
✳✳✳=✳✳✳ un−1 =un+1−un
un =un+2−un+1
❋❛③❡♥❞♦ ❛ s♦♠❛ t❡r♠♦ ❛ t❡r♠♦ ♥♦s ❞♦✐s ♠❡♠❜r♦s ♦❜t❡♠♦s✿
u1+u2+u3+· · ·+un= (u3−u2) + (u4−u3) + (u5−u4) +·+ (un+1−un) + (un+2−un+1)
❈❛♥❝❡❧❛♥❞♦ ♦s t❡r♠♦s ❞♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ q✉❡ s❡ ❛♥✉❧❛♠ ♦❜t❡♠✲s❡✿
u1+u2 +u3+· · ·+un =un+2−u2
❯♠❛ ✈❡③ q✉❡ u2 = 1 ❡♥tã♦
u1+u2+u3+· · ·+un=un+2−1 ✭✷✳✹✮
Pr♦❜❧❡♠❛ ❜✳ ❉ê ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ♦r❞❡♠ í♠♣❛r
❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✳
❙♦❧✉çã♦✿ ❈♦♠♦ ❥á é s❛❜✐❞♦✱
u2 =u1
u4 =u3+u2 ⇒u3 =u4 −u2
u6 =u5+u4 ⇒u5 =u6 −u4
✳✳✳=✳✳✳
u2n =u2n−1+u2n−2 ⇒u2n−1 =u2n−u2n−2
❆ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❞❡ ♦r❞❡♠ í♠♣❛r ❡♥tã♦ ✜❝❛ ❛ss✐♠
u1+u3+u5+· · ·+u2n−1
❡ ❢❛③❡♥❞♦ ❛ s✉❜st✐t✉✐çã♦ ♣❡❧♦s ♥ú♠❡r♦s ❡♥❝♦♥tr❛❞♦s ❛❝✐♠❛
u2+u4 −u2 +u6−u4+· · ·+u2n−u2n−2
q✉❡ ❝❛♥❝❡❧❛♥❞♦ ♦s ♥ú♠❡r♦s s♦❜r❛ ❛♣❡♥❛s ♦ t❡r♠♦ u2n ♥♦s ❞❛♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡
u1+u3+u5+· · ·+u2n−1 =u2n ✭✷✳✺✮
Pr♦❜❧❡♠❛ ❜✳ ❉ê ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ♦r❞❡♠ ♣❛r ❞❛
s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✳
❙♦❧✉çã♦✿ ❙❡ ❛ s♦♠❛ ❞♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ❛té ❛ ♦r❞❡♠ 2n é✿
u1+u2+· · ·+u2n−1+u2n=u2n+2−1
❡ ❞♦s í♠♣❛r❡s ❛té ❛ ♦r❞❡♠ 2n−1✜❝❛ ❛ss✐♠✿
u1 +u3+u5+· · ·+u2n−1 =u2n.
❙✉❜tr❛✐♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ❛ ♣r✐♠❡✐r❛ ✐❣✉❛❧❞❛❞❡ ❞❛ s❡❣✉♥❞♦ ♦❜t❡♠✲s❡ ❛ s♦♠❛ ❞♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ♦r❞❡♠ ♣❛r ❥á ♥♦ s❡❣✉♥❞♦ ♠❡♠❜r♦
u2n+2−u2n−1
❯♠❛ ✈❡③q✉❡ u2n+1 =u2n+2−u2n t❡♠♦s
u2+u4+· · ·+u2n =u2n+1−1
❚❡♥❞♦ ♠♦str❛❞♦ ❛ s♦❧✉çã♦ ♣❛r❛ ♦s três ♣r♦❜❧❡♠❛s✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r ✉♠ ❡①❡♠♣❧♦ ♣❛r❛ ❝❛❞❛ ✉♠ ❞❡❧❡s ❡ ❡♥tã♦ ❢♦r♠❛❧♠❡♥t❡ ❛♣r❡s❡♥t❛r ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞❛s ❢ór♠✉❧❛s ❛tr❛✈és ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✳
