❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás
■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛
Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠
▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
❈❘■P❚❖●❘❆❋■❆
❚✉❧✐♦ ❋❡r♥❛♥❞♦ ❞❛ ▼❛t❛
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: [X] Dissertação [ ] Tese 2. Identificação da Tese ou Dissertação
Autor (a): Tulio Fernando da Mata
E-mail: [email protected]
Seu e-mail pode ser disponibilizado na página? [X]Sim [ ] Não
Vínculo empregatício do autor Mestrando Bolsista e Professor De matemática Agência de fomento: Coor. de Aperf. Pessoal de Nível Superior
Sec. de Educação do Estado de Goiás
Sigla: CAPES SEDUC País: Brasil UF: GO CNPJ:
Título: Criptografia
Palavras-chave: Criptografia, matemática, sigilo, codificação, decodificação.
Título em outra língua: Cryptographic
Palavras-chave em outra língua: Cryptography, mathematics, confidentiality, coding, deco-ding.
Área de concentração: Matemática do Ensino Básico Data defesa:(dd/mm/aaaa) 06/08/2015
Programa de Pós-Graduação: Mestrado em Matemática-PROFMAT Orientador (a): Prof. Dra. Ivonildes Ribeiro Martins Dias
E-mail: [email protected] 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.
O sistema da Biblioteca Digital de Teses e Dissertações garante aos autores, que os ar-quivos contendo eletronicamente as teses e ou dissertações, antes de sua disponibilização, receberão procedimentos de segurança, criptografia (para não permitir cópia e extração de conteúdo, permitindo apenas impressão fraca) usando o padrão do Acrobat.
________________________________________ Data: 27 / 08 / 2015 Assinatura do (a) autor (a)
1
❚✉❧✐♦ ❋❡r♥❛♥❞♦ ❞❛ ▼❛t❛
❈❘■P❚❖●❘❆❋■❆
❚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❛✳ ■✈♦♥✐❧❞❡s ❘✐❜❡✐r♦ ▼❛rt✐♥s ❉✐❛s
Ficha catalográfica elaborada automaticamente
com os dados fornecidos pelo(a) autor(a), sob orientação do Sibi/UFG.
da Mata, Tulio Fernando
Criptografia [manuscrito] / Tulio Fernando da Mata. - 2015. LXI, 61 f.
Orientador: Prof. Dr. Ivonildes Ribeiro Martins Dias.
Dissertação (Mestrado) - Universidade Federal de Goiás, Instituto de Matemática e Estatística (IME) , Programa de Pós-Graduação em Matemática, Goiânia, 2015.
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❡t✉❞♦ ❡ ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ q✉❡r♦ ❛❣r❛❞❡❝❡r ❛♦ ♠❡✉ ❜♦♠ ❉❡✉s✱ ❏❡♦✈á✱ q✉❡ ♠❡ ❞❡✉ ❝❛♣❛❝✐❞❛❞❡✱ ✐♥t❡❧✐❣ê♥❝✐❛✱ ❢♦rç❛ ❡ ♣❡rs❡✈❡r❛♥ç❛ ♣❛r❛ ❝♦♥❝❧✉✐r ❡st❡ ❝✉rs♦✳
❚❛♠❜é♠ q✉❡r♦ ❞❡✐①❛r r❡❣✐str❛❞♦ ♦ ♠❡✉ ❛❣r❛❞❡❝✐♠❡♥t♦ ❛ ♠✐♥❤❛ q✉❡r✐❞❛ ❡ ❛♠❛❞❛ ❡s♣♦s❛✱ P♦❧❧②❛♥❛ ❈❧❛✉❞✐♥❛✱ q✉❡ ♠❡ ❛♣♦✐♦✉✱ ♠❡ ❡♥❝♦r❛❥♦✉ ❡ ❢❡③ ❞♦s ♠❡✉s ❣r❛♥❞❡s ♣r♦❜❧❡✲ ♠❛s✱ ♣❡q✉❡♥♦s ♦❜stá❝✉❧♦s✳ ❉❡✈♦ t❛♠❜é♠ ✉♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ ❛♦s ♠❡✉s ✜❧❤♦s✱ P❛✉❧♦ ❋❡r♥❛♥❞♦ ❡ ❊❞✉❛r❞❛✱ q✉❡ ❝♦♠♣r❡❡♥❞❡r❛♠ ♦s ♠❡✉s ♦❜❥❡t✐✈♦s✱ ♠❡s♠♦ ❛s ✈❡③❡s t❡♥t♦ q✉❡ s❛❝r✐✜❝❛r ♦ ❧❛③❡r ❞❛ ❢❛♠í❧✐❛✳
❆❣r❛❞❡ç♦ ❛✐♥❞❛ ♠✐♥❤❛ ♠ã❡✱ ▼❛r✐❛ ▲✉✐③❛ ❡ ♠✐♥❤❛ t✐❛ ❱❛❧❞❡❧✐ ✭❩✐♥❤❛✮ q✉❡ s❡♠♣r❡ ❛❝r❡❞✐t❛r❛♠ ♥❛ ♠✐♥❤❛ ❝❛♣❛❝✐❞❛❞❡ ❡ ♠❡ ❞❡r❛♠ ❢♦rç❛ ♥❛s ❤♦r❛s ❞✐❢í❝❡✐s✳
❆❣r❛❞❡ç♦ ❛♦s ❛♠✐❣♦s ●✐s❡❧❡ ▲✉✐③❛✱ ❆♥❛ ❈❧❡②✱ ▼❛r✐❛ ❆♣❛r❡❝✐❞❛ ✭❇r❛♥❝❛✮✱ ❙✉❡❧❡♥❡✱ ❆❣❧❡②❞s♦♥✱ ❱❛❧❞❡❝✐✱ ❙✉③❛♥❛✱ ▼❛r❝♦s ❆♥tô♥✐♦ ✭▼❛r❦ã♦✮ ❡ ♠✉✐t♦s ♦✉tr♦s ♣♦r ♠❡ ✐♥❝❡♥✲ t✐✈❛r❡♠✱ ✐♥t✉✐t✐✈❛♠❡♥t❡ ❛ s❡r ♠❡❧❤♦r ♣❛r❛ ♣♦❞❡r ❛❝♦♠♣❛♥❤❛✲❧♦s✳
❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦s ❝♦❧❡❣❛s✱ q✉❡ sã♦ ♠✉✐t♦s✱ ♣♦r ♠❡ ❡♥t❡♥❞❡r❡♠ ❡ ♠❡ ❛❥✉❞❛r❡♠ t♦❞❛s ✈❡③❡s q✉❡ ♣r❡❝✐s❡✐✳
❉❡✐①♦ ✉♠ ❛❜r❛ç♦ ❞❡ ❛❣r❛❞❡❝✐♠❡♥t♦ ❛♦s ❛♠✐❣♦s ❞❡ s❛❧❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ❍✉❣❧❡✐❞s♦♥✱ ❈❧❡✉❜❡r✱ ●❡❛♥✱ ❇r✉♥♦ ❡ ❙❛♠✐r❛ ♣❡❧❛s ❡①♣❡r✐ê♥❝✐❛s tr♦❝❛❞❛s✱ ❞ú✈✐❞❛s t✐r❛❞❛s ❡ ❛♠✐③❛❞❡s ❝♦♥str✉í❞❛s✳
❉❡✈♦ ✉♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ❝❛r✐♥❤♦s♦ ❛♦ ❛♠✐❣♦ ❏♦ã♦ ▲✉✐s✱ ❛♠✐❣♦ ♠✉✐t♦ ❡s♣❡❝✐❛❧ q✉❡ ✜③ ♥♦ ❝✉rs♦✱ ♠❛s ♣♦r ♠♦t✐✈♦ ♣❛rt✐❝✉❧❛r❡s ❡❧❡ ❞❡s✐st✐✉✳
❆ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ Pr♦❢✳ ❉r❛✳ ■✈♦♥✐❧❞❡s ❘✐❜❡✐r♦ ▼❛rt✐♥s ❉✐❛s✱ ❞❡✈♦ ❛ ❡❧❛ ♠❛✐s q✉❡ ✉♠ ❛❣r❛❞❡❝✐♠❡♥t♦✱ ♣♦✐s ❛♣❡s❛r ❞❡ ❡✉ ♥ã♦ t❡r s✐❞♦ tã♦ ♣♦♥t✉❛❧✱ ❡❧❛ ❢♦✐✳
❆❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ♠❡ ❛❥✉❞❛r❛♠ ♦✉ ♠❡s♠♦ t♦r❝❡r❛♠ ♣♦r ♠✐♠✳
❘❡s✉♠♦
