❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ tr✐â♥❣✉❧♦ ▼✐♥❡✐r♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✱ ◆❛t✉r❛✐s ❡ ❊❞✉çã♦
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
❯♠ ❡st✉❞♦ s♦❜r❡ ♦ ✐♥✜♥✐t♦✿ ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡
❡ ❞❡♥s✐❞❛❞❡ ❞♦s ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s
❲②s♥❡r ▼❛① ❞❡ ▲✐♠❛ ❙✐❧✈❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r✲ ❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡
❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ▲❡♦♥❛r❞♦ ❞❡ ❆♠♦r✐♠ ❡ ❙✐❧✈❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❚r✐â♥❣✉❧♦ ▼✐♥❡✐r♦
❯▼ ❊❙❚❯❉❖ ❙❖❇❘❊ ❖ ■◆❋■◆■❚❖✿ ❊◆❯▼❊❘❆❇■▲■❉❆❉❊ ❊ ❉❊◆❙■❉❆❉❊ ❉❖❙ ❈❖◆❏❯◆❚❖❙ ◆❯▼➱❘■❈❖❙
❲❨❙◆❊❘ ▼❆❳ ❉❊ ▲■▼❆ ❙■▲❱❆
❉✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ à ❝♦♠✐ssã♦ ❆❝❛❞ê♠✐❝❛ ■♥st✐t✉❝✐♦♥❛❧ ❞♦ P❘❖❋▼❆❚✲❯❋❚▼ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❖r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ▲❡♦♥❛r❞♦ ❞❡ ❆♠♦r✐♠ ❡ ❙✐❧✈❛
❯❜❡r❛❜❛ ✲ ▼✐♥❛s ●❡r❛✐s ❏✉♥❤♦ ❞❡ ✷✵✶✻
C a t a l o g a ç ã o n a f o n t e : B i b l i o t e c a d a U n i v e r s i d a d e F e d e r a l d o T r i â n g u l o M i n e i r o
Silva, Wysner Max de Lima
S58e Um estudo sobre o infinito: densidade e enumerabilidade dos conjuntos numéricos / Wysner Max de Lima Silva. -- 2016. 73 f. : il., fig., graf.
Dissertação (Mestrado Profissional em Matemática em Rede Nacional) -- Universidade Federal do Triângulo Mineiro, Uberaba, MG, 2016
Orientador: Prof. Dr. Leonardo de Amorim e Silva
1. Matemática - Estudo e ensino. 2. Teoria dos conjuntos. 3. Números cardinais. 4. Números primos. 5. Funções (Matemática). 6. Geometria. I. Silva, Leonardo de Amorim e. II. Universidade Fe- deral do Triângulo Mineiro. III. Título.
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦ ✜♥❛❧✐③❛r ❡st❡ tr❛❜❛❧❤♦✱ ❞❡✐①♦ ❛q✉✐ ♠❡✉s ♠❛✐s ♣r♦❢✉♥❞♦s ❛❣r❛❞❡❝✐♠❡♥t♦s✿
• ➚ ♠✐♥❤❛ ♠ã❡✱ q✉❡ ♠❡ ❡♥s✐♥♦✉ ✈❛❧♦r❡s✱ q✉❡ s❡♠♣r❡ ❜❛t❛❧❤♦✉ ♣❛r❛ q✉❡ ❡✉ ❡st✉❞❛ss❡ ♥❛s ♠❡❧❤♦r❡s ❡s❝♦❧❛s ❡ ❥❛♠❛✐s ❤❡s✐t♦✉ ❡♠ ❝✉st❡❛r ♠❡✉s ❡st✉❞♦s✳
• ❆♦ ♠❡✉ ♣❛✐✱ q✉❡ tr❛❜❛❧❤♦✉ ❞✉r♦ ❛ ✈✐❞❛ t♦❞❛✱ ✐♥❝❡♥t✐✈❛♥❞♦ ❡ ♠♦str❛♥❞♦ q✉❡ ♦ tr❛❜❛❧❤♦ ❞✐❣♥✐✜❝❛ ♦ ❤♦♠❡♠✳