❊①❡♠♣❧♦ ✺✳ ❆ s♦♠❛ ❞♦s ❞❡③ ♣r✐♠❡✐r♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ s❡r✐❛
u1+u2+u3+u4+u5+u6+u7+u8+u9+u10= 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55
u1+u2+u3+u4+u5+u6+u7 +u8+u9+u10=u12−1
u1+u2 +u3+u4+u5+u6+u7+u8+u9 +u10 = 144−1
u1+u2+u3 +u4+u5+u6+u7+u8+u9+u10= 143
❊①❡♠♣❧♦ ✻✳ ❆ s♦♠❛ ❞♦s s❡t❡ ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞❡ ♦r❞❡♠ í♠♣❛r ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ s❡r✐❛
u1+u3+u5+u7+u9 +u11+u13 = 1 + 2 + 5 + 13 + 34 + 89 + 233
u1+u3+u5+u7+u9+u11+u13=u14
u1+u3+u5+u7+u9+u11+u13= 377
❊①❡♠♣❧♦ ✼✳ ❆ s♦♠❛ ❞♦s ♦✐t♦ ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞❡ ♦r❞❡♠ ♣❛r ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦✲ ♥❛❝❝✐ s❡r✐❛
u2 +u4+u6+u8+u10+u12+u14+u16= 1 + 3 + 8 + 21 + 55 + 144 + 377 + 987
u2+u4+u6+u8 +u10+u12+u14+u16 =u17−1
u2 +u4+u6+u8+u10+u12+u14+u16= 1577−1
u2+u4+u6+u8+u10+u12+u14+u16= 1576
Pr♦❜❧❡♠❛ ✹✳ ❉❡♠♦♥str❡ q✉❡ ❛ s♦♠❛ ❞♦s n♣r✐♠❡✐r♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐
é ❞❛❞♦ ♣♦r un+2−1✳
❉❡♠♦♥str❛çã♦✳ ❙♦❧✉çã♦✿ P❡❧♦ P IM é ❢á❝✐❧ ✈❡r q✉❡ é ✈á❧✐❞❛ ♣❛r❛ n= 1✳ ❙✉♣♦♥❞♦ ✈á❧✐❞♦
♣❛r❛ ❛❧❣✉♠ k✱ q✉❡r ❞✐③❡r
u1+u2+u3+· · ·+uk =uk+2−1
✈❛♠♦s s♦♠❛r uk+1 ♥♦s ❞♦✐s ♠❡♠❜r♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ♦❜t❡♠♦s
u1+u2+u3+· · ·+uk+uk+1 =uk+2+uk+1−1
❈♦♠♦ uk+3 =uk+2+uk+1 ❡♥tã♦ t❡♠♦s ✐♠❡❞✐❛t❛♠❡♥t❡ q✉❡
u1+u2+u3+· · ·+uk+uk+1 =uk+3−1
Pr♦❜❧❡♠❛ ❜✳ ❉❡♠♦♥str❡ q✉❡ ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ♦r❞❡♠ í♠♣❛r ❞❛
s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ é ❞❛❞♦ ♣♦r un+1✳
❉❡♠♦♥str❛çã♦✳ ❙♦❧✉çã♦✿ P❡❧♦ P IM é ❢á❝✐❧ ❞❡ ✈❡r q✉❡ é ✈á❧✐❞❛ ♣❛r❛ n = 1✳ ❙✉♣♦♥❞♦
✈á❧✐❞❛ ♣❛r❛ ❛❧❣✉♠ k✱ q✉❡r ❞✐③❡r✱
u1+u3+u5+· · ·+u2k−1 =u2k
❙♦♠❛♥❞♦ u2k+1 ❛♦s ❞♦✐s ♠❡♠❜r♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ❡♥❝♦♥tr❛♠♦s
u1+u3+u5+· · ·+u2k−1+u2k+1 =u2k+u2k+1
❈♦♠♦ u2k+u2k+1 =u2k+2 ❡♥tã♦ é ✐♠❡❞✐❛t♦ ✐♥❢❡r✐r q✉❡