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢❛③ ✉♠❛ ❜r❡✈❡ ❧❡✐t✉r❛ s♦❜r❡ ❛ s✐t✉❛çã♦ ❞❡ ❝r✐s❡ ✈✐✈✐❞❛ ♣❡❧♦ ❡♥✲ s✐♥♦ ❞❡ ♠❛t❡♠át✐❝❛ ♥❛ ❡s❝♦❧❛ ♣ú❜❧✐❝❛ ❜r❛s✐❧❡✐r❛ ♥♦s ú❧t✐♠♦s ❛♥♦s✱ ❛♥❛❧✐s❛♥❞♦ ❛❧❣✉♠❛s ❞❛s r❛③õ❡s ❡ ♠♦t✐✈♦s ❞❡ss❛ ❝r✐s❡ ❡ t❡♥t❛♥❞♦ ❛♣♦♥t❛r ❝❛♠✐♥❤♦s ♣❛r❛ q✉❡ ❡ss❡ ❡♥s✐♥♦ s❡ t♦r♥❡ ♠❡❧❤♦r ❡ ♠❛✐s ❡✜❝✐❡♥t❡ ❡ ❛❝✐♠❛ ❞❡ t✉❞♦ ❛♣r❡s❡♥t❛♥❞♦ ❛ ♠❛t❡♠át✐❝❛ ❝♦♠♦ ✉♠❛ ❝✐ê♥❝✐❛ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ♠✉♥❞♦ ❛t✉❛❧✱ ❛♣♦♥t❛♥❞♦ s✉❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❡♠ s✐t✉❛çõ❡s ❝♦t✐❞✐❛♥❛s q✉❡ ♣♦❞❡♠ ❞❡s♣❡rt❛r ♦ ✐♥t❡r❡ss❡ ❞♦s ❛❧✉♥♦s ❡ s❡ ❛♣r❡s❡♥t❛♠ ❝♦♠♦ ❞❡s❛✜♦s ♣❛r❛ s✉❛s ❝❛♣❛❝✐❞❛❞❡s ❞❡ r❛❝✐♦❝✐♥❛r ❡ ❞❡ ❛♣r❡♥❞❡r✳ ◆❛ ♣❡rs♣❡❝t✐✈❛ ❞❡ ♠♦str❛r ❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❡ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ♠❛t❡♠át✐❝❛ ❛♣r❡s❡♥t❛✲s❡ ❛ ❝r✐♣t♦❣r❛✜❛✱ ❡❧❡❣❡♥❞♦✲s❡ ❝♦♠♦ ❜❛s❡ ❞❡ ❛♥á❧✐s❡ ♦ ♠ét♦❞♦ ❝r✐♣t♦❣rá✜❝♦ ❘❙❆✱ ✉♠ ❞♦s ♠❛✐s ❞✐❢✉♥❞✐❞♦s ❡♠ t♦❞♦ ♦ ♠✉♥❞♦✱ ♣❛r❛ ❛♣r❡s❡♥tá✲❧♦ ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ q✉❡ ✉s❛ ❛❧❣♦r✐t♠♦s ♠❛t❡♠á✲ t✐❝♦s ♣❛r❛ ❛ ❝♦❞✐✜❝❛çã♦ ❡✴♦✉ ❞❡❝♦❞✐✜❝❛çã♦ ❞❡ ❞❛❞♦s✱ ♣❡r♠✐t✐♥❞♦ ❛ ♣r♦t❡çã♦ ❡ ♦ s✐❣✐❧♦ ❞❡ ✐♥❢♦r♠❛çõ❡s ❝♦♥✜❞❡♥❝✐❛✐s tr❛♥s♠✐t✐❞❛s ✈✐❛ s❛té❧✐t❡ ❡♠ tr❛♥s❛çõ❡s ❝♦♠❡r❝✐❛✐s ❡ ❜❛♥✲ ❝ár✐❛s✳ ❖ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ❛ ✐♠♣♦rtâ♥❝✐❛ ❣❡r❛❧ ❞❛ ❝r✐♣t♦❣r❛✜❛ ♥♦ ❝♦♥t❡①t♦ ❞♦ ♠✉♥❞♦ ❣❧♦❜❛❧✐③❛❞♦ ❡ t❡❝♥♦❧ó❣✐❝♦ ❞❛ ❛t✉❛❧✐❞❛❞❡✱ ❛ss✐♠ ❝♦♠♦ ❞❡♥tr♦ ❞❛ ❤✐stór✐❛✱ r❡ss❛❧t❛♥❞♦✱ ✐♥❝❧✉s✐✈❡✱ s✉❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡❝✐s✐✈❛ ♣❛r❛ ❛ ✈✐tór✐❛ ❞♦s ❆❧✐❛❞♦s ❝♦♥tr❛ ♦s ❛❧❡♠ã❡s ♥❛ ✷✳❛ ●✉❡rr❛ ▼✉♥❞✐❛❧✳ ❆♣r❡s❡♥t❛✲s❡ ❛✐♥❞❛ ❛ ❜❛s❡ ♠❛t❡♠át✐❝❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ❝r✐♣t♦❣r❛✲
✜❛✱ ♦✉ s❡❥❛✱ ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♣ré✈✐♦s ❞❡ ♠❛t❡♠át✐❝❛ q✉❡ sã♦ ✉s❛❞♦s ♣❛r❛ ❝r✐♣t♦❣r❛❢❛r ♠❡♥s❛❣❡♥s ❡ ❞❛❞♦s✱ ❜❡♠ ❝♦♠♦ s❡ ❛♣r❡s❡♥t❛ ❡①❡♠♣❧♦s ❞❡ ❝♦❞✐✜❝❛çã♦ ❡ ❞❡❝♦❞✐✜❝❛çã♦ ❞❡ ♠❡♥s❛❣❡♥s s✐♠♣❧❡s✳ P♦r ✜♠✱ ❢❛③✲s❡ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❣❡r❛❧ s♦❜r❡ ❛ ❝r✐♣t♦❣r❛✜❛✱ ❞❡s❞❡ s❡✉ ❝♦♥❝❡✐t♦ ❡ ❞❡✜♥✐çõ❡s ♠❛✐s ❝♦♠✉♥s✱ ❛♣♦♥t❛♥❞♦ ♦s r❡q✉✐s✐t♦s ❣❛r❛♥t✐❞♦s ♣❡❧♦ s❡✉ ✉s♦ ❡ ♦s ♣r✐♥❝✐♣❛✐s t✐♣♦s ❞❡ ❛❧❣♦r✐t♠♦s ♣❛r❛ ❝✐❢r❛❣❡♠ ❡ ❞❡❝✐❢r❛❣❡♠ ❞❡ ❝ó❞✐❣♦s✱ t❛♠❜é♠ ❝❤❛♠❛❞❛s ❞❡ ❝❤❛✈❡s ❝r✐♣t♦❣rá✜❝❛s✳
P❛❧❛✈r❛s✲❝❤❛✈❡ ❈r✐♣t♦❣r❛✜❛✱ ♠❛t❡♠át✐❝❛✱ s✐❣✐❧♦✱ ❝♦❞✐✜❝❛çã♦✱ ❞❡❝♦❞✐✜❝❛çã♦✳
❆❜str❛❝t
❚❤✐s ♣❛♣❡r ♠❛❦❡s ❛ ❜r✐❡❢ r❡❛❞✐♥❣ ♦♥ t❤❡ ❝r✐s✐s ❡①♣❡r✐❡♥❝❡❞ ❜② t❤❡ ♠❛t❤❡♠❛t✐❝s ❡❞✉✲ ❝❛t✐♦♥ ✐♥ ❇r❛③✐❧✐❛♥ ♣✉❜❧✐❝ s❝❤♦♦❧s ✐♥ r❡❝❡♥t ②❡❛rs✱ ❛♥❛❧②③✐♥❣ s♦♠❡ ♦❢ t❤❡ r❡❛s♦♥s ❛♥❞ ♠♦t✐✈❡s ♦❢ t❤✐s ❝r✐s✐s ❛♥❞ tr②✐♥❣ t♦ ♣♦✐♥t ♦✉t ✇❛②s t❤❛t t❤✐s t❡❛❝❤✐♥❣ ✇✐❧❧ ❜❡❝♦♠❡ ❜❡t✲ t❡r✱ ♠♦r❡ ❡✣❝✐❡♥t ❛♥❞ ❛❜♦✈❡ ❛❧❧ ♣r❡s❡♥t✐♥❣ ♠❛t❤❡♠❛t✐❝s ❛s ❛ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rt❛♥❝❡ ❢♦r t♦❞❛②✬s ✇♦r❧❞ s❝✐❡♥❝❡✱ ♣♦✐♥t✐♥❣ t❤❡✐r ❛♣♣❧✐❝❛❜✐❧✐t② ✐♥ ❡✈❡r②❞❛② s✐t✉❛t✐♦♥s t❤❛t ♠❛② ❛r♦✉s❡ t❤❡ ✐♥t❡r❡st ♦❢ st✉❞❡♥ts ❛♥❞ ♣r❡s❡♥t t❤❡♠s❡❧✈❡s ❛s ❝❤❛❧❧❡♥❣❡s t♦ t❤❡✐r ❛❜✐❧✐t② t♦ t❤✐♥❦ ❛♥❞ ❧❡❛r♥✳ ■♥ ♦r❞❡r t♦ s❤♦✇ t❤❡ ❛♣♣❧✐❝❛❜✐❧✐t② ❛♥❞ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ ♠❛t❤❡♠❛t✐❝s s❤♦✇s t❤❡ ❡♥❝r②♣t✐♦♥✱ ✐❢ ❡❧❡❝t❡❞ ❛s ❛ ❜❛s✐s ❢♦r ❛♥❛❧②s✐s t❤❡ ❝r②♣t♦❣r❛♣❤✐❝ ♠❡t❤♦❞ ❘❙❆✱ ♦♥❡ ♦❢ t❤❡ ♠♦st ✇✐❞❡s♣r❡❛❞ t❤r♦✉❣❤♦✉t t❤❡ ✇♦r❧❞✱ t♦ ♣r❡s❡♥t ✐t ❛s ❛ t♦♦❧ t❤❛t ✉s❡s ♠❛t❤❡♠❛t✐❝❛❧ ❛❧❣♦r✐t❤♠s t♦ ❡♥❝♦❞✐♥❣ ❛♥❞✴♦r ❞❡❝♦❞✐♥❣ ♦❢ ❞❛t❛✱ ❡♥❛❜❧✐♥❣ t❤❡ ♣r♦t❡❝✲ t✐♦♥ ❛♥❞ ❝♦♥✜❞❡♥t✐❛❧✐t② ♦❢ s❡♥s✐t✐✈❡ ✐♥❢♦r♠❛t✐♦♥ tr❛♥s♠✐tt❡❞ ✈✐❛ s❛t❡❧❧✐t❡ ✐♥ ❝♦♠♠❡r❝✐❛❧ ❛♥❞ ❜❛♥❦✐♥❣ tr❛♥s❛❝t✐♦♥s✳ ❚❤❡ ♣❛♣❡r ♣r❡s❡♥ts t❤❡ ❣❡♥❡r❛❧ ✐♠♣♦rt❛♥❝❡ ♦❢ ❝r②♣t♦❣r❛♣❤② ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❛ ❣❧♦❜❛❧✐③❡❞ ❛♥❞ t❡❝❤♥♦❧♦❣✐❝❛❧ ✇♦r❧❞ ♦❢ t♦❞❛②✱ ❛s ✇❡❧❧ ❛s ✇✐t❤✐♥ t❤❡ st♦r②✱ ♥♦t✐♥❣ ❡✈❡♥ ✐ts ❞❡❝✐s✐✈❡ ❝♦♥tr✐❜✉t✐♦♥ t♦ t❤❡ ❆❧❧✐❡❞ ✈✐❝t♦r② ❛❣❛✐♥st t❤❡ ●❡r♠❛♥s ✐♥ t❤❡ ✷♥❞ ❲♦r❧❞ ❲❛r✳ ■t ♣r❡s❡♥ts ❛❧s♦ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❜❛s✐s ❢♦r ❡♥❝r②♣t✐♦♥✱ ♣r✐♦r ❦♥♦✇❧❡❞❣❡ ♦❢ ♠❛t❤❡♠❛t✐❝s t❤❛t ❛r❡ ✉s❡❞ t♦ ❡♥❝r②♣t ♠❡ss❛❣❡s ❛♥❞ ♦r ❞❛t❛✱ ❛♥❞ ♣r❡s❡♥ts ❡①❛♠♣❧❡s ♦❢ ❡♥❝♦❞✐♥❣ ❛♥❞ ❞❡❝♦❞✐♥❣ s✐♠♣❧❡ ♠❡ss❛❣❡s✳ ❋✐♥❛❧❧②✱ ❛ ❣❡♥❡r❛❧ ❛♣♣r♦❛❝❤ ♦♥ t❤❡ ❡♥❝r②♣t✐♦♥ ✐s ❞♦♥❡✱ s✐♥❝❡ ✐ts ❝♦♥❝❡♣t ❛♥❞ ❝♦♠♠♦♥ ❞❡✜♥✐t✐♦♥s✱ ♣♦✐♥t✐♥❣ s❡❝✉r❡❞ r❡✲ q✉✐r❡♠❡♥ts ❢♦r ✐ts ✉s❡ ❛♥❞ t❤❡ ♠❛✐♥ t②♣❡s ♦❢ ❛❧❣♦r✐t❤♠s ❢♦r ❡♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ❝♦❞❡s✱ ❛❧s♦ ❝❛❧❧❡❞ ❝r②♣t♦❣r❛♣❤✐❝ ❦❡②s✳
❑❡②✇♦r❞s
❝r②♣t♦❣r❛♣❤②✱ ♠❛t❤❡♠❛t✐❝s✱ ❝♦♥✜❞❡♥t✐❛❧✐t②✱ ❝♦❞✐♥❣✱ ❞❡❝♦❞✐♥❣✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶ P❡❞r❡✐r♦ tr❛❜❛❧❤❛♥❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷ ▼áq✉✐♥❛ ❞❡ ❈r✐♣t♦❣r❛t✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✸ ❆❧❛♥ ❚✉r✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✶ ❊✈♦❧✉çã♦ ❞♦s ◆ú♠❡r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ P✐❡rr❡ ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✸ ▼❛r✐♥ ▼❡rs❡♥♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✹ ◆ú♠❡r♦s ✐♥❞♦✲❛rá❜✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷✳✶ ❆✈❛♥ç♦ ❚❡❝♥♦❧ó❣✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ ❈♦♥❤❡❝✐♠❡♥t♦ Pré✈✐♦ ♣❛r❛ ❈r✐♣t♦❣r❛✜❛ ✼