• ❆ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s q✉❡ s♦♠❛r❛♠ ❢❡❧✐❝✐❞❛❞❡ ❝♦♠✐❣♦ ♣♦r s❛❜❡r ❞❛ ❝♦♥❝❧✉sã♦ ❞❡st❛ ❡t❛♣❛ ❡♠ ♠✐♥❤❛ ✈✐❞❛✳
• ❛♦s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ ❝❧❛ss❡ q✉❡✱ ❞❡s❞❡ ❡♥tã♦✱ ❢❛③❡♠ ♣❛rt❡ ❞❡ ♠✐♥❤❛ ❤✐stór✐❛✳ • ❆♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❞❛ ❡❞✉❝❛çã♦ ✐♥❢❛♥t✐❧✱ ❡♥s✐♥♦s ❢✉♥❞❛♠❡♥t❛❧ ❡ ♠é❞✐♦ ❛♦ ❧♦♥❣♦
❞❡ ♠✐♥❤❛ ✈✐❞❛ ❡st✉❞❛♥t✐❧ ❜ás✐❝❛✳ ❙❡♠ ✈♦❝ês✱ ❡✉ ♥ã♦ t❡r✐❛ ❛ ❝♦♠♣❡tê♥❝✐❛ ♣❛r❛ t❡r♠✐♥❛r ❡st❡ tr❛❜❛❧❤♦✳
• ❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❋❆▼❆❚ ✲ ❯❋❯✱ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ❞❡ ♠❛t❡♠át✐❝❛ ❡ ♣♦r ♠❡ ❝♦♥❞✉③✐r❡♠ ❛♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡st❛ tã♦ ♠❛r❛✈✐❧❤♦s❛ ❡ ✐♠♣♦rt❛♥t❡ ❝✐ê♥❝✐❛ q✉❡ é ❛ ♠❛t❡♠át✐❝❛✳
• ❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ❉r✳ ▲❡♦♥❛r❞♦ ❆♠♦r✐♠✱ ♣❡❧❛ ❛t❡♥çã♦✱ ♦r✐❡♥t❛çã♦ ❡ ❡♥s✐♥❛♠❡♥t♦s✳
• ❆♦s ♣r♦❢❡ss♦r❡s ❱❛♥❡ss❛ ❡ ❊❞s♦♥ q✉❡ ❛❝❡✐t❛r❛♠ ♦ ❝♦♥✈✐t❡ ♣❛r❛ ❝♦♠♣♦r❡♠ ❛ ❜❛♥❝❛ ❞❡ ♠✐♥❤❛ ❞❡❢❡s❛✳
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ t❡♠ ♣♦r ✜♥❛❧✐❞❛❞❡ ❡①♣ôr ❛ ❤✐stór✐❛ ❡ ♦ s✉r❣✐♠❡♥t♦ ❞♦s ❝♦♥❥✉♥t♦s ♥✉♠é✲ r✐❝♦s✱ ❜❡♠ ❝♦♠♦ ❛♥❛❧✐s❛r ❝❛❞❛ ✉♠ ❞❡❧❡s ♠❡❞✐❛♥t❡ ♦ ❡st✉❞♦ ❞♦ ✐♥✜♥✐t♦✳ ❚❛❧ ❝♦♥❝❡✐t♦ é ✉♠❛ ❜❛rr❡✐r❛ ♣❛r❛ ♠✉✐t♦s ♣♦r s❡ tr❛t❛r ❞❡ ✉♠ ❛ss✉♥t♦ ❛❜str❛t♦ ❡ ❝♦♠♣❧❡①♦✱ ♣♦ré♠✱ s❡ ❝♦♠♣r❡❡♥❞✐❞♦✱ ♣♦❞❡✲s❡ ♣❡r❝❡❜❡r ♦ q✉ã♦ ✐♠♣♦rt❛♥t❡ é ❡st❛ ❢❡rr❛♠❡♥t❛ ❞❡♥tr♦ ❞❛ ♠❛t❡✲ ♠át✐❝❛✳ P❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❢♦✐ ♥❡❝❡ssár✐❛ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ❝♦♥❝❡✐t♦s ✐♠✲ ♣♦rt❛♥t❡s ❞❛ ♠❛t❡♠át✐❝❛ t❛✐s ❝♦♠♦ ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡✱ ❝❛r❞✐♥❛❧✐❞❛❞❡✱ ♥ú♠❡r♦s ♣r✐♠♦s✱ ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s✱ ❢✉♥çõ❡s✱ ❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥✱ ❣❡♦♠❡tr✐❛✱ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛ ❡ ❛✐♥❞❛ ♦ ❡st✉❞♦ s♦❜r❡ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r ❡ ❝❛r❞✐♥❛✐s tr❛♥s✜♥✐t♦s✳
❆❜str❛❝t