u1+u3+u5+· · ·+u2k−1+u2k+1 =u2k+3
Pr♦❜❧❡♠❛ ✻✳ ❉❡♠♦♥str❡ q✉❡ ❛ s♦♠❛ ❞♦s n♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ♦r❞❡♠ ♣❛r ❞❛ s❡q✉ê♥❝✐❛
❞❡ ❋✐❜♦♥❛❝❝✐ é ❞❛❞♦ ♣♦r u2n+1−1✳
❉❡♠♦♥str❛çã♦✳ ❙♦❧✉çã♦✿ P❡❧♦ P IM é ❢á❝✐❧ ❞❡ ✈❡r q✉❡ é ✈á❧✐❞❛ ♣❛r❛ n = 1✳ ❙✉♣♦♥❞♦
✈á❧✐❞❛ ♣❛r❛ ❛❧❣✉♠ k✱ q✉❡r ❞✐③❡r✱
u2+u4+u6+· · ·+u2k =u2k+1−1
❙♦♠❛♥❞♦ u2n+2 ❛♦s ❞♦✐s ♠❡♠❜r♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ♦❜t❡♠✲s❡
u2+u4+u6+· · ·+u2k+u2n+2 =u2k+1+u2n+2−1
❈♦♠♦ u2k+1 +u2n+2 = u2k+3 ♣♦❞❡♠♦s s✉❜st✐t✉✐r ♥♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ ❞❛ ✐❣✉❛❧❞❛❞❡
❛❝✐♠❛ ❡ ♦❜t❡r
u2+u4+u6+· · ·+u2k+u2n+2 =u2k+3−1
✷✳✸ ❋r❛çõ❡s ❈♦♥tí♥✉❛s ❡ ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐
◆❡st❛ s❡çã♦✱ ✈❛♠♦s ✐♥tr♦❞✉③✐r ♠❛✐s ✉♠❛ r❡❧❛çã♦ ❜❡❧íss✐♠❛ ❡♥tr❡ ❞♦✐s ❝❛♠♣♦s ❞❛ ♠❛t❡♠át✐❝❛ ❛♣❛r❡♥t❡♠❡♥t❡ ❡st❛♥q✉❡s✱ ❡ q✉❡ ♠❛✐s ✉♠❛ ✈❡③ tr❛rá ✉♠❛ s✉r♣r❡❡♥❞❡♥t❡r❡❧❛çã♦ ❝♦♠ ♦s ♥ú♠❡r♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✳ P❛r❛ t❛❧✱ s❡rá ❞❡✜♥✐❞❛ ❛s ❢r❛çõ❡s ❝♦♥tí♥✉❛s✱ q✉❡ ❛q✉✐ s❡❣✉✐rá ❝♦♥❢♦r♠❡ ♦✉tr❛s tr❛❞✉çõ❡s ♣❛r❛ ♥♦ss❛ ❧í♥❣✉❛✱ ♣♦ré♠✱ ❝♦♠ ❛ r❡ss❛❧✈❛ q✉❡ ♥❛ ❧í♥❣✉❛ ♣♦rt✉❣✉❡s❛ s❡r✐❛ ♠❛✐s ❝♦❡r❡♥t❡ tr❛❞✉③✐r ♣♦r ❢r❛çõ❡s ❝♦♥t✐♥✉❛❞❛s✱ ♠❛s ❝♦♠♦ ❛ ❡①♣r❡ssã♦ ❥á ❛❞❡♥tr♦✉ ❛ss✐♠✱ s❡rá ❝♦♥s❡r✈❛❞❛ ❡ss❛ tr❛❞✉çã♦ ♠❡s♠♦ q✉❡ ♥ã♦ tr❛❣❛ ♦ r❡❛❧ s✐❣♥✐✜❝❛❞♦✱ ❥á q✉❡ ♦ ♣r♦❝❡ss♦ r❡❛❧♠❡♥t❡ ❢❛③ é ❝♦♥t✐♥✉❛r ❛ ❢r❛çã♦ ❝♦♠❢♦r♠❡
s❡rá ♠♦str❛❞❛ ❡ ♠♦t✐✈❛❞❛ ♣♦r ✉♠ ❡①❡♠♣❧♦ ♥✉♠ér✐❝♦ ❝♦♠ ❛ ❢r❛çã♦ 79
29✳
❊①❡♠♣❧♦ ✽✳ ❋❛③❡♥❞♦ ✉s♦ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s s✉❝❡ss✐✈❛♠❡♥t❡✱ t❡♠✲s❡✿
79 =2·29 + 21
29 =1·21 + 8
21 =2·8 + 5
8 = 1·5 + 3
5 = 1·3 + 2
3 = 1·2 + 1
2 = 2·1 + 0
◗✉❡r ❞✐③❡r✱ 79
29 =2+ 21
29✱ ♦♥❞❡ ♦ ♥ú♠❡r♦ ✷ ❡ ❛ ❢r❛çã♦
21
29 ♦❜t❡♠✲s❡ ✐♠❡❞✐❛t❛♠❡♥t❡
❞❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛ ❞❛s ❞✐✈✐sõ❡s s✉❝❡ss✐✈❛s✱ ❡ ✉s❛♥❞♦ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ♥❛ ❢r❛çã♦ 21
29
t❡♠✲s❡
79
29 =2+ 1 29 21