✶✳✶ ◆ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✶✳✶ ❉❡st❛q✉❡ ♣❛r❛ ♦ ❆①✐♦♠❛ ❞❛ ■♥❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷ ◆ú♠❡r♦s Pr✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷✳✶ ◗✉❛♥t♦s ◆ú♠❡r♦s Pr✐♠♦s ❊①✐st❡♠❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷✳✷ ❈♦♠♦ ❘❡❝♦♥❤❡❝❡r ✉♠ ◆ú♠❡r♦ Pr✐♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷✳✸ ❖s ◆ú♠❡r♦s ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷✳✹ ❖s ◆ú♠❡r♦s ❞❡ ▼❡rs❡♥♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✸ ◆ú♠❡r♦s ■♥t❡✐r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳✶ ◆ú♠❡r♦s ◆❡❣❛t✐✈♦s✿ ❖r✐❣❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳✷ ❖s ■♥t❡✐r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✺ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✻ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✻✳✶ ❋❛t♦r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✼ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✼✳✶ ❖ ❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✼✳✷ ❖ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✼✳✸ ❈r✐tér✐♦s ❞❡ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✽ ◆ú♠❡r♦s P❡r❢❡✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✾ ❈♦♥❣r✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✾✳✶ ❚❡♦r❡♠❛ ❞❡ ■♥✈❡rtí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✳✶✵ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✷ ❈r✐♣t♦❣r❛✜❛✱ ❯s♦s✱ ■♠♣♦rtâ♥❝✐❛ ❡ ▲✐♠✐t❡s ✸✵ ✷✳✶ ❈♦♥❝❡✐t✉❛♥❞♦ ❛ ❈r✐♣t♦❣r❛✜❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷ ❆ ❈r✐♣t♦❣r❛✜❛ ❡ s✉❛ ■♠♣♦rtâ♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✸ ❘❡q✉✐s✐t♦s ●❛r❛♥t✐❞♦s ♣❡❧❛ ❈r✐♣t♦❣r❛✜❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✸ ❈r✐♣t♦❣r❛✜❛ ✸✺
✸✳✶ ❈r✐♣t♦❣r❛✜❛ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷ ❈♦❞✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✸ ❉❡❝♦❞✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✹ P♦r q✉❡ ❋✉♥❝✐♦♥❛❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✺ ❘❙❆ ❘❡❛❧♠❡♥t❡ é ❙❡❣✉r♦❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✻ ❖s Pr✐♠♦s P❡r❢❡✐t♦s P❛r❛ ♦ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✹✹
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✻
■♥tr♦❞✉çã♦
❖ ❡♥s✐♥♦ ❞❡ ♠❛t❡♠át✐❝❛ ♥❛ ❡s❝♦❧❛ ♣ú❜❧✐❝❛ ✈❡♠ s❡♥❞♦ ❞✐s❝✉t✐❞♦ ❤á ✈ár✐♦s ❛♥♦s ❡ s❡ ❝♦♥st✐t✉✐ ♥✉♠ ❣r❛♥❞❡ ❞❡s❛✜♦ ❛ s❡r s✉♣❡r❛❞♦ ♣❡❧♦s ❛t♦r❡s ❡♥✈♦❧✈✐❞♦s ♥❡ss❡ ♣r♦❝❡ss♦✳
▼✉✐t♦ s❡ t❡♠ ❢❛❧❛❞♦ s♦❜r❡ ♦s ♦❜❥❡t✐✈♦s✱ ❛s ♠❡t♦❞♦❧♦❣✐❛s ❡ ❞✐✈❡rs♦s ❡st✉❞♦s ❡ ♣❡s✲ q✉✐s❛s tê♠ s✐❞♦ ❢❡✐t❛s ♣❛r❛ ❡♥❝♦♥tr❛r ❛ r❛✐③ ♦✉ ❛s r❛í③❡s ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ❜❛✐①❛ ❛♣r❡♥❞✐✲ ③❛❣❡♠ ♠❛t❡♠át✐❝❛ ❞♦s ❛❧✉♥♦s✱ ♠❛s ♦s r❡s✉❧t❛❞♦s ❞❡ ❛✈❛❧✐❛çõ❡s ♦✜❝✐❛✐s ♠♦str❛♠ q✉❡ ❛s ❛çõ❡s ❡♠♣r❡❡♥❞✐❞❛s s❡ ♠♦str❛♠ ✐♥❡✜❝✐❡♥t❡s ❡ ❛ ♠❛t❡♠át✐❝❛✱ ❝♦♠♦ ❞✐s❝✐♣❧✐♥❛ ❡s❝♦❧❛r✱ ❝♦♥t✐♥✉❛ s❡♥❞♦ ♦ ❝❛❧❝❛♥❤❛r ❞❡ ❆q✉✐❧❡s ❞❛ ❡❞✉❝❛çã♦ ❜r❛s✐❧❡✐r❛✳
▼✉✐t❛s sã♦ ❛s r❛③õ❡s ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ ❡ ❛s ❥✉st✐✜❝❛t✐✈❛s ♣❛r❛ s✉❛ ♣r❡✲ s❡♥ç❛ ♥♦s ❝✉rrí❝✉❧♦s ❡s❝♦❧❛r❡s✳ ❆ ❛♣r❡♥❞✐③❛❣❡♠ ❡ ❛s ❛q✉✐s✐çõ❡s ❞❡ ❝♦♠♣❡tê♥❝✐❛s ♠❛t❡✲ ♠át✐❝❛s ❛❧é♠ ❞❡ ❢♦r♥❡❝❡r ❛♦ ✐♥❞✐✈í❞✉♦ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ♣❛r❛ ❡①♣r❡ss❛r s❡✉ ♣❡♥s❛♠❡♥t♦✱ ♦ ❞♦t❛ ❞❡ ❢❡rr❛♠❡♥t❛s ❝♦♠ ❛s q✉❛✐s ❡❧❡ ♣♦❞❡ ❣❡r❛r ♥♦✈♦s ♣❡♥s❛♠❡♥t♦s ❡ ❞❡s❡♥✈♦❧✈❡r r❛❝✐♦❝í♥✐♦s✱ ❝♦♥❢♦r♠❡ ❛❝❡♥t✉❛ ◆❯◆❊❙✱ ❬✶✶❪❀
✏❆ ▼❛t❡♠át✐❝❛ ♥ã♦ é s✐♠♣❧❡s♠❡♥t❡ ✉♠❛ ❞✐s❝✐♣❧✐♥❛✱ ♠❛s t❛♠❜é♠ ✉♠❛ ❢♦r♠❛ ❞❡ ♣❡♥s❛r✳ ➱ ♣♦r ✐ss♦ q✉❡ ❛ ▼❛t❡♠át✐❝❛✱ ❛ss✐♠ ❝♦♠♦ ❛ ❛❧❢❛❜❡t✐③❛çã♦✱ é ❛❧❣♦ q✉❡ ❞❡✈❡ s❡r t♦r♥❛❞♦ ♣♦ssí✈❡❧ ♣❛r❛ t♦❞♦s✑✳ ✭✶✾✾✼✱ ♣✳ ✾✺✮
❊♠ ❞❡❢❡s❛ ❞♦ ❡♥s✐♥♦ ❡ ❞❛ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡st❛ ❝✐ê♥❝✐❛ ♦✉ ❞✐s❝✐♣❧✐♥❛ ❡s❝♦❧❛r✱ ✈✐st♦ q✉❡ ♥♦ â♠❜✐t♦ ❡s❝♦❧❛r ❡❧❛ ❛ss✉♠❡ ✉♠ ❛s♣❡❝t♦ ♠❡♥♦s ❢♦r♠❛❧ ❡ ❛s ✈❡③❡s ♣❡r❞❡♥❞♦ ♦ ❡①tr❡♠♦ r✐❣♦r ❛❝❛❞❡♠✐❝✐st❛✱ ❝❛❜❡ r❡ss❛❧t❛r ❛✐♥❞❛ q✉❡ ❛s ❢♦r♠❛s ❞❡ ♣❡♥s❛♠❡♥t♦ ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ♠❛t❡♠át✐❝❛ ♣♦❞❡♠ ❡①♣❛♥❞✐r✲s❡ ♣❛r❛ ♦✉tr♦s r❛❝✐♦❝í♥✐♦s✱ ✐♠♣✉❧s✐♦♥❛♥❞♦ ❛ ❝❛♣❛❝✐❞❛❞❡ ❣❧♦❜❛❧ ❞❡ ❛♣r❡♥❞✐③❛❞♦✳ ❆♦ tr❛❜❛❧❤❛r ♠❛t❡♠át✐❝❛ ❡①❡r❝✐t❛✲s❡ ❛ ♠❡♥t❡ ❝♦♠ ✉♠❛ sér✐❡ ❞❡ ❛çõ❡s q✉❡ ❡①✐❣❡♠ ♠✉✐t♦ r❛❝✐♦❝í♥✐♦ ❡ ❛t✐t✉❞❡s ✐♥t❡❧❡❝t✉❛✐s ❝♦♠♣❧❡①❛s✱ ♣♦✐s é ❝♦♠✉♠ ❢✉♥❞❛♠❡♥t❛r ♦ ♣❡♥s❛♠❡♥t♦ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❛①✐♦♠❛s✱ ♥❛ ❣❡r❛çã♦ ❡ ✈❛❧✐❞❛çã♦ ❞❡ ❤✐✲ ♣ót❡s❡s✱ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❛❧❣♦r✐t♠♦s ❡ ♣r♦❝❡❞✐♠❡♥t♦s ❞❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✱ ❡st❛❜❡❧❡❝❡♥❞♦ ❝♦♥❡①õ❡s ❡ ❢❛③❡♥❞♦ ❡st✐♠❛t✐✈❛s✳ ❊ss❛s ❢❡rr❛♠❡♥t❛s sã♦ ❛♣❧✐❝á✈❡✐s ❛ ✉♠