❚❤✐s st✉❞② ❛✐♠s t♦ ❡①♣♦s❡ t❤❡ ❤✐st♦r② ❛♥❞ t❤❡ ❡♠❡r❣❡♥❝❡ ♦❢ ♥✉♠❡r✐❝❛❧ s❡ts✱ ❛♥❞ t♦ ❡①❛♠✐♥❡ ❡❛❝❤ ♦❢ t❤❡♠ t❤r♦✉❣❤ t❤❡ st✉❞② ♦❢ ✐♥✜♥✐t②✳ ❚❤✐s ❝♦♥❝❡♣t ✐s ❛ ❜❛rr✐❡r ❢♦r ♠❛♥② ❞✉❡ t♦ ✐ts ❝♦♠♣❧❡①✐t② ❛♥❞ ❛❜str❛❝t✐♦♥✱ ❤♦✇❡✈❡r✱ ✐❢ ✇❡❧❧ ✉♥❞❡rst♦♦❞✱ ❝❛♥ ❜❡ ♥♦t❡❞ ❤♦✇ ✐♠♣♦rt❛♥t t♦♦❧ ✐♥ ♠❛t❤❡♠❛t✐❝s✳ ❋♦r t❤✐s ✇♦r❦✱ ✐t ❤❛❞ ❜❡❡♥ ♥❡❝❡ss❛r② t❤❡ ✉s❡ ♦❢ ✐♠✲ ♣♦rt❛♥t ❝♦♥❝❡♣ts ✐♥s✐❞❡ ✐♥ ♠❛t❤❡♠❛t✐❝s ❛s ❡♥✉♠❡r❛❜✐❧✐t②✱ ❝❛r❞✐♥❛❧✐t②✱ ♣r✐♠❡ ♥✉♠❜❡rs✱ ♥✉♠❡r✐❝❛❧ s❡ts✱ ❢✉♥❝t✐♦♥s✱ ◆❡✇t♦♥✬s ❇✐♥♦♠✐❛❧✱ ❣❡♦♠❡tr②✱ ❛♥❛❧②t✐❝❛❧ ❣❡♦♠❡tr② ❛s ✇❡❧❧ ❛s t❤❡ st✉❞✐❡s ❛❜♦✉t ❈❛♥t♦r s❡ts ❛♥❞ tr❛♥s✜♥✐t❡ ❝❛r❞✐♥❛❧ ♥✉♠❜❡rs✳
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✼
✷ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛ ✶✾
✷✳✶ ❖ s✉r❣✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❖ s✉r❣✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ◆❡❣❛t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸ ❆ ❞❡s❝♦❜❡rt❛ ❞♦s ♥ú♠❡r♦s ❘❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✹ ❖ s✉r❣✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ■rr❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✺ ❆ ❢✉♥❞❛♠❡♥t❛çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❘❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✻ ❖s ♥ú♠❡r♦s ❈♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸ ❖ ■♥✜♥✐t♦ ✸✶
✸✳✶ P❛r❛❞♦①♦s✿ ✉♠❛ ♠❛♥❡✐r❛ ❝♦♥tr❛❞✐tór✐❛ ❞❡ ❡♥①❡r❣❛r ♦ ✐♥✜♥✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷ ❈❛r❞✐♥❛❧✐❞❛❞❡ ❡ ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✸ N✱ Z❡ Q sã♦ ❝♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✸✳✹ R ❡C ♥ã♦ sã♦ ❡♥✉♠❡rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸✳✺ ❊q✉✐♣♦tê♥❝✐❛ ❡ ❈❛r❞✐♥❛✐s ❚r❛♥s✜♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✻ ❖♣❡r❛çõ❡s ❝♦♠ ❈❛r❞✐♥❛✐s ❚r❛♥s✜♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✼ ❖ ❈♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✹ ❘❡❧❛çõ❡s ❞❡ ✐♥❝❧✉sã♦ ❡ ❞❡♥s✐❞❛❞❡ ✹✼
✹✳✶ Z,Q,R❡ R−Q ♥ã♦ sã♦ ❞❡♥s♦s ❡♠N ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✹✳✷ N,Q,R❡ R−Q ♥ã♦ sã♦ ❞❡♥s♦s ❡♠Z ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✹✳✸ N ❡Z ♥ã♦ sã♦ ❞❡♥s♦s ❡♠ Q ♦✉R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✹✳✹ Q é ❞❡♥s♦ ❡♠ R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✹✳✺ R−Q é ❞❡♥s♦ ❡♠R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✹✳✻ R⊂C ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✺ ❖s ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s ✺✶