❡ ❝♦♥t✐♥✉❛♥❞♦ ❡ss❡ ♣r♦❝❡ss♦ ❛té ♦ ✜♥❛❧✱ ♦❜t❡♠✲s❡ ❡♠ ❝❛❞❛ ♣❛ss♦ ❛s s❡❣✉✐♥t❡s ❢r❛çõ❡s✿
79
29 =2+ 1
1+ 8 21
79
29 =2+ 1
1+ 1 21
8 79
29 =2+
1
1+ 1
2+ 18
5
79
29 =2+
1
1+ 1
2+ 1 1+ 3
5 79
29 =2+
1
1+ 1
2+ 1
1+ 1 1+2
3 79
29 =2+
1
1+ 1
2+ 1
1+ 1
1+ 1 1+1
2
❉❡ss❛ ú❧t✐♠❛ ❡①♣r❡ssã♦ ❞❡❝♦rr❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❢r❛çã♦ ❝♦♥tí♥✉❛ q✉❡ r❡♣r❡s❡♥t❛ ♦
♥ú♠❡r♦ r❛❝✐♦♥❛❧ 79
29 ❡ ♦ ♦s ♥ú♠❡r♦s ✭✷✱✶✱✷✱✶✱✶✱✶✱✷✮ sã♦ ♦s q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s ❞❡ss❛
❢r❛çã♦✳
❉❡ ✉♠❛ ♠❛♥❡✐r❛ ❣❡r❛❧✿
❉❡✜♥✐çã♦ ✷✳✸✳✶✳ ❆ ❡①♣r❡ssã♦
q0+
1
q1+
1
q2+ 1
q3+✳✳✳
+1 qn
♦♥❞❡ q0 ∈Z ❡ q1, q2,· · · , qn∈N∗✱ é ❝❤❛♠❛❞❛ ❞❡ ❢r❛çã♦ ❝♦♥tí♥✉❛ ❡ ♦s q′is r❡❝❡❜❡♠ ♦ ♥♦♠❡ ❞❡ q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s✳
❈❛s♦ ❛ ❢r❛çã♦ s❡❥❛ ♥❡❣❛t✐✈❛✱ ❢❛③✲s❡ t♦❞♦ ♦ ♣r♦❝❡ss♦ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ♦♣♦st♦ ❞❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ ❡✈✐t❛♥❞♦ ❛ss✐♠ ✐♥❝♦❡rê♥❝✐❛ ❝♦♠ ❛s r❡str✐çõ❡s ✐♠♣♦st❛s ♣❛r❛ ♦s q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s✳
❖ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ❞❡ss❛ s❡çã♦ ❡♥✈♦❧✈❡rá ❛♣❡♥❛s ❛s ❢r❛çõ❡s ❝♦♥tí♥✉❛s ❞❡ q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s ✐♥t❡✐r♦s✱ ❝❧❛ss✐✜❝❛❞❛s ❝♦♠♦ s✐♠♣❧❡s✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ❛♦ s❡r ❞✐t♦ ❢r❛çõ❡s ❝♦♥tí✲ ♥✉❛s✱ ✜❝❛ s❡♠♣r❡ s✉❜t❡♥❞✐❞♦ tr❛t❛r✲s❡ ❞♦ ❝❛s♦ ❞❡❧❛s s❡r❡♠ s✐♠♣❧❡s ❡ s❡rã♦ ❛❜♦r❞❛❞❛s ❛♣❡♥❛s ❛s ❝♦♥tí♥✉❛s ✜♥✐t❛s✳
P❛r❛ ❡♥❝♦♥tr❛r ♦s q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s ♣❛r❛ ❛ ❢r❛çã♦ a
b✱ ❝♦♥❢♦r♠❡ ❡①❡♠♣❧✐✜❝❛❞♦✱
❡❢❡t✉❛✲s❡ ❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ❞♦ ♥✉♠❡r❛❞♦r ♣❡❧♦ ❞❡♥♦♠✐♥❛❞♦r✱ q✉❡r ❞✐③❡r✱ ♥❛ ❢r❛çã♦ a
b
❢❛③✲s❡
a=bq0+r1
♦♥❞❡ 0 r0 q0. ❊♠ s❡❣✉✐❞❛✱ ❡❢❡t✉❛✲s❡ ♥♦✈❛♠❡♥t❡ ❛ ❞✐✈✐sã♦✱ ❞❡ss❛ ✈❡③❞❡ b ❞✐✈✐❞✐❞♦
♣♦r r1 ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡ ✜❝❛♥❞♦ ❛ss✐♠✱
b =r1q1+r2
r1 =r2q2+r3
✳✳✳=✳✳✳
rn−2 =rn−1qn−1+rn
rn−1 =rnqn.