❝♦♥❥✉♥t♦ ❛♠♣❧♦ ❞❡ s✐t✉❛çõ❡s s✐♠✐❧❛r❡s✱ ♣♦✐s ♣❡r♠✐t❡♠ ❛ ❛♥á❧✐s❡ ❞❡ s✐t✉❛çõ❡s ♣❛rt✐❝✉❧❛✲ r❡s ❡ s✉❛ ✐♥s❡rçã♦ ♥❛s ❡str✉t✉r❛s ♠❛✐s ❛♠♣❧❛s ❡ ❣❧♦❜❛✐s ❡ ✐ss♦ t♦r♥❛ ♣♦ssí✈❡❧ ❝♦♥str✉✐r ❡str✉t✉r❛s ❞❡ ♣❡♥s❛♠❡♥t♦ q✉❡ ♣♦❞❡♠ s❡r út❡✐s ❡ ❝♦♥tr✐❜✉✐r ♠✉✐t♦ ❡♠ s✐t✉❛çõ❡s ♥ã♦ r❡❧❛❝✐♦♥❛❞❛s ❛ ♠❛t❡♠át✐❝❛ ❡①♣❡r✐♠❡♥t❛❞❛ ♥❛ ✈✐❞❛ ❡♠ s♦❝✐❡❞❛❞❡✳
❖s P❈◆✬s ✲ P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ❞❡❢❡♥❞❡♠ ❛ ✐❞❡✐❛ ❞❡ q✉❡ ❛ ❛♣r❡♥✲ ❞✐③❛❣❡♠ ♠❛t❡♠át✐❝❛ ❝♦♥tr✐❜✉✐ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞♦ ❝✐❞❛❞ã♦ q✉❛♥❞♦ ❛♣♦♥t❛ q✉❡ ✉♠ ❝✉rrí❝✉❧♦ ❞❡ ♠❛t❡♠át✐❝❛ ❞❡✈❡✿
✏❬✳✳✳❪ ♣r♦❝✉r❛r ❝♦♥tr✐❜✉✐r✱ ❞❡ ✉♠ ❧❛❞♦✱ ♣❛r❛ ❛ ✈❛❧♦r✐③❛çã♦ ❞❛ ♣❧✉r❛✲ ❧✐❞❛❞❡ s♦❝✐♦❝✉❧t✉r❛❧✱ ✐♠♣❡❞✐♥❞♦ ♦ ♣r♦❝❡ss♦ ❞❡ s✉❜♠✐ssã♦ ♥♦ ❝♦♥✲ ❢r♦♥t♦ ❝♦♠ ♦✉tr❛s ❝✉❧t✉r❛s❀ ❞♦ ♦✉tr♦✱ ❝r✐❛r ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ tr❛♥s❝❡♥❞❛ ✉♠ ♠♦❞♦ ❞❡ ✈✐❞❛ r❡str✐t♦ ❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❡s♣❛ç♦ s♦❝✐❛❧ ❡ s❡ t♦r♥❡ ❛t✐✈♦ ♥❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ s❡✉ ❛♠❜✐✲ ❡♥t❡✳✑✭❇r❛s✐❧✱ ✶✾✾✽✱ ♣✳ ✸✵✮
◗✉❡r s❡ ❣♦st❡ ❞❡❧❛ ♦✉ ♥ã♦✱ ❛ ▼❛t❡♠át✐❝❛ é ✉♠❛ ❝✐ê♥❝✐❛ q✉❡ ❡stá ♣r❡s❡♥t❡ ♥♦ ❝♦t✐✲ ❞✐❛♥♦ ❞❡ t♦❞♦ ❝✐❞❛❞ã♦✳ ❊ss❛ ♣r❡s❡♥ç❛ ♣♦❞❡ s❡r ❞❡ ❢♦r♠❛ ❝❧❛r❛ ❡ ❡①♣❧í❝✐t❛ ❡ ❡♠ ❝❡rt❛s s✐t✉❛çõ❡s ❞❡ ❢♦r♠❛ s✉tí❧✳
❱❡r✐✜❝❛r ♦ ♣r❡ç♦ ❞❡ ✉♠ ♣r♦❞✉t♦ ♥♦ s✉♣❡r♠❡r❝❛❞♦✱ ❝♦♥❢❡r✐r ♦ tr♦❝♦ ❡♠ ✉♠❛ ❝♦♠♣r❛✱ ✈❡r✐✜❝❛r ❛s ❤♦r❛s ♥♦ r❡❧ó❣✐♦ q✉❛♥❞♦ ♥❡❝❡ssár✐♦✱ q✉❛♥❞♦ ❛❝❤❛♠♦s q✉❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❡♥❞❡r❡ç♦ ❡stá ❧♦♥❣❡ ♦✉ ♣❡rt♦✱ ❛♦ ❞✐③❡r q✉❡ ❛ ❜♦❧❛ ❝❤✉t❛❞❛ ♣❡❧♦ ❥♦❣❛❞♦r ❞♦ t✐♠❡ ❡♥tr♦✉ ♥♦ â♥❣✉❧♦✱ ❡st❛♠♦s ✏❧❡♥❞♦✑ ❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛✱ ❡①❡r❝✐t❛♥❞♦ ♥♦ss❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❛❜str❛çã♦ ❡ ✉t✐❧✐③❛♥❞♦ ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ❝♦♥str✉í❞♦s ❛♦ ❧♦♥❣♦ ❞❡ ✈ár✐♦s sé❝✉❧♦s ♣❡❧❛ ❤✉♠❛♥✐❞❛❞❡✳ ❆té ♠❡s♠♦ ♣❛r❛ ❝♦♠♣r❡❡♥❞❡r ❝❡rt❛s ♥♦tí❝✐❛s ✈❡✐❝✉❧❛❞❛s ♥❛s ♣á❣✐♥❛s ❞❡ ✉♠ ❥♦r♥❛❧ ♦✉ ❞❡ ✉♠❛ r❡✈✐st❛✱ é ✐♥❞✐s♣❡♥sá✈❡❧ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❞♦s❡ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❡ ✉♠ ❞♦♠í♥✐♦ ♠í♥✐♠♦ ❞❛s ❧✐♥❣✉❛❣❡♥s ✐♥❡r❡♥t❡s ❛ ❡ss❛ ❝✐ê♥❝✐❛✳
❆ ♠❛t❡♠át✐❝❛ é ❝❛❞❛ ✈❡③ ♠❛✐s ♣r❡s❡♥t❡ ❡♠ ❞✐✈❡rs❛s s✐t✉❛çõ❡s ❡ é ❝❛❞❛ ✈❡③ ♠❛✐s ❡♠♣r❡❣❛❞❛ ♣❛r❛ ❞❡s❝r❡✈❡r✱ ♠♦❞❡❧❛r ❡ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♥❛s ♠❛✐s ❞✐✈❡rs❛s ár❡❛s ❞❛ ❛t✐✈✐❞❛❞❡ ❤✉♠❛♥❛✳
❘❡♣r❡s❡♥t❛çõ❡s ❡♠ ❢♦r♠❛ ❞❡ ♣♦r❝❡♥t❛❣❡♥s✱ ❣rá✜❝♦s ♦✉ t❛❜❡❧❛s sã♦ ✉s✉❛✐s ❡♠ ❞✐✲ ✈❡rs❛s s✐t✉❛çõ❡s ❝♦♠♦ ♥❛ ❞❡s❝r✐çã♦ ❡ ❛♥á❧✐s❡ ❞❡ ✈ár✐♦s ❛ss✉♥t♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ♣r❡ç♦s✱ ♣❡sq✉✐s❛s ❞❡ ♦♣✐♥✐ã♦ ❡ ❞❡ ✐♥t❡♥çã♦ ❞❡ ✈♦t♦s✱ s✐t✉❛çõ❡s ❞❡ ♠❡r❝❛❞♦ ✜♥❛♥❝❡✐r♦✱ ✐♥✢❛çã♦✱ ❛✉♠❡♥t♦ ❞❡ s❛❧ár✐♦s✱ t❛①❛s ❡ ✐♠♣♦st♦s✱ ❡t❝✳
❋✐❣✉r❛ ✶✿ P❡❞r❡✐r♦ tr❛❜❛❧❤❛♥❞♦
❙ã♦ s✐t✉❛çõ❡s q✉❡ ✉s❛♠ ❡❧❡♠❡♥t♦s ❞❛ ♠❛t❡♠át✐❝♦s ♣♦rt❛♥t♦✱ ✉♠ ♠é❞✐❝♦ q✉❡ ✐♥✲ t❡r♣r❡t❛ ✉♠ ❡❧❡tr♦❝❛r❞✐♦❣r❛♠❛✱ ❡stá ❢❛✲ ③❡♥❞♦ ❛ ❧❡✐t✉r❛ ❞❡ ✉♠ ❣rá✜❝♦ q✉❡ ♣♦❞❡ s❡r ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ ❢✉♥çã♦❀ ✉♠ ♠❡s✲ tr❡ ❞❡ ♦❜r❛s ♦✉ ♠❡s♠♦ ✉♠ ♣❡❞r❡✐r♦ q✉❡ s❡♠ ❝♦♥❤❡❝✐♠❡♥t♦s ❞❡ ár❡❛✱ ✈♦❧✉♠❡✱ â♥✲ ❣✉❧♦s ❝♦♥st❛♥t❡♠❡♥t❡ ❛♣❧✐❝❛ t❛✐s s❛❜❡r❡s ❡♠ s❡✉ tr❛❜❛❧❤♦ ✭❋✐❣✉r❛ ✶✮✱ ✉♠ ♠❡♥✐♥♦ ♦✉ ✉♠ ❛♣♦s❡♥t❛❞♦ q✉❡ ✈❡♥❞❡ ♣✐❝♦❧és ♥❛ r✉❛ ❡ ❝❛❧❝✉❧❛ q✉❛♥t♦ ❣❛♥❤❛ ♣❡❧❛s ✈❡♥❞❛s ❞♦ ❞✐❛ ✉s❛♥❞♦ ❝♦♥❤❡❝✐♠❡♥t♦s ❞❡ ♣♦r❝❡♥✲ t❛❣❡♠✳ ❊ss❡ ❡ ✐♥✜♥✐t♦s ♦✉tr♦s ❡①❡♠♣❧♦s ❞❛ r❡❛❧✐❞❛❞❡ ♣r♦✈❛♠ t❛♥t♦ ❛ ♣r❡s❡♥ç❛ ❞❛ ♠❛t❡♠át✐❝❛ ♥♦ ❝♦t✐❞✐❛♥♦ ❞❛s ♣❡ss♦❛s q✉❛♥t♦ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡st❛ ❝✐ê♥❝✐❛ ♣❛r❛ ❛ ✈✐❞❛✱ ❛ s♦❝✐❡❞❛❞❡ ❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ♠✉♥❞♦✳
❊♠ ❢❛❝❡ ❞❛ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ▼❛t❡♠át✐❝❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ✐♥t❡❧❡❝t✉❛❧✱ ♣❡ss♦❛❧ ❡ ♣❛r❛ ♦ ❡①❡r❝í❝✐♦ ❞❛ ❝✐❞❛❞❛♥✐❛✱ ❡st❛ ❝✐ê♥❝✐❛ t❡♠ ❛✐♥❞❛ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ t❡r ❝♦♥tr✐❜✉í❞♦ ❞❡ ❢♦r♠❛ ❞❡❝✐s✐✈❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❤✉♠❛♥✐❞❛❞❡✱ s❡♥❞♦ ✉♠ ❞♦s ♣✐❧❛r❡s ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❝✐❡♥tí✜❝♦ ❡ t❡❝♥♦❧ó❣✐❝♦ q✉❡ ♦ ♠✉♥❞♦ ✈❡♠ ❡①♣❡r✐♠❡♥t❛♥❞♦ ❛♦ ❧♦♥❣♦ ❞❡ s✉❛ ❤✐stór✐❛✳ ▼❡s♠♦ ❞✐❛♥t❡ ❞❡ tã♦ ♥♦tá✈❡✐s ❝♦♥tr✐❜✉✐çõ❡s ❡ tã♦ s♦❜❡❥❛ ✐♠♣♦rtâ♥❝✐❛ ❛ ♠❛t❡♠át✐❝❛ é ❛❧✈♦ ❞♦ ❞❡s♣r❡③♦ ❞❛ q✉❛s❡ ❛❜s♦❧✉t❛ ♠❛✐♦r✐❛ ❞♦s ❛❧✉♥♦s✱ ♦ q✉❡ ❣❡r❛ ❛♥❣úst✐❛ ❡ ❢r✉str❛çã♦ ♥♦s ♣r♦❢❡ss♦r❡s ❞❡ss❛ ❞✐s❝✐♣❧✐♥❛✱ q✉❡ ♥ã♦ ❞❡✈❡♠ s❡ ❞❡✐①❛r ✈❡♥❝❡r ♣❡❧♦s ❛r❣✉♠❡♥t♦s ❝♦♥trár✐♦s ❡ s❡♠ ❢✉♥❞❛♠❡♥t♦s q✉❡ ✈✐s❛♠ ❝♦❧♦❝á✲❧❛ ❝♦♠♦ ❛ ❣r❛♥❞❡ ✈✐❧ã ❞♦ ❡♥s✐♥♦ ❡s❝♦❧❛r✳