❊st❡ ♣r♦❝❡❞✐♠❡♥t♦ ♥ã♦ ♣♦❞❡ ❝♦♥t✐♥✉❛r ✐♥❞❡✜♥✐❞❛♠❡♥t❡✱ ♣♦✐s t❡rí❛♠♦s ✉♠❛ s❡q✉ê♥✲
❝✐❛ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s r1 > r2 > · · · q✉❡ ♥ã♦ ♣♦ss✉✐ ♠❡♥♦r ❡❧❡♠❡♥t♦✱ ♦ q✉❡ ♥ã♦
é ♣♦ssí✈❡❧ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❛ ❜♦❛ ♦r❞❡♥❛çã♦✳ ▲♦❣♦✱ ♣❛r❛ ❛❧❣✉♠ n, rn−1 = rnqn✳ ❉❛
♣r✐♠❡✐r❛ ❞❡st❛s ❡q✉❛çõ❡s✱ ♦❜t❡♠✲s❡
a
b =q0+ r1
b =q0+ 1 b r1
❊ ❞❛ s❡❣✉♥❞❛✱
b r1
=q1+ r2 r1
=q1+ 1 r1 r2
❉❛ t❡r❝❡✐r❛ é ♣♦ssí✈❡❧ ✐♠♣❧✐❝❛r
r1 r2
=q2+ 1 r2 r3
❊♥tã♦✱ ♣♦❞❡✲s❡ ❡s❝r❡✈❡r ❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ ❛♣❡♥❛s ♣❡❧♦s s❡✉s q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s✱ ❛ss✐♠✿
q0+
1
q1 +
1
q2+ 1
q3+✳✳✳
= (q0, q1,· · · , qn)
+ 1 qn
➱ ♣❡rt✐♥❡♥t❡ r❡ss❛❧t❛r✱ ♥❡st❡ ♣♦♥t♦✱ ♦ ❢❛t♦ ❞❡ qn ✭ú❧t✐♠♦ q✉♦❝✐❡♥t❡ ♣❛r❝✐❛❧✮ s❡r
s❡♠♣r❡ ♠❛✐♦r ❞♦ q✉❡ ✶✱ ♣♦✐s s❡ ❢♦ss❡ ✐❣✉❛❧ ❛ ✶✱ rn−1 s❡r✐❛ ✐❣✉❛❧ ❛ rn ❡ rn−2 ❞✐✈✐sí✲
✈❡❧ ♣♦r rn−1 ❡ ❛ ❞✐✈✐sã♦ t❡r✐❛ ❡♥❝❡rr❛❞♦ ♥♦ ♣❛ss♦ ❛♥t❡r✐♦r✳ P♦r ✐ss♦✱ ♣❡♥s❛♥❞♦ ❡♠
qn = (qn −1) +
1
1 ❡ q✉❡ ❞❡ ❢❛t♦✱ qn −1 ❡ 1 sã♦ ♦s ú❧t✐♠♦s q✉♦❝✐❡♥t❡s ✐♥❝♦♠♣❧❡t♦s
❞❡ q✉❛❧q✉❡r ❢r❛çã♦ ❝♦♥tí♥✉❛✱ ❧♦❣♦✱ s❡♠♣r❡ q✉❡ rn > 1 ♣♦❞❡✲s❡ ❡s❝r❡✈❡r ♦s q✉♦❝✐❡♥t❡s
♣❛r❝✐❛✐s (r0, r1,· · ·, rn) = (r0, r1,· · · , rn−1,1).