❊♠ ♠❡✐♦ ❛ t❛♥t♦s ❞❡s❛✜♦s ❡♥❝♦♥tr❛❞♦s ♥❛ ár❞✉❛ t❛r❡❢❛ ❞❡ ❡♥s✐♥❛r ♠❛t❡♠át✐❝❛✱ ✉♠ ❞♦s ♠❛✐♦r❡s ❡♥❢r❡♥t❛❞♦s ♣❡❧♦s ♣r♦❢❡ss♦r❡s é ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝♦♥s❝✐❡♥t✐③❛r ♦s ❞✐s❝❡♥t❡s s♦❜r❡ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦s ❝♦♥t❡ú❞♦s ❡st✉❞❛❞♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ❞❡✐①❛♥❞♦ ❝❧❛r♦ ♣❛r❛ ❡❧❡s q✉❡ t❛✐s ❝♦♥t❡ú❞♦s ♣♦❞❡♠ ♣❡r❢❡✐t❛♠❡♥t❡ tr❛♥s♣♦r ❡ss❡ ❧✐♠✐t❡✱ ♠♦str❛♥❞♦ q✉❡ ♦s ❝♦♥t❡ú❞♦s ❣❛♥❤❛♠ s✐❣♥✐✜❝❛❞♦ ❡♠ ❞✐✈❡rs♦s ❝♦♥t❡①t♦s ❡ s✐t✉❛çõ❡s ♣rát✐❝❛s ❡ ♣♦❞❡♠ s❡r ♣❡r❢❡✐t❛♠❡♥t❡ ❛♣❧✐❝á✈❡✐s ♥♦ ❝♦t✐❞✐❛♥♦✳
❙✉♣❡r❛r s✐t✉❛çõ❡s ♥❛s q✉❛✐s ♦s ♣r♦❢❡ss♦r❡s ❞❛ ár❡❛ ❞❡ ❝✐ê♥❝✐❛s ❡①❛t❛s✱ ♠❛✐s ♣r❡❝✐✲ s❛♠❡♥t❡ ♦ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛✱ ♦✉✈❡ ❝♦rr✐q✉❡✐r❛♠❡♥t❡ ♥❛s ❛✉❧❛s ✐♥❞❛❣❛çõ❡s t❛✐s ❝♦♠♦✿ ✏❖♥❞❡ ✈♦✉ ✉s❛r ❡st❡ ❝♦♥t❡ú❞♦✱ ♣r♦❢❡ss♦r❄✑✱ ✏P♦r q✉❡ ❡✉ ♣r❡❝✐s♦ ❛♣r❡♥❞❡r ✐st♦❄✑✱ ✏❊ss❛ ♠❛tér✐❛ s❡r✈❡ ♣r❛ ❛❧❣✉♠❛ ❝♦✐s❛ ♥❛ ♠✐♥❤❛ ✈✐❞❛❄✑ é ✉♠ ❞♦s ♣♦♥t♦s ❝r✉❝✐❛✐s ♣❛r❛ s❡ ❛❧❛✈❛♥❝❛r ♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ ♥❛s ♥♦ss❛s ❡s❝♦❧❛s✳ ◆❡ss❡ s❡♥t✐❞♦✱ ❞♦t❛r ♦s ❝♦♥✲
t❡ú❞♦s ❞❡ s✐❣♥✐✜❝❛❞♦✱ ✈✐♥❝✉❧❛♥❞♦✲♦s ❛ r❡❛❧✐❞❛❞❡ ❡ ♠♦str❛♥❞♦ s✉❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ♣rát✐❝❛ é✱ ♣♦ss✐✈❡❧♠❡♥t❡✱ ❛ ♠❛✐♦r ❜❛rr❡✐r❛ ❛ s❡r s✉♣❡r❛❞❛ ♥♦ tr❛❜❛❧❤♦ ❞❡ s❡ ♣r♦❞✉③✐r ♠❡❧❤♦r❡s r❡s✉❧t❛❞♦s ♥♦ ❡♥s✐♥♦✴❛♣r❡♥❞✐③❛❣❡♠ ❞❡ss❛ ❝✐ê♥❝✐❛✳
❉✐❛♥t❡ ❞❡ss❛ ♣r♦❜❧❡♠át✐❝❛✱ ❛ ♣r❡s❡♥t❡ ♣❡sq✉✐s❛ ✈✐s❛ ❛♠❡♥✐③❛r ❡ss❡ ♣r♦❜❧❡♠❛ ❞❛ ❞❡s✲ ✈✐♥❝✉❧❛çã♦ ❞♦s ❝♦♥t❡ú❞♦s ❝✉rr✐❝✉❧❛r❡s ❞❡ ♠❛t❡♠át✐❝❛ ❞❛ r❡❛❧✐❞❛❞❡ ♣rát✐❝❛ ✈✐✈❡♥❝✐❛❞❛ ♣❡❧♦s ❛❧✉♥♦s ♣r♦✈❛♥❞♦ q✉❡ ❛s ✈❡r❞❛❞❡s ♠❛t❡♠át✐❝❛s ♣♦❞❡♠ s❡r ❛♣❧✐❝❛❞❛s ❛♦ ♠✉♥❞♦ r❡❛❧ ❞❛♥❞♦✲❧❤❡ ❢✉♥❞❛♠❡♥t❛çã♦ ❧ó❣✐❝❛ ❡ ❝♦♥❝r❡t❛✳
◆❡ss❡ ♣❛rt✐❝✉❧❛r é ✐♠♣♦rt❛♥t❡ ❧❡♠❜r❛r q✉❡ ♦s ♥ú♠❡r♦s s✉r❣✐r❛♠ ♥❛t✉r❛❧♠❡♥t❡✱ ❡ q✉❡ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❡✈♦❧✉✐✉ ♣❛✉t❛❞♦ ♣❡❧❛s ♥❡❝❡ss✐❞❛❞❡s ❞❛ ❤✉♠❛♥✐❞❛❞❡✱ ❞❡♥tr♦ ❞❡ ❝❛❞❛ ❝♦♥t❡①t♦ ❡ ♣❡rí♦❞♦ ❞❛ ❤✐stór✐❛✱ ❧❡✈❛♥❞♦ ❛s s♦❝✐❡❞❛❞❡s q✉❡ ❛ ❡st✉❞❛♠ ❡ ❡♠♣r❡❣❛♠ s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ♣❛r❛ s✉♣r✐r s✉❛s ♥❡❝❡ss✐❞❛❞❡s ❡ ✐♥t❡r❡ss❡s ❛ ❣r❛♥❞❡s ❝♦♥q✉✐st❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ✈❡♥❝❡r ✉♠❛ ❣✉❡rr❛✱ ❛tr❛✈és ❞♦ ✉s♦ ❞❛ ❝r✐♣t♦❣r❛✜❛✳
❋✐❣✉r❛ ✷✿ ▼áq✉✐♥❛ ❞❡ ❈r✐♣t♦❣r❛t✐❛
❉✉r❛♥t❡ ❛ s❡❣✉♥❞❛ ●✉❡rr❛ ▼✉♥❞✐❛❧✱ ♦s ❛❧✐❛❞♦s ♣❡r❝❡❜❡r❛♠ q✉❡ ❛ ❚❡♦r✐❛ ❞❛ ▲ó✲ ❣✐❝❛ ▼❛t❡♠át✐❝❛ ♣♦❞❡r✐❛ s❡r ✉s❛❞❛ ♣❛r❛ ❞❡❝✐❢r❛r ❛s ♠❡♥s❛❣❡♥s ❞♦s ❛❧❡♠ã❡s ✭❋✐✲ ❣✉r❛ ✷✮✱ ❛♣❡♥❛s ❛tr❛✈és ❞❡ ❝á❧❝✉❧♦s rá✲ ♣✐❞♦s ❡ ♣r❡❝✐s♦s✳ ◗✉❡♠ ♠❛✐s ❝♦♥tr✐❜✉✐✉ ♣❛r❛ ❡ss❛ q✉❡❜r❛ ❞❡ ❝ó❞✐❣♦s ❢♦✐ ♦ ♠❛t❡✲ ♠át✐❝♦ ✐♥❣❧ês ❆❧❛♥ ❚✉r✐♥❣ ✭❋✐❣✉r❛ ✸✮✳ ❊❧❡ ❢♦✐ ❝♦♥✈♦❝❛❞♦ ♣❡❧❛ ❊s❝♦❧❛ ❞❡ ❈✐❢r❛s ❡ ❈ó✲ ❞✐❣♦s ❞♦ ●♦✈❡r♥♦ ♣❛r❛ ❞❡❝✐❢r❛r ❛s ♠❡♥s❛✲ ❣❡♥s ❝♦❞✐✜❝❛❞❛s ❞♦ ✐♥✐♠✐❣♦ ❛❧❡♠ã♦✱ ❡st❡ ♣♦ss✉í❛ ✉♠ s♦✜st✐❝❛❞♦ ❡ ♠♦❞❡r♥♦ s✐st❡♠❛ ❞❡ ❝♦❞✐✜❝❛çã♦✱ ❝❤❡❣❛♥❞♦ ❛ ❛❝r❡❞✐t❛r q✉❡ ❡ss❡ s✐st❡♠❛ ❡r❛ ✐♥❢❛❧í✈❡❧✳ ❚✉r✐♥❣ ♣r❡✲ ❝✐s❛✈❛ ❞❡❝✐❢r❛r ❛s ♠❡♥s❛❣❡♥s ❡♠ t❡♠♣♦ ❤á❜✐❧✱ ♣♦r ❡①❡♠♣❧♦ s❡ ❛ ♠❡♥s❛❣❡♠ ❢♦ss❡ ♣❛r❛ ❞❡str✉✐r ✉♠ ♥❛✈✐♦✱ ❛ ♠❡♥s❛❣❡♠ ❞❡✲ ✈❡r✐❛ s❡r ❞❡❝✐❢r❛❞❛ ❛♥t❡s q✉❡ ♦ ♥❛✈✐♦ ❢♦ss❡ ❞❡str✉í❞♦✱ ❝❛s♦ ❝♦♥trár✐♦✱ ❡ss❛ ♠❡♥s❛❣❡♠ ♥ã♦ s❡r✐❛ ♠❛✐s út✐❧✳ ❆❧é♠ ❞✐ss♦✱ ♦s ❆❧✐❛✲ ❞♦s t✐♥❤❛♠ q✉❡ ❞✐s❢❛rç❛r ❛ ♣♦♥t♦ ❞♦ ✐♥✐♠✐❣♦ ❛❧❡♠ã♦ ♥❡♠ s✉s♣❡✐t❛r q✉❡ ❛ ♠❡♥s❛❣❡♠ ❤❛✈✐❛ s✐❞♦ ❞❡❝✐❢r❛❞❛✳
❋✐❣✉r❛ ✸✿ ❆❧❛♥ ❚✉r✐♥❣
❊ss❛ ❢♦✐ ❝❡rt❛♠❡♥t❡ ✉♠❛ ❞❛s ♠❛✐♦r❡s ❡ ♠❡❧❤♦r❡s ❛r♠❛s ❞❡ ❣✉❡rr❛ ❞♦s ❆❧✐❛❞♦s ✉s❛❞❛ ❢♦r❛ ❞♦ ❝❛♠♣♦ ❞❡ ❜❛t❛❧❤❛✳ ❉✐❛♥t❡ ❞❡ t❛✐s r❡❛❧✐③❛çõ❡s✱ ♥ã♦ é ♥❡♥❤✉♠ ❛❜s✉r❞♦ ❞✐③❡r q✉❡ ❛ s❡❣✉♥❞❛ ●✉❡rr❛ ▼✉♥❞✐❛❧ t❛♠✲ ❜é♠ ❢♦✐ ❛ ✏❣✉❡rr❛ ❞♦s ♠❛t❡♠át✐❝♦s✑✳
❆ ❝r✐♣t♦❣r❛✜❛ é ✉♠❛ ❜❛t❛❧❤❛ ❡♥tr❡ ♦ ❝r✐❛❞♦r ❞♦ ❝ó❞✐❣♦ ❡ ❛q✉❡❧❡ q✉❡ t❡♥t❛ ❞❡❝✐❢rá✲❧♦✱ ❛❧é♠ ❞♦ ❞❡st✐♥❛tár✐♦✳ ❖ ❞❡s❛✲ ✜♦ ♣❛r❛ ♦ ❝♦❞✐✜❝❛❞♦r é ♠✐st✉r❛r ❛ ♠❡♥✲ s❛❣❡♠ ❛té ✉♠ ♣♦♥t♦ ❡♠ q✉❡ ❡❧❛ s❡❥❛✱ t❡✲ ♦r✐❝❛♠❡♥t❡✱ ✐♥❞❡❝✐❢rá✈❡❧✱ ♣❛r❛ ✉♠ ♣♦ssí✲ ✈❡❧ ✐♥t❡r♠❡❞✐❛❞♦r ✐♥❞❡s❡❥❛❞♦✱ s❡♥❞♦ ❢á❝✐❧ ❛ ❞❡❝✐❢r❛çã♦ ❞❛ ♠❡♥s❛❣❡♠ ♣❛r❛ ♦ ❞❡st✐♥❛tár✐♦✳ P❛r❛ ♠❛✐♦r❡s ✐♥❢♦r♠❛çõ❡s ❝♦♥s✉❧t❡ ❛s r❡❢❡rê♥❝✐❛s ❬✷❪ ❡ ❬✶✸❪
❆t✉❛❧♠❡♥t❡ ✈✐✈❡♠♦s ✉♠❛ ❣✉❡rr❛ ❞❡ q✉❡❜r❛ ❞❡ ❝ó❞✐❣♦s ♠✉✐t♦ ✐♥t❡♥s❛ ❡①❡♠♣❧✐✜❝❛❞❛s ♥❛s ❝♦♠♣r❛s r❡❛❧✐③❛❞❛s ❝♦♠ ❝❛rtõ❡s ❞❡ ❝ré❞✐t♦✱ ❛tr❛✈és ❞❡ s✐t❡s ♥❛ ✐♥t❡r♥❡t✱ ❡t❝✳ ▼✉✐t♦s ♠❛❧ ✐♥t❡♥❝✐♦♥❛❞♦s t❡♥t❛♠ ❞❡❝✐❢r❛r ♦s ❝ó❞✐❣♦s✱ ❡ ❛ss✐♠ ♦❜t❡r❡♠ ✐♥❢♦r♠❛çõ❡s ♣r✐✈✐❧❡❣✐❛✲ ❞❛s ❡ ❝♦♥✜❞❡♥❝✐❛✐s ❞♦ ❝❧✐❡♥t❡ ♣❛r❛ ♣♦st❡r✐♦r♠❡♥t❡ ✉sá✲❧❛s ❝❛✉s❛♥❞♦ ♣r❡❥✉í③♦ ✜♥❛♥❝❡✐r♦ ❡ ✉♠❛ sér✐❡ ❞❡ ♦✉tr♦s tr❛♥st♦r♥♦s✳ ❊♠ ❝♦♥tr❛♣❛rt✐❞❛ ♦s ❜❛♥❝♦s ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛s ❧♦❥❛s ✈✐rt✉❛✐s t❡♥t❛♠ t♦r♥❛r s✉❛s ❝♦❞✐✜❝❛çõ❡s ❝❛❞❛ ✈❡③ ♠❛✐s ❝♦♠♣❧❡①❛s✱ ♦❜❥❡t✐✈❛♥❞♦ ♠❛✐♦r s❡❣✉r❛♥ç❛✱ ❡ ♣❛r❛ ✐ss♦ é ❝r✉❝✐❛❧ ♦ ✉s♦ ❞❛ ♠❛t❡♠át✐❝❛✳