◆♦ ❝❛s♦ ❞♦ ❡♣❡❝í✜❝♦ ❞♦ ❊①❡♠♣❧♦ ✼ ❡♥tã♦ t❡♠✲s❡✿
79
29 =2+
1
1+ 1
2+ 1
1+ 1
1+ 1
1+ 1 1+1
1 79
29 ❂ ✭✷✱✶✱✷✱✶✱✶✱✶✱✶✱✶✮✳
❆♣❧✐❝❛♥❞♦ ❡ss❛ ✐❞❡✐❛ ♥❛ ❢r❛çã♦ 35
8 ♦❜t❡♠♦s
35 =4·8 + 3
✉♠❛ ✈❡③ q✉❡ ♦ q✉♦❝✐❡♥t❡ ❞❛ ❞✐✈✐sã♦ é ✹ ❡ ♦ r❡st♦ ✸✳ ❈♦♥t✐♥✉❛♥❞♦ ❛s ❞✐✈✐sõ❡s
8 = 2·3 + 2
3 = 1·2 + 1
2 = 2·1 + 0
❢❛③❡♥❞♦ ❝♦♠ q✉❡ t❡♥❤❛✲s❡ ♥❛ ❡①♣r❡ssã♦
q0+
1
q1 + 1
q2+ 1 q3
♦ ❝♦❡✜❝✐❡♥t❡ q0 = 4, q1 = q2 = 1 ❡ q3 = 2 ❛❧é♠ ❞❡ r1 = 8, r2 = 3 ❡ r3 = 2 ♦❜t❡♥❞♦ ❛
❢r❛çã♦ ❝♦♥tí♥✉❛
4+ 1
2+ 1 1+1
2
❊①❡♠♣❧♦ ✾✳ ❈♦♥str✉✐r ❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ q✉❡ ♣♦ss✉✐ ♦s q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s ✭✹✱✷✱✶✱✸✮✳
4+ 1
2+ 1 1+1
3
❊ss❡ ❢r❛çã♦ ❝♦♥tí♥✉❛ ♣♦❞❡✲s❡ ❞❡s❡♥✈♦❧✈❡r ♣❛r❛
4+ 1 2+3
4
4+ 4 11 =
48 11
❆❣♦r❛✱ ❤❛✈❡rá ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝♦♥❤❡❝❡r ❛s r❡❞✉③✐❞❛s ❞❡ ✉♠❛ ❢r❛çã♦ ❝♦♥tí♥✉❛✳
❉❡✜♥✐çã♦ ✷✳✸✳✷✳ ❙❡❥❛ α ✉♠❛ ❢r❛çã♦ ❝♦♥tí♥✉❛✱ q✉❡r ❞✐③❡r✱
α=q0+
1
q1+
1
q2+ 1
q3+✳✳✳
+1 qn ❈♦♥s✐❞❡r❛♥❞♦ ♦s ♥ú♠❡r♦s
q0, q0+ 1 q1
, q0+ 1
q1+ 1 q2
, q0+
1
q1+ 1
q2+ 1 q3
,· · ·
❝❤❛♠❛✲s❡ r❡❞✉③✐❞❛s ❞❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ α ❛s r❛③õ❡s ♥❛ ❢♦r♠❛ ✐rr❡❞✉tí✈❡❧ ❝♦♥❢♦r♠❡
s❡❣✉❡✿
α0 = q0
1 = P0 Q0
α1 =q0 + 1 q1
= q1q0+ 1 q1
= P1 Q1
α2 =q0 + 1
q1+ 1 q2
= q2q1q0+q0+q2 q1q2+ 1
= P2 Q2
α3 = P3 Q3
=q0+
1
q1+ 1
q2+ 1 q3
✳✳✳ ❂ ✳✳✳ ❂ ✳✳✳
αn=· · ·= Pn Qn
❖❜s❡r✈❡ q✉❡αn =α ❡ ♣❛r❛ ♦❜t❡r
Pn+1
Qn+1 ❜❛st❛ s✉❜st✐t✉✐r ❡♠
Pn Qn
♦ ú❧t✐♠♦ q✉♦❝✐❡♥t❡
qn ♣♦r
qn+
1 qn+1
.
❘❡s✉❧t❛❞♦ q✉❡ s❡rá ✉s❛❞♦ ♥♦ ♣ró①✐♠♦ t❡♦r❡♠❛✱ q✉❡ ❥á ❛♥t❡❝❡❞❡ ♦ t❡♦r❡♠❛ ❝❡♥tr❛❧ ❞❡st❛ s❡çã♦✱ q✉❛♥❞♦ s❡rá tr❛t❛❞♦ ❞❡ ❢r❛çõ❡s ❝♦♥tí♥✉❛s ❡ r❛③ã♦ ❡♥tr❡ t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ s✉r♣r❡❡♥❞❡♥t❡✳
❚❡♦r❡♠❛ ✷✳✸✳✶✳ ❙❡❥❛ αi ❛ ✐✲és✐♠❛ r❡❞✉③✐❞❛ ❞❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ (q0, q1,· · · , qn−1,1)✱
❡♥tã♦ ✈❛❧❡♠ ♣❛r❛ ♦ ♥✉♠❡r❛❞♦r Pi ❡ ❞❡♥♦♠✐♥❛❞♦r Qi ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s
i) Pi+1 =qiPi+Pi−1
ii) Qi+1 =qiQi+Qi−1
iii) Pi+1Qi−PiQi+1 = (−1)i
❝♦♠ n ∈N∗.