❖s ♠❛t❡♠át✐❝♦s ❝♦♥s✐❞❡r❛♠ ✐♠♣♦rt❛♥tíss✐♠❛s ♠❛tér✐❛s ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s t❛✐s ❝♦♠♦ ♥ú♠❡r♦s ♣r✐♠♦s✱ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✱ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❡ ❢❛t♦r❛çã♦✱ ❝♦♥❝❡✐t♦s ❝♦♠✉♥s ♣♦ré♠ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ s✐st❡♠❛ ❞❡ ❝r✐♣t♦❣r❛✜❛ t❛❧ ❝♦♠♦ ♦ ❘❙❆✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❡st❛ ♣❡sq✉✐s❛ s❡ ❞✐r✐❣❡ ❛♦s ❡st✉❞❛♥t❡s ❝♦♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❜ás✐❝♦ s♦✲ ❜r❡ ❛ ❢❛t♦r❛çã♦ ❞❡ ✐♥t❡✐r♦s ❡ ♣r✐♠♦s✱ q✉❡ t❡♥❤❛ ❝❡rt❛ ❢❛❝✐❧✐❞❛❞❡ ♥♦ ❝á❧❝✉❧♦ ❝♦♠ ❢ór♠✉❧❛s ❡❧❡♠❡♥t❛r❡s ❡ q✉❡ t❡♥❤❛ ✐♥t❡r❡ss❡ ♠❛t❡♠át✐❝♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❛♣r❡❝✐❛r ❛r❣✉♠❡♥t♦s ❞❡ ❞❡♠♦♥str❛çõ❡s ❜❛st❛♥t❡ ❜ás✐❝❛s✱ ❞❡s♣❡rt❛♥❞♦ ❛ ❝✉r✐♦s✐❞❛❞❡ ❞♦ ❛❧✉♥♦ s❛❜❡♥❞♦ q✉❡ ❝✉✲ r✐♦s✐❞❛❞❡ é ❝❛♣❛③ ❞❡ ❧❡✈❛r ♦ ✐♥❞✐✈í❞✉♦ ♠✉✐t♦ ❛❧é♠ ❞♦ q✉❡ s❡ ♣♦❞❡ ✐♠❛❣✐♥❛r✳
❖ ♥♦ss♦ tr❛❜❛❧❤♦ ❢♦✐ ❞✐✈✐❞✐❞♦ ❞❡ ✉♠ ♠♦❞♦ q✉❡ ♥♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ sã♦ ❢❡✐t❛s r❡✲ ❢❡rê♥❝✐❛s ❡ ❞❡♠♦♥str❛çõ❡s ❞❡ tó♣✐❝♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ♠❛t❡♠át✐❝❛ q✉❡ sã♦ r❡q✉✐s✐t♦s ♣❛r❛ ❛ ❝r✐♣t♦❣r❛✜❛✳ ❊♥tr❡ ❡ss❡s ❝♦♥❤❡❝✐♠❡♥t♦s ♣ré✈✐♦s ❞❡ ♠❛t❡♠át✐❝❛ ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ ❝r✐♣t♦❣r❛✜❛ ❡stã♦ ❞❡st❛❝❛❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ♥ú♠❡r♦s ♣r✐♠♦s✱ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ✭▼▼❈✮ ❡ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✱ ❢❛t♦r❛çã♦✱ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✱
♥ú♠❡r♦s ♣❡r❢❡✐t♦s✱ ❝♦♥❣r✉ê♥❝✐❛ ❡ ♦ ♣❡q✉❡♥♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ ♠♦str❛♥❞♦ s✉❛ ❛❞❡✲ q✉❛çã♦ ❡ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ♥❛s ❛✉❧❛s ❞❡ ♠❛t❡♠át✐❝❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❖ ❈❛♣ít✉❧♦ ✷ é ❞❡❞✐❝❛❞♦ à ❞❡✜♥✐çã♦ ❞❛ ❝r✐♣t♦❣r❛✜❛✱ ❡ ❛✐♥❞❛ s❡ ♦❝✉♣❛ ❡♠ ❛♣r❡s❡♥t❛r ♦s r❡q✉✐s✐t♦s ❞❡ s❡❣✉r❛♥ç❛ ❣❛r❛♥t✐❞♦s ♣❡❧❛ ❝r✐♣t♦❣r❛✜❛ ❡ ♦s ♣r✐♥❝✐♣❛✐s t✐♣♦s ❞❡ ❝❤❛✈❡s ♣❛r❛ ❝✐❢r❛❣❡♠ ❡ ❞❡❝✐❢r❛❣❡♠ ❞❡ ❝ó❞✐❣♦s ❝r✐♣t♦❣r❛❢❛❞♦s✳
❆ ❞❡✜♥✐çã♦ ❡ ✉♠ ❜r❡✈❡ ❤✐stór✐❝♦ ❞❛ ❝r✐♣t♦❣r❛✜❛ sã♦ ♦ t❡♠❛ ❞♦ ❈❛♣ít✉❧♦ ✸✱ r❡ss❛❧✲ t❛♥❞♦ s✉❛ ❝♦♥tr✐❜✉✐çã♦ ♣❛r❛ ❛ ✈✐tór✐❛ ❞♦s ❆❧✐❛❞♦s ❝♦♥tr❛ ♦s ❛❧❡♠ã❡s ♥❛ ❙❡❣✉♥❞❛ ●r❛♥❞❡ ●✉❡rr❛✱ ❞❡st❛❝❛ ♦ ♠ét♦❞♦ ❞❡ ❝r✐♣t♦❣r❛✜❛ ❘❙❆ ❡ ❛♣r❡s❡♥t❛ ❡①❡♠♣❧♦s ❞❡ ❝♦❞✐✜❝❛çã♦ ❡ ❞❡❝♦❞✐✜❝❛çã♦ ❞❡ ♠❡♥s❛❣❡♥s ♣♦r ♠❡✐♦ ❞❡ss❡ ♠ét♦❞♦✳
❈❛♣ít✉❧♦ ✶
❈♦♥❤❡❝✐♠❡♥t♦ Pré✈✐♦ ♣❛r❛
❈r✐♣t♦❣r❛✜❛
✶✳✶ ◆ú♠❡r♦s ◆❛t✉r❛✐s
❋✐❣✉r❛ ✶✳✶✿ ❊✈♦❧✉çã♦ ❞♦s ◆ú♠❡r♦s
❆ ❝✐✈✐❧✐③❛çã♦ ❤✉♠❛♥❛ ❧❡♥t❛♠❡♥t❡ ❢♦✐ ❞♦♠✐♥❛♥❞♦ ♦ ♠♦❞❡❧♦ ❞❡ ❝♦♥t❛❣❡♠✱ ✭✉♠✱ ❞♦✐s✱ três✱✳✳✳✮ q✉❡ sã♦ ♦s ♥ú♠❡r♦s ♥❛t✉✲ r❛✐s✳ ❈♦♥❢♦r♠❡ ❛s ♥❡❝❡ss✐❞❛❞❡s ❞♦ ❤♦✲ ♠❡♠ ❢♦r❛♠ s❡ t♦r♥❛♥❞♦ ❝♦♠♣❧❡①❛s ✭❋✐✲ ❣✉r❛ ✶✳✶✮✱ ❢♦✐ s❡ ❛✉♠❡♥t❛♥❞♦ ❛s ♥❡❝❡ss✐✲ ❞❛❞❡s ❡ r❡✢❡①õ❡s s♦❜r❡ ❛ ❡✈♦❧✉çã♦ ❞♦ s✐s✲ t❡♠❛ ❞❡ ❝♦♥t❛❣❡♠✱ ♣♦st❡r✐♦r♠❡♥t❡ s♦❜r❡ ❛ ♠❛t❡♠át✐❝❛✳
◆❛ t❡♦r✐❛ ♠❛t❡♠át✐❝❛✱ t♦❞❛s ❛s ✈❡③❡s q✉❡ ❤á ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❞❡✜♥✐r ❛❧❣♦✱ ✐ss♦ é ❢❡✐t♦ ❝♦♠ ❜❛s❡ ❡♠ ❝♦♥❝❡✐t♦s ❛♥t❡r✐♦r❡s ♦✉ ❛ ♣❛rt✐r ❞❡ ❞❡✜♥✐çõ❡s ❛♥t❡r✐♦r❡s q✉❡ s✉s✲ t❡♥t❛♠✱ ❝♦♠♣r♦✈❛♠ ♦✉ ❝♦rr♦❜♦r❛♠ ❝♦♠ ♦ ♥♦✈♦ ❝♦♥❝❡✐t♦ q✉❡ s❡ q✉❡r ❞❡♠♦♥str❛r✳ ❍❛✈❡♥❞♦ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s✱ ♦✉ s❡❥❛✱ s❡♠ ✉♠❛ ❡①♣❧✐❝❛çã♦ ❞❡✜♥✐❞❛✱
✐♥tr♦❞✉③✲s❡ ❛❧❣✉♥s ❛①✐♦♠❛s✱ ♦✉ s❡❥❛✱ ❝❡rt❛s ♣r♦♣♦s✐çõ❡s q✉❡ s❡ t♦♠❛♠ ❝♦♠♦ ✈❡r❞❛❞❡✐r❛s ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ q✉❛❧q✉❡r ❞❡♠♦♥str❛çã♦✳
P♦❞❡✲s❡ ❞✐③❡r q✉❡ ❤♦❥❡ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ✭r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦ sí♠❜♦❧♦ N✮ s❡ ❝❛✲ r❛❝t❡r✐③❛♠ ❡ss❡♥❝✐❛❧♠❡♥t❡ ♣❡❧♦ s✐❣♥✐✜❝❛❞♦ ❞❛ ♣❛❧❛✈r❛ s✉❝❡ss♦r✱ ♦✉ s❡❥❛ q✉❡ s❡❝❡❞❡✱ q✉❡ ✈❡♠ ❞❡♣♦✐s✱ s❡❣✉♥❞♦ ♦ ❞✐❝✐♦♥ár✐♦ ❝♦♠t❡♣♦r❛♥ê♦ ❞❛ ❧í♥❣✉❛ ♣♦rt✉❣✉❡s❛✳ ❆✜r♠❛r♠♦s q✉❡ s❡n✱n′ ∈N✱ ❡ s❡n′ é s✉❝❡ss♦r ❞❡n✱ ♣♦❞❡♠♦s ❝♦♠ ❝❡rt❡③❛ ❛✜r♠❛r q✉❡ ♥ã♦ ❡①✐st❡ ♦✉tr♦
♥ú♠❡r♦ ♥❛t✉r❛❧ ❡♥tr❡n ❡n′✳ P♦r❡♠ ❡ss❛ ❝❛r❛❝t❡ríst✐❝❛ ♥ã♦ ❞❡✜♥❡ ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♦s
♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❢♦r❛♠ ❢♦r♠✉❧❛❞♦ ♣♦r P❡❛♥♦ ♦s s❡❣✉✐♥t❡s ❛①✐♦♠❛s✿
i) ❩❡r♦ ✭r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ sí♠❜♦❧♦ ✵✮ é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧
ii) ❚♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ t❡♠ ✉♠ ú♥✐❝♦ s✉❝❡ss♦r❀
iii) ◆ú♠❡r♦s ♥❛t✉r❛✐s ❞✐❢❡r❡♥t❡s tê♠ s✉❝❡ss♦r❡s ❞✐❢❡r❡♥t❡s❀
vi) ❊①✐st❡ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ ♥❛t✉r❛❧✱ ♦ ③❡r♦✱ q✉❡ ♥ã♦ é s✉❝❡ss♦r ❞❡ ♥❡♥❤✉♠ ♦✉tr♦ ♥ú♠❡r♦❀
v) ❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s ✭✐st♦ é✱ X ⊂ N✮✳ ❙❡ 0 ∈X ❡ s❡✱ ❛❧é♠ ❞✐ss♦✱ ♦ s✉❝❡ss♦r ❞❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ X ❛✐♥❞❛ ♣❡rt❡♥❝❡ ❛ X✱ ❡♥tã♦ X=N✳
P❛r❛ ❡s❝❧❛r❡❝✐♠❡♥t♦s ♠❛✐♦r❡s ❝♦♥s✉❧t❡ ❛s r❡❢❡rê♥❝✐❛s ❬✼❪
✶✳✶✳✶ ❉❡st❛q✉❡ ♣❛r❛ ♦ ❆①✐♦♠❛ ❞❛ ■♥❞✉çã♦
❖ ú❧t✐♠♦ ❞♦s ❛①✐♦♠❛s ❞❡ P❡❛♥♦✱ ❆①✐♦♠❛ ❞❛ ■♥❞✉çã♦✱ é ❛ ❜❛s❡ ❡✜❝✐❡♥t❡ ❞❡ ❞❡✲ ♠♦♥str❛çã♦ ❞❡ ♣r♦♣♦s✐çõ❡s r❡❢❡r❡♥t❡s ❛ ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ q✉❡ ❡st❛❜❡❧❡❝❡ ♦ s❡❣✉✐♥t❡ ♣r✐♥❝í♣✐♦❀ Pr✐♥❝í♣✐♦ ❞❛ ✐♥❞✉çã♦✿ s❡❥❛ Pn ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ r❡❧❛t✐✈❛ ❛♦ ♥ú♠❡r♦ ♥❛t✉r❛❧
n✳ ❙✉♣♦♥❤❛ q✉❡❀
i) P1 é ✈á❧✐❞❛❀
ii) P❛r❛ t♦❞♦ n ∈ N✱ ❛ ✈❛❧✐❞❡③ ❞❡ Pn ✐♠♣❧✐❝❛ ❛ ✈❛❧✐❞❡③ ❞❡ Pn′✱ ♦♥❞❡n′ é ♦ s✉❝❡ss♦r ❞❡ n❀
❊♥tã♦Pn é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ q✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ n✳
❖ ❛①✐♦♠❛ ❞❛ ✐♥❞✉çã♦ é ✉♠❛ ❢♦r♠❛ ❞❡ ❞✐③❡r q✉❡ q✉❛❧q✉❡r ♥ú♠❡r♦ ♥❛t✉r❛❧n♣♦❞❡ s❡r ❛❧❝❛♥ç❛❞♦ s❡ ♣❛rt✐r♠♦s ❞❡ ✶ ❡ s❡ ❝♦♥t✐♥✉❛r♠♦s ❛ r❡♣❡t✐r ❛ ♦♣❡r❛çã♦ t♦♠❛r ♦ s✉❝❡ss♦r ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✱ s❡❣✉♥❞♦ ❍②❣✐♥♦ ❍✳ ❉♦♠✐♥❣✉❡s✿
❙❡❥❛L={x∈N/x≥a ❡ Pa é ❢❛❧s❛⑥✳ ❇❛st❛ ♣r♦✈❛r ❡♥tã♦ q✉❡L=∅✳ ❙✉♣♦♥❤❛♠♦s q✉❡L6=∅❡ s❡❥❛m ♦ ♠❡♥♦r ❡❧❡♠❡♥t♦ ❞❡L✳ ▲♦❣♦Pm é ❢❛❧s❛ ❡ ❝♦♠♦✱ ♣♦r ❤✐♣ót❡s❡✱Pa é
✈❡r❞❛❞❡✐r❛✱ ❡♥tã♦ m > a✳ ❉❡st❛ ú❧t✐♠❛ r❡❧❛çã♦ s❡❣✉❡ q✉❡ m >0❀ ♣♦rt❛♥t♦ m = 1 +u✱ ♣❛r❛ ❛❧❣✉♠u∈N✱ ❡ ❞❛í u < m✳
▼❛sm > a✐♠♣❧✐❝❛ q✉❡m ≥a+ 1✳ ❆ss✐♠m= 1 +u≥a+ 1✱ ❞♦ q✉❡ r❡s✉❧t❛u≥a✳
❊♠ r❡s✉♠♦✿ m > u ≥ a✳ ▼❛s ✐st♦ ♦❜r✐❣❛ P(u) ❛ s❡r ✈❡r❞❛❞❡✐r❛ ✭s❡ ❢♦ss❡ ❢❛❧s❛✱ u
❡st❛r✐❛ ❡♠L✱ ♦ q✉❡ ♥ã♦ é ♣♦ssí✈❡❧ ♣♦✐s u < m=♠✐♥ L✮✳ ❊♥tã♦✱ ❞❡✈✐❞♦ ❛ Pm+1❂Pm é
✈❡r❞❛❞❡✐r❛✳ ❆❜s✉r❞♦✳✭❋✉♥❞❛♠❡♥t♦s ❞❡ ❆r✐t♠ét✐❝❛ ♣❣ ✷✸✮
❊①❡♠♣❧♦ ✶✳✶✳✶✳ ❯s❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ ✐♥❞✉çã♦✱ ♠♦str❡♠♦s ❛ ✐❣✉❛❧❞❛❞❡ 32n+ 7 = 8k✱
♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ n❀ P❛r❛ n = 1 t❡♠♦s q✉❡❀ 32n+ 7 = 32.1+ 7
32n+ 7 = 32+ 7
32n+ 7 = 9 + 7
32n+ 7 = 16
32n+ 7 = 8.k
1✳ ❈♦♠ k1 = 2 ❧♦❣♦ ❛ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❡✐r❛✳
❙✉♣♦♥❤❛♠♦s q✉❡ ♣❛r❛ ❛❧❣✉♠ p >1✱ p∈N ❛ ❛✜r♠❛çã♦ s❡❥❛ ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛❀ 32p+ 7 = 8k
p
◗✉❡r❡♠♦s ♣r♦✈❛r ❛ ✈❛❧✐❞❡③ ♣❛r❛ n=p+ 1 ❝♦♠♦ 32p+ 7 = 8k
p ♠✉❧t✐♣❧✐❝❛♠♦s ♦s ❞♦✐s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ♣♦r ✾ ♦❜t❡♥❞♦
9.(32p+ 7) = 9.8k p
9.32p+ 63 = 72k
p s✉❜tr❛✐♥❞♦s ✺✻ ❡♠ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡
9.32p+ 63−56 = 72k
p −56 ❢❛t♦r❛♥❞♦s 9 = 32✱ ♥♦ ♣r✐♠❡✐r♦ ❧❛❞♦ ❞❛ ✐❣✉❛❧❞❛❞❡ t❡♠♦s
32.32p + 7 = 72k
p−56 ❢❛t♦r❛♠♦s 72kp−56 = 8(9kp−7) ❛ss✐♠
32p+2+ 7 = 8(9k
p−7) ❢❛③❡♥❞♦ 9kp−7 = kq✱ ♦✉ s❡❥❛
32(p+1)+ 7 = 8k
q ❝q❞✳
▲♦❣♦✱ ❛ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ n∈N✳
P❛r❛ ♠❛✐♦r❡s ✐♥❢♦r♠❛çõ❡s ❝♦♥s✉❧t❡ ❛ r❡❢❡rê♥❝✐❛ ❬✸❪✳
✶✳✷ ◆ú♠❡r♦s Pr✐♠♦s
❙❡ a ❡ b sã♦ ♥❛t✉r❛✐s✱ ❞✐③❡♠♦s q✉❡ a ❞✐✈✐❞❡ b✱♣♦❞❡♥❞♦ ❡s❝r❡✈❡r a | b✱ s❡ ❡①✐st✐r ✉♠ ♥❛t✉r❛❧k1 t❛❧ q✉❡b =k1.a✱ ♣♦❞❡♥❞♦ ❛✜r♠❛r q✉❡a é ❞✐✈✐s♦r ❞❡b ❡ q✉❡ b é ♠✉❧t✐♣❧♦ ❞❡
a✳
❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❯♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ n(n >1) ♣♦ss✉✐♥❞♦ s♦♠❡♥t❡ ❞♦✐s ❞✐✈✐s♦r❡s ♣♦s✐✲ t✐✈♦s n ❡ ✶ é ❝❤❛♠❛❞♦ ♣r✐♠♦✳
❙❡❣✉♥❞♦ ❡ss❛ ❞❡✜♥✐çã♦ ♦ ♥ú♠❡r♦ ✶ ♥ã♦ é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✱ ♣♦✐s ♦ ♠❡s♠♦ ♥ã♦ ❛♣r❡s❡♥t❛ ❞♦✐s ❞✐✈✐s♦r❡s ❞✐st✐♥t♦s✳
❖ ♥ú♠❡r♦ ✷ é ♦ ú♥✐❝♦ ♥ú♠❡r♦ ♣r✐♠♦ ♣❛r✱ ❥á q✉❡ t♦❞♦s ♦s ❞❡♠❛✐s ♥ú♠❡r♦s ♣❛r❡s ♣♦ss✉❡♠ ❛♦ ♠❡♥♦s ✸ ❞✐✈✐s♦r❡s✱ ❞❡♥tr❡ ❡❧❡s ♦ ♥ú♠❡r♦ ✶✱ ♦ ♣ró♣r✐♦ ♥ú♠❡r♦ ❡ ♦ ♥ú♠❡r♦ ✷✳ ◆ú♠❡r♦s ♥❛t✉r❛✐s ♥ã♦ ♥✉❧♦s q✉❡ ♣♦ss✉❡♠ ♠❛✐s ❞❡ ❞♦✐s ❞✐✈✐s♦r❡s sã♦ ❝❤❛♠❛❞♦s ❞❡ ♥ú♠❡r♦s ❝♦♠♣♦st♦s✳
Pr♦♣♦s✐çã♦ ✶✳ ❙❡ p|a.b✱ s❡♥❞♦ p ♣r✐♠♦✱ ❡♥tã♦ p|a ♦✉ p|b✳
❉❡♠♦♥str❛çã♦✳ ❙❡p|b✱ ♣♦r ❞❡✜♥✐çã♦ b=k1.p✱ s❡♥❞♦k1 ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✱ s✉♣♦♥❞♦
q✉❡p6 |a✭a ♥ã♦ é ❞✐✈✐ssí✈❡❧ ♣♦rp✮✱ ❧♦❣♦a=k2.p+d ♦♥❞❡ k2 ❡d t❛♠❜é♠ sã♦ ♥ú♠❡r♦s
♥❛t✉r❛✐s✱ ♣♦❞❡♠♦s ♠✉❧t✐♣❧✐❝❛r ❛♠❜♦s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ♣♦rb✜❝❛♥❞♦a.b=k2.p.b+d.b✱
❧❡♠❜r❛♥❞♦ q✉❡b =k1.p ❡ ❢❛③❡♥❞♦ ❛ s✉❜st✐t✉✐çã♦ ❛♣❡♥❛s ♥♦ s❡❣✉♥❞♦ ❧❛❞♦ ❞❛ ✐❣✉❛❧❞❛❞❡
a.b = k2.p.k1.p+d.k1.p✱ ❝♦❧♦❝❛♥❞♦ p ❡♠ ❡✈✐❞ê♥❝✐❛ ♥♦ s❡❣✉♥❞♦ ❧❛❞♦ ❞❛ ✐❣✉❛❧❞❛❞❡
a.b= (k2.p.k1+d.k1).p✱ ❡ ❝❤❛♠❛♥❞♦k2.p.k1+d.k1 =k3✱ ❧♦❣♦a.b=k3.p✱ ♦ q✉❡ ✐♠♣❧✐❝❛
p|a.b✳
✶✳✷✳✶ ◗✉❛♥t♦s ◆ú♠❡r♦s Pr✐♠♦s ❊①✐st❡♠❄
◆♦ ❧✐✈r♦ ■❳✱ ❇❖❨❊❘ ❬♣ ✼✾❪ ♥♦s ❞✐③ q✉❡✿
✏❖ ▲✐✈r♦ ■❳✱ ♦ ú❧t✐♠♦ ❞♦s três s♦❜r❡ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ❝♦♥tê♠ ✈ár✐♦s r❡s✉❧t❛❞♦s ✐♥t❡r❡ss❛♥t❡s✳ ❉❡ss❡s✱ ♦ ♠❛✐s ❝é❧❡❜r❡ é ❛ Pr♦♣♦s✐✲ çã♦✿ ♥ú♠❡r♦s ♣r✐♠♦s sã♦ ♠❛✐s ❞♦ q✉❡ q✉❛❧q✉❡r q✉❛♥t✐❞❛❞❡ ✜①❛❞❛ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✳ ■st♦ é✱ ❊✉❝❧✐❞❡s ❞á ❛q✉✐ ❛ ♣r♦✈❛ ❡❧❡♠❡♥t❛r ❜❡♠ ❝♦♥❤❡❝✐❞❛ ❞♦ ❢❛t♦ ❞❡ q✉❡ ❤á ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐♠♦s✳ ❆ ♣r♦✈❛ é ✐♥❞✐r❡t❛✱ ♣♦✐s ♠♦str❛✲s❡ q✉❡ ❛ ❤✐♣ót❡s❡ ❞❡ ❤❛✈❡r s♦♠❡♥t❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣r✐♠♦s ❧❡✈❛ ❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦✑✳
❙❡❣✉♥❞♦ ♦ t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧
❚❡♦r❡♠❛ ✶✳✷✳✷✳ ✏❊①✐st❡♠ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✑✳
❖ q✉❡ é ♠✉✐t♦ ❜❡♠ ❞❡♠♦♥str❛❞♦ ♣♦r ❊✉❝❧✐❞❡s✿