❯♠❛ ✈❡③ q✉❡ ♦ ✐♥t✉✐t♦ ❞❡st❡ tr❛❜❛❧❤♦ ❡♥✈♦❧✈❡ ❛♣❧✐❝❛çõ❡s ❞♦ P IM ❡ ♥❡st❡ ❝❛♣ít✉❧♦
❡ss❛s ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ s❡rá ♦♠✐t✐❞❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ú❧t✐♠♦ t❡♦r❡♠❛✱ ❢❛❝✐❧♠❡♥t❡ ✈❡r✐✜❝á✈❡✐s ❡♠ ❬✾❪✱ s❡♥❞♦ ❛q✉✐ ❛♣r❡s❡♥t❛❞♦ ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❝❛s♦
♣❛rt✐❝✉❧❛r ❞❛s ❢r❛çõ❡s ❝♦♥tí♥✉❛s ❡ s✉❛s r❡❧❛çõ❡s ❝♦♠ ♦s ♥ú♠❡r♦s un ❞❛ s❡q✉ê♥❝✐❛ ❞❡
❋✐❜♦♥❛❝❝✐✳
❚❡♦r❡♠❛ ✷✳✸✳✷✳ ❙❡❥❛ α ✉♠❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ ❝♦♠ n q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s t♦❞♦s ✐❣✉❛✐s ❛
✶✱ ❡♥tã♦ α= un+1
un ✱ ♦♥❞❡
uné ♦ ❡♥és✐♠♦ t❡r♠♦ ❞❛ s❡q✉ê♥❝✐❛ ❞♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳
❉❡♠♦♥str❛çã♦✳ ❋❛③❡♥❞♦ ✉s♦ ❞❛s r❡❞✉③✐❞❛s αn✱ t❡♠♦s
αi = Pi Qi
♦♥❞❡ α1 = 1 = 1
1, α2 = 1 + 1 1 =
2
1 ❞❡ ♠❛♥❡✐r❛ q✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞❡ ✐✮ ❡♠ ❚❡♦r❡♠❛
✷✳✸✳✶✳
Pi+1 =qiPi+Pi−1 =Pi+Pi−1
✉♠❛ ✈❡③ q✉❡ t♦❞♦s q✉♦❝✐❡♥t❡s ♣❛r❝✐❛s sã♦ ✉♥✐tár✐♦s✱ ❡✱ ♣❡❧❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ r❡❝♦rrê♥❝✐❛
❡ ♦s t❡r♠♦s ✐♥✐❝✐❛✐s✱ ♣♦❞❡✲s❡ ❛✜r♠❛r Pn = un+1✱ ❥á ❡♠ r❡❧❛çã♦ ❛♦s ❞❡♥♦♠✐♥❛❞♦r❡s✱
Q1 =Q2 = 1 ❡ t❛♠❜é♠
Qi+1 =Qiqi+ 1 +Qi−1 =Qi+Qi−1
❡ ♥♦✈❛♠❡♥t❡✱ ♣❡❧❛ ♣❛rt✐❝✉❧❛r✐❞❛❞❡ ❞❡ss❛ r❡❝♦rrê♥❝✐❛ ❡ ♠❛✐s ✉♠❛ ✈❡③ ♣❡❧♦s ✈❛❧♦r❡s ✐♥✐✲
❝✐❛✐s✱ Qi =ui ❢❛③❡♥❞♦ ❝♦♠ q✉❡
αi = ui+1
ui
❊ss❡ t❡♦r❡♠❛ ❛✜r♠❛ q✉❡ ✉♠❛ ✈❡③ q✉❡ ♦s q✉♦❝✐❡♥t❡s ♣❛r❝✐❛s ❞❡ ✉♠❛ ❢r❛çã♦ ❝♦♥tí♥✉❛
sã♦ t♦❞♦s ✉♥✐tár✐♦s✱ ❡♥tã♦✱ ♣❛r❛ n q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s✱ t❡♠✲s❡ q✉❡ ❛ ❢r❛çã♦ é ✐❣✉❛❧ à