❉❡♠♦♥str❛çã♦✳ ✏❙✉♣♦♥❤❛♠♦s q✉❡ ❛ s✉❝❡ssã♦p1❂✷✱p2❂✸✱ ✳✳✳✳✱pr✱ ❞♦sr♥ú♠❡r♦s ♣r✐♠♦s
s❡❥❛ ✜♥✐t❛✳ ❋❛ç❛♠♦sP =p1✳p2✳✳✳✳✳pr+ 1✱ ❡ s❡❥❛ p✉♠ ♥ú♠❡r♦ ♣r✐♠♦ q✉❡ ❞✐✈✐❞❡ P✳ ❊ss❡
♥ú♠❡r♦ p ♥ã♦ ♣♦❞❡ s❡r ✐❣✉❛❧ ❛ q✉❛❧q✉❡r ✉♠ ❞♦s ♥ú♠❡r♦sp1✱p2✱✳✳✳✱pr✱ ♣♦rq✉❡ ❡♥tã♦ ❡❧❡
❞✐✈✐❞✐r✐❛ ❛ ❞✐❢❡r❡♥ç❛ P −p1✳p2✳✳✳✳✳pr❂✶✱ ♦ q✉❡ é ✐♠♣♦ssí✈❡❧✳ ❆ss✐♠ p♥ã♦ é ✉♠ ♥ú♠❡r♦
♣r✐♠♦ q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ ❛ s✉❝❡ssã♦ p1✱ p2✱ ✳✳✳✱ pr✱ ❡ ♣♦r ❝♦♥s❡q✉ê♥❝✐❛ ♣♦❞❡♥❞♦ ❛✜r♠❛r
q✉❡r ♥ã♦ r❡♣r❡s❡♥t❛ t♦❞♦s ♦s ♥ú♠❡r♦s ♣r✐♠♦s✳✑ ✭◆ú♠❡r♦s Pr✐♠♦s ❱❡❧❤♦s ▼✐stér✐♦s ❡ ◆♦✈♦s ❘❡❝♦r❞❡s✱ ♣❣ ✶✮
❊①❡♠♣❧♦ ✶✳✷✳✸✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ p1 = 2✱ p2 = 3✱ p3 = 5✱ p4 = 7 ❡ p5 = 11
❡ ❝♦♠♦ P =p1✳p2✳✳✳✳✳pr+ 1 ❧♦❣♦✱
P = 2.3.5.7.11 + 1 P = 2310 + 1 P = 2311
♣♦rt❛♥t♦✱ P ♣♦ss✉✐ ✉♠ ♣r✐♠♦ p q✉❡ ♥♦ ❝❛s♦ p = 2311✱ ❞✐❢❡r❡♥t❡ ❞♦s ❛♣r❡s❡♥t❛❞♦ ♥❛ s❡q✉ê♥❝✐❛ p1.p2.p3.p4.p5
●♦❧❞❜❛❝❤ t❛♠❜é♠ ❢❡③ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❜❛st❛♥t❡ s✐♠♣❧❡s❀
❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✷✳✷ s❡❣✉♥❞♦ ●♦❧❞❜❛❝❤✳ ✏❇❛st❛ ❛❝❤❛r ✉♠❛ s✉❝❡ssã♦ ✐♥✜✲ ♥✐t❛ a1❁ a2❁ a3❁ ✳✳✳❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ♣r✐♠♦s ❡♥tr❡ s✐✱ ♦✉ s❡❥❛ s❡♠ ❢❛t♦r ❝♦♠✉♠
❡♥tr❡ s✐✳ ❙❡ p1 é ❢❛t♦r ♣r✐♠♦ ❞❡ a1✱ p2 ✉♠ ❢❛t♦r ♣r✐♠♦ ❞❡ a2✱ ✳✳✳✱ pn ❢❛t♦r ♣r✐♠♦ ❞❡ an✱
✳✳✳✱ ❡♥tã♦p1✱ p2✱p3✱ ✳✳✳pn✱ ✳✳✳ sã♦ ❞✐st✐♥t♦s✳✑ ✭◆ú♠❡r♦s Pr✐♠♦s ❱❡❧❤♦s ▼✐stér✐♦s ❡ ◆♦✈♦s
❘❡❝♦r❞❡s✱ ♣❣ ✹✮
✶✳✷✳✷
❈♦♠♦ ❘❡❝♦♥❤❡❝❡r ✉♠ ◆ú♠❡r♦ Pr✐♠♦
◆♦ ❛rt✐❣♦ ✸✷✾ ❞❛s ❉✐sq✉✐s✐s✐t✐♦♥❡s ❆r✐t❤♠❡t✐❝❛❡✱ ●❛✉ss ❡s❝r❡✈❡✉✿
✏❖ ♣r♦❜❧❡♠❛ ❞❡ ❞✐st✐♥❣✉✐r ♦s ♥ú♠❡r♦s ♣r✐♠♦s ❞♦s ♥ú♠❡r♦s ❝♦♠✲ ♣♦st♦s ❡ ❞❡ ❡①♣r✐♠✐r ❡st❡s ú❧t✐♠♦s à ❝✉st❛ ❞❡ s❡✉s ❢❛t♦r❡s ♣r✐♠♦s ❞❡✈❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠ ❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❡ ❞♦s ♠❛✐s út❡✐s ❡♠ ❛r✐t♠ét✐❝❛✳ ❆ ♣ró♣r✐❛ ❞✐❣♥✐❞❛❞❡ ❞❛ ❝✐ê♥❝✐❛ r❡q✉❡r q✉❡ t♦❞♦s ♦s ♠❡✐♦s ♣♦ssí✈❡✐s s❡❥❛♠ ❡①♣❧♦r❛❞♦s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ tã♦ ❡❧❡❣❛♥t❡ ❡ tã♦ ❢❛♠♦s♦✑✳✭◆ú♠❡r♦s Pr✐♠♦s ❱❡❧❤♦s ▼✐stér✐♦s ❡ ◆♦✈♦s ❘❡❝♦r❞❡s✱ ♣❣ ✶✸✮
❖ ♣r✐♠❡✐r♦ ♣❡♥s❛♠❡♥t♦ ♣❛r❛ ✈❡r✐✜❝❛r s❡ ✉♠ ♥ú♠❡r♦né ♣r✐♠♦ é ♦ ❞❛ ❢❛t♦r❛çã♦✱ ♠❛s ❡①✐st❡ ♦✉tr♦ ♣r♦❝❡❞✐♠❡♥t♦ ❛♣❧✐❝á✈❡❧ ❛ t♦❞♦ ♥❛t✉r❛❧ N✱ q✉❡ ✐♥❞✐❝❛rá✱ ❝♦♠ ✉♠ ♥ú♠❡r♦
✜♥✐t♦ ❞❡ ❝á❧❝✉❧♦✱ s❡ n é ♣r✐♠♦ ♦✉ ❝♦♠♣♦st♦✳ ❙❡♥❞♦ ♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ N✱ ❜❛st❛ t❡♥t❛r ❞✐✈✐❞í✲❧♦ s✉❝❡ss✐✈❛♠❡♥t❡ ♣♦r n = 2,3,5,7, ...., k✱ s❡♥❞♦ k ♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ✐❣✉❛❧ ♦✉ ✐♥❢❡r✐♦r ❛n✳
❖ ❈r✐✈♦ ❞❡ ❊r❛tóst❡♥❡s
❈♦♠♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ é ♦♣❡r❛çã♦ ♠❛✐s ❢á❝✐❧ ❞❡ ❡①❡❝✉t❛r ❞♦ q✉❡ ❛ ❞✐✈✐sã♦✱ ❊r❛tós✲ t❡♥❡s ♥♦ sé❝✉❧♦ ■■■ ❆✳❈✳✱ t❡✈❡ ❛ ✐❞❡✐❛ ❞❡ ♦r❣❛♥✐③❛r ♦s ❝á❧❝✉❧♦s s♦❜ ❛ ❢♦r♠❛ ❞❡ ❝r✐✈♦✱ q✉❡ ❧❡✈❛ s❡✉ ♥♦♠❡✳ ❖ ❝r✐✈♦ s❡r✈❡ ♣❛r❛ ❛ ❞❡t❡r♠✐♥❛r t♦❞♦s ♦s ♥ú♠❡r♦s ♣r✐♠♦s ❡ t❛♠❜é♠ ♦s ❢❛t♦r❡s ❞♦s ♥ú♠❡r♦s ❝♦♠♣♦st♦s ❛ ✉♠ ♥ú♠❡r♦N ❞❛❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡✳
❱❛♠♦s ✐❧✉str❛r ♦ ♣r♦❝❡ss♦ t♦♠❛♥❞♦ ❝♦♠♦ ❡①❡♠♣❧♦N = 131 ❖ ♣r♦❝❡ss♦ é r❡❛❧✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
i) ❡s❝r❡✈❛ t♦❞♦s ♦s ♥ú♠❡r♦s ❛té ✶✸✶❀
ii) r✐s❝❛✲s❡ t♦❞♦s ♦s ♠ú❧t✐♣❧♦s ❞❡ ✷✱ s✉♣❡r✐♦r ❛ ✷
iii) ❡♠ ❝❛❞❛ ♥♦✈❛ ❡t❛♣❛✱ sã♦ r✐s❝❛❞♦s t♦❞♦s ♦s ♠ú❧t✐♣❧♦s ❞♦ ♠❡♥♦r ✐♥t❡✐r♦ p✱ ❛✐♥❞❛ ♥ã♦ r✐s❝❛❞♦s ❡ q✉❡ sã♦ ♠❛✐♦r❡s q✉❡ p✳
iv) ❜❛st❛ ❝❤❡❣❛r ❛♦ ♥ú♠❡r♦ pt❛❧ q✉❡ p2 ❥á ✉❧tr❛♣❛ss❛ ✶✸✶
✷
✸
✴✹
✺
✴✻
✼
✴✽
✴✾
✴✴✴
✶✵
✶✶ ✴✴✴
✶✷
✶✸ ✴✴✴
✶✹ ✴✴✴
✶✺ ✴✴✴
✶✻
✶✼ ✴✴✴
✶✽
✶✾ ✴✴✴
✷✵
✴✴✴
✷✶ ✴✴✴
✷✷
✷✸ ✴✴✴
✷✹ ✴✴✴
✷✺ ✴✴✴
✷✻ ✴✴✴
✷✼ ✴✴✴
✷✽
✷✾ ✴✴✴
✸✵
✸✶ ✴✴✴
✸✷ ✴✴✴
✸✸ ✴✴✴
✸✹ ✴✴✴
✸✺ ✴✴✴
✸✻
✸✼ ✴✴✴
✸✽ ✴✴✴
✸✾ ✴✴✴
✹✵
✹✶ ✴✴✴
✹✷
✹✸ ✴✴✴
✹✹ ✴✴✴
✹✺ ✴✴✴
✹✻
✹✼ ✴✴✴
✹✽ ✴✴✴
✹✾ ✴✴✴
✺✵
✴✴✴
✺✶ ✴✴✴
✺✷
✺✸ ✴✴✴
✺✹ ✴✴✴
✺✺ ✴✴✴
✺✻ ✴✴✴
✺✼ ✴✴✴
✺✽
✺✾ ✴✴✴
✻✵
✻✶ ✴✴✴
✻✷ ✴✴✴
✻✸ ✴✴✴
✻✹ ✴✴✴
✻✺ ✴✴✴
✻✻
✻✼ ✴✴✴
✻✽ ✴✴✴
✻✾ ✴✴✴
✼✵
✼✶ ✴✴✴
✼✷
✼✸ ✴✴✴
✼✹ ✴✴✴
✼✺ ✴✴✴
✼✻ ✴✴✴
✼✼ ✴✴✴
✼✽
✼✾ ✴✴✴
✽✵
✴✴✴
✽✶ ✴✴✴
✽✷
✽✸ ✴✴✴
✽✹ ✴✴✴
✽✺ ✴✴✴
✽✻ ✴✴✴
✽✼ ✴✴✴
✽✽
✽✾ ✴✴✴
✾✵
✴✴✴
✾✶ ✴✴✴
✾✷ ✴✴✴
✾✸ ✴✴✴
✾✹ ✴✴✴
✾✺ ✴✴✴
✾✻
✾✼ ✴✴✴
✾✽ ✴✴✴
✾✾ ✴✴✴✴✴
✶✵✵
✶✵✶ ✴✴✴✴✴
✶✵✷ ✶✵✸ ✴✴✴✴✴
✶✵✹ ✴✴✴✴✴
✶✵✺ ✴✴✴✴✴
✶✵✻ ✶✵✼ ✴✴✴✴✴
✶✵✽ ✶✵✾ ✴✴✴✴✴
✶✶✵
✴✴✴✴✴
✶✶✶ ✴✴✴✴✴
✶✶✷ ✶✶✸ ✴✴✴✴✴
✶✶✹ ✴✴✴✴✴
✶✶✺ ✴✴✴✴✴
✶✶✻ ✴✴✴✴✴
✶✶✼ ✴✴✴✴✴
✶✶✽ ✴✴✴✴✴
✶✶✾ ✴✴✴✴✴
✶✷✵
✴✴✴✴✴
✶✷✶ ✴✴✴✴✴
✶✷✷ ✴✴✴✴✴
✶✷✸ ✴✴✴✴✴
✶✷✹ ✴✴✴✴✴
✶✷✺ ✴✴✴✴✴
✶✷✻ ✶✷✼ ✴✴✴✴✴
✶✷✽ ✴✴✴✴✴
✶✷✾ ✴✴✴✴✴
✶✸✵
✶✸✶
❆ss✐♠ t♦❞♦s ♦s ♠ú❧t✐♣❧♦s ❞❡ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸ ❁√131sã♦ ❡①♣✉r❣❛❞♦s✳ ❖ ♥ú♠❡r♦ ✺✸✱ ♣♦r ❡①❡♠♣❧♦✱ é ♣r✐♠♦ ♣♦rq✉❡ ♥ã♦ ❢♦✐ ❡①♣✉r❣❛❞♦✳ ❊♥tã♦✱ ♦s ♥ú♠❡r♦s ♣r✐♠♦s ✐♥❢❡r✐♦r❡s ♦✉ ✐❣✉❛✐s ❛ ✶✸✶ sã♦ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ✶✼✱ ✶✾✱ ✷✸✱ ✷✾✱ ✸✶✱ ✸✼✱ ✹✶✱ ✹✸✱ ✹✼✱ ✺✾✱ ✻✶✱ ✻✼✱ ✼✶✱ ✼✸✱ ✼✾✱ ✽✸✱ ✽✾✱ ✾✼✱ ✶✵✶✱ ✶✵✸✱ ✶✵✼✱ ✶✵✾✱ ✶✶✸✱ ✶✷✼✱ ✶✸✶✳
P❛r❛ ♠❛✐♦r❡s ✐♥❢♦r♠❛çã♦ ✈❡❥❛ ❛s r❡❢❡rê♥❝✐❛s ❬✶✵❪✱ ❬✶✷❪ ❡ ❬✶✹❪