r❛③ã♦ ❡♥tr❡ ♦ t❡r♠♦ n+ 1❡n ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ❝♦♥❢♦r♠❡ ♣♦❞❡✲s❡ ❡①❡♠♣❧✐✜❝❛r
❛❜❛✐①♦✿
❊①❡♠♣❧♦ ✶✵✳ ❙❡❥❛ ❛ ❢r❛çã♦ ❝♦♥tí♥✉❛ α = (1,1,1,1,1)✱ q✉❡r ❞✐③❡r✱ ❝✐♥❝♦ q✉♦❝✐❡♥t❡s
♣❛r❝✐❛✐s ✐❣✉❛✐s ❛ ✶✱
1 + 1
1 + 1
1 + 1 1 + 1
1
❊❢❡t✉❛♥❞♦ ♦s ❝á❧❝✉❧♦s ♣❛ss♦ ❛ ♣❛ss♦✱ ✈✐s✉❛❧✐③❛♠♦s
1 + 1
1 + 1
1 + 1 1 + 1
1
= 1 + 1
1 + 1 1 + 1
2
= 1 + 1 1 + 2
3
= 1 + 3 5 =
8 5 =
u6 u5
❊①❡♠♣❧♦ ✶✶✳ ❖✉tr❛ ❢r❛çã♦ ❝♦♥tí♥✉❛✱ ❛❣♦r❛ ❝♦♠ ♦✐t♦ q✉♦❝✐❡♥t❡s ♣❛r❝✐❛✐s ✉♥✐tár✐♦s✱
❡s♣❡r❛✲s❡ q✉❡ s❡❥❛ ✐❣✉❛❧ à ❢r❛çã♦ u9
u8❂ ✭✶✱✶✱✶✱✶✱✶✱✶✱✶✱✶✮❂
1 + 1
1 + 1
1 + 1
1 + 1
1 + 1
1 + 1
1 + 11
1
= 1 + 1
1 + 1
1 + 1
1 + 1
1 + 1 1 + 1
2 =
= 1 + 1
1 + 1
1 + 1
1 + 1 1 + 2
3
= 1 + 1
1 + 1
1 + 1 1 + 3
5
= 1 + 1 1 + 8
3 =
= 1 + 1 21 13
= 1 +13 21 =
34 21
q✉❡ s❡ tr❛t❛ ❡①❛t❛♠❡♥t❡ ❞❛ r❛③ã♦ ❡s♣❡r❛❞❛✳
❈❛♣ít✉❧♦ ✸
❯s♦ ❞♦
P IM
❞❡ ♠❛♥❡✐r❛ ❧ú❞✐❝❛
✸✳✶ ❚♦rr❡ ❞❡ ❍❛♥♦✐
❆ t♦rr❡ ❞❡ ❍❛♥♦✐✱ ❥♦❣♦ ❞❡ q✉❡❜r❛✲❝❛❜❡ç❛ q✉❡ ❝♦♥s✐st❡ ❞❡ ✉♠❛ ❜❛s❡ ❝♦♠ três ♣✐♥♦s✱ ❡♠ ✉♠ ❞❡❧❡s✱ ❞✐s❝♦s ❢✉r❛❞♦s ♥♦ ❝❡♥tr♦ ❝♦♠ ❞✐â♠❡tr♦s ❞✐st✐♥t♦s ❡ ❡♠ ♦r❞❡♠ ❝r❡s❝❡♥t❡ ❞❡ ♠❡♥♦r ❞✐â♠❡tr♦ ♣❛r❛ ♠❛✐♦r ❞✐â♠❡tr♦ ❡stã♦ ❡♠♣✐❧❤❛❞♦s ✉♠ s♦❜r❡ ♦ ♦✉tr♦✳
❆ ✜❣✉r❛ ❛❜❛✐①♦ ✐❧✉str❛ ✐ss♦✳
❋✐❣✉r❛ ✸✳✶✿ ❚♦rr❡ ❞❡ ❍❛♥♦✐✳
❆ ❧❡♥❞❛ ♣♦r trás ❞♦ ❥♦❣♦ ❬✺❪ ❞✐③ q✉❡ ♥♦ ❝❡♥tr♦ ❞♦ ♠✉♥❞♦✱ ❤❛✈✐❛ ✉♠❛ ❜❛♥❞❡❥❛ ❞❡ ❜r♦♥③❡ ❝♦♠ três ❛❣✉❧❤❛s ❞❡ ❞✐❛♠❛♥t❡s✱ ❡ q✉❡ ❞✉r❛♥t❡ ❛ ❝r✐❛çã♦✱ ❉❡✉s ❝♦❧♦❝❛r❛ ✻✹ ❞✐s❝♦s ❞❡ ♦✉r♦ ♣✉r♦ ❡♠♣✐❧❤❛❞❛s ❡♠ ✉♠❛ ❛❣✉❧❤❛ ❝♦♠ ♦ ♠❛✐♦r ❞♦s ❞✐s❝♦s s♦❜r❡ ❛ ❜❛♥❞❡❥❛ ❡ ♦ r❡st❛♥t❡ ❝❛❞❛ ✈❡③ ♠❡♥♦r s♦❜r❡ ❡ss❛ ❛❣✉❧❤❛✳ ❊ss❛ ❡♥tã♦ ✜❝❛r✐❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❛ ❚♦rr❡ ❞❡ ❍❛♥♦✐✳