✺✳✶ ❋♦r♠❛ ♣♦❧❛r ❞♦s ♥ú♠❡r♦s ❈♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✺✳✷ ❖ ❝♦r♣♦ ❞♦s ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✺✳✸ C ♥ã♦ é ♦r❞❡♥❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✻ ❉❡♥s✐❞❛❞❡ ❡ ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡ ❛♦s ♦❧❤♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✻✸ ✻✳✶ ❯♠❛ ♣r♦♣♦st❛ ❞❡ ❞❡♠♦♥str❛çã♦ ✐♥❢♦r♠❛❧ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ Q❡♠ R♣❛r❛
❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✻✳✷ ❯♠❛ ♣r♦♣♦st❛ ❞❡ ❞❡♠♦♥str❛çã♦ ✐♥❢♦r♠❛❧ s♦❜r❡ ❞❡♥s✐❞❛❞❡ ❞❡ R−Q ❡♠
R ♣❛r❛ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
✻✳✸ ❯♠❛ ❛♥❛❧♦❣✐❛ s♦❜r❡ ♦ ❡st✉❞♦ ❞♦s ❝❛r❞✐♥❛✐s tr❛♥s✜♥✐t♦s ❛♦s ♦❧❤♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾
✼ ❈♦♥❝❧✉sã♦ ✼✶
✶ ■♥tr♦❞✉çã♦
◆❛ ♠❛t❡♠át✐❝❛✱ ❛ ❞✐s❝✐♣❧✐♥❛ q✉❡ ❡st✉❞❛ ❛s r❡❧❛çõ❡s ❡♥tr❡ ♥ú♠❡r♦s✱ ♣♦r ✐♥t❡r♠é❞✐♦ ❞❡ ❡①♣r❡ssõ❡s s✐♠❜ó❧✐❝❛s ❣❡r❛✐s✱ é ❞❡♥♦♠✐♥❛❞❛ ❞❡ ➪▲●❊❇❘❆ ❡ é ❝♦♠ ❡st❛ á❧❣❡❜r❛ q✉❡ ❞❡t❡r♠✐♥❛❞♦s ❝❛♠✐♥❤♦s ♥♦ ✉♥✐✈❡rs♦ ❢♦r❛♠ tr❛ç❛❞♦s✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❡st✉❞♦ ❞❛ ❛str♦♥♦♠✐❛✳ P❛r❛ q✉❡ ❢♦ss❡ ❢✉♥❞❛♠❡♥t❛❞❛ ❞❛ ♠❛♥❡✐r❛ ❝♦♠♦ t❡♠♦✲❧❛ ❤♦❥❡✱ ❤♦✉✈❡ ✉♠ ✧❛♠❛❞✉r❡❝✐♠❡♥t♦✧❞❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ♣♦r sé❝✉❧♦s ✭t❛❧✈❡③ ♠✐❧ê♥✐♦s✮ ❡✱ ❞✉r❛♥t❡ ❡ss❡ ♣r♦❝❡ss♦✱ ❢♦✐✲s❡ ❞❡s❡♥✈♦❧✈❡♥❞♦ ❛ ♠❛t❡♠át✐❝❛ ♣✉r❛ ❡ ❛❜str❛t❛✳
❖ ❡st✉❞♦ ❞♦s ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s é ❛ ❜❛s❡ ❞♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ ♠❛t❡♠át✐❝❛ ♣✉r❛ q✉❡✱ ❛♣❡s❛r ❞❡ ❛❜str❛t❛✱ ❡stá ♣r❡s❡♥t❡ ❡♠ ✈ár✐❛s ár❡❛s ❝♦♥❝r❡t❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳
❊st❡ tr❛❜❛❧❤♦ ❢♦✐ ❡❧❛❜♦r❛❞♦ ❝♦♠ ❜❛s❡ ♥♦ ❡st✉❞♦ ❞❛ ❤✐stór✐❛ ❡ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❡ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛♣r❡s❡♥t❛r ✉♠ ❡st✉❞♦ s♦❜r❡ ♦s ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s✱ r❡s❣❛t❛♥❞♦ ❢✉♥❞❛♠❡♥t❛çõ❡s ❡ ❢❛t♦s ❤✐stór✐❝♦s ❞❡ s✉❛s ❞❡s❝♦❜❡rt❛s ❡ ❞❡♠♦♥str❛♥❞♦ t❡♦r❡♠❛s ✐♠♣♦r✲ t❛♥t❡s ❞❡♥tr♦ ❞❛ ♠❛t❡♠át✐❝❛ ♣✉r❛✳ ❆❜♦r❞❛r❡♠♦s ♦ s✉r❣✐♠❡♥t♦ ❞❡ ❝❛❞❛ ✉♠ ✲ ♥❛t✉r❛✐s✱ ✐♥t❡✐r♦s✱ r❛❝✐♦♥❛✐s✱ ✐rr❛❝✐♦♥❛✐s✱ r❡❛✐s ❡ ❝♦♠♣❧❡①♦s ✲ q✉❡ sã♦ ❛ ❜❛s❡ ❞❡ ♥♦ss❛ ❡str✉t✉✲ r❛çã♦ ♠❛t❡♠át✐❝❛ ❡✱ ♣♦st❡r✐♦r♠❡♥t❡✱ ❡st✉❞❛r❡♠♦s s✉❛ ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡✱ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❡ ❞❡♥s✐❞❛❞❡✳ ❆♣r❡s❡♥t❛r❡♠♦s t❛♠❜é♠ ❛❧❣✉♥s ♣❛r❛❞♦①♦s q✉❡ s❡r✈❡♠ ❝♦♠♦ â♥❝♦r❛ ♥♦ ❡st✉❞♦ ❞♦ ✐♥✜♥✐t♦ q✉❡ é✱ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ ♠❛t❡♠át✐❝♦ ❡ ✜❧♦só✜❝♦✱ ❜❡♠ ❝♦♠♦ ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ❞♦ ❈♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r ❡ ❞♦s ◆ú♠❡r♦s ❈❛r❞✐♥❛✐s ❚r❛♥s✜♥✐t♦s✳
❈♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ♣r♦♣♦r❝✐♦♥❛r ✉♠❛ ✈✐sã♦ ❣❡r❛❧ ❞❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❜r❡✈❡ ❞❡s❝r✐çã♦ ❞♦s ❛ss✉♥t♦s ❛❜♦r❞❛❞♦s✿
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❡①♣♦♠♦s ❢❛t♦s ❤✐stór✐❝♦s ❞♦ s✉r❣✐♠❡♥t♦ ❡ ❞❛ ❝r✐❛çã♦ ❞♦s ❝♦♥✲ ❥✉♥t♦s ♥✉♠ér✐❝♦s ❡st✉❞❛❞♦s ❛té ♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❜❡♠ ❝♦♠♦ ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s ✐♠♣♦rt❛♥t❡s ❝♦♠♦ ❛ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡ ❞❡ √2❡ ❛ ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦✳
◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ❢❛③❡♠♦s ✉♠❛ ❜r❡✈❡ ❞❡s❝r✐çã♦ ✜❧♦só✜❝♦✲♠❛t❡♠át✐❝❛ ❞♦ ✐♥✜♥✐t♦ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ❛❧❣✉♥s ♣❛r❛❞♦①♦s ❢❛♠♦s♦s✳ ❚❛♠❜é♠ sã♦ ❛❜♦r❞❛❞♦s ♦s ❛ss✉♥t♦s ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡ ❡ ❝❛r❞✐♥❛❧✐❞❛❞❡✱ ❈♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r ❡ ❈❛r❞✐♥❛✐s ❚r❛♥s✜✲ ♥✐t♦s✳
◆♦ q✉❛rt♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛s r❡❧❛çõ❡s ❞❡ ✐♥❝❧✉sã♦ ❡ ❞❡♥s✐❞❛❞❡ ❡♥tr❡ ❝♦♥❥✉♥t♦s✱ ❜❡♠ ❝♦♠♦ ❞❡♠♦♥str❛çõ❡s ❛❧❣é❜r✐❝❛s✳
◆♦ q✉✐♥t♦ ❝❛♣ít✉❧♦ ❢❛③❡♠♦s ✉♠ ❡st✉❞♦ ❡s♣❡❝í✜❝♦ ❞♦ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❛♣r❡s❡♥t❛♥❞♦ ❛ ❢♦r♠❛ ♣♦❧❛r ❞❡ ❝❛❞❛ ♥ú♠❡r♦✱ s✉❛ ♥ã♦ ♦r❞❡♥❛çã♦ ❡ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s✳ ◆♦ s❡①t♦ ❝❛♣ít✉❧♦ s❡❣✉✐♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ ✈♦❧t❛❞❛ ♣❛r❛ ♦ ❡♥s✐♥♦ ♠é❞✐♦ ✉t✐❧✐③❛♥❞♦
✶✽ ■♥tr♦❞✉çã♦
❛ ✈✐sã♦ ❣❡♦♠étr✐❝❛ ♥❛s ❞❡♠♦♥str❛çõ❡s ❞❡ ✐♥❝❧✉sã♦ ❡ ❞❡♥s✐❞❛❞❡ ❞❡ ❝♦♥❥✉♥t♦s✳
✷ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛
●r❛♥❞❡ ♣❛rt❡ ❞❛s ✐♥❢♦r♠❛çõ❡s ❛♣r❡s❡♥t❛❞❛s ❛❜❛✐①♦ ❢♦r❛♠ ❜❛s❡❛❞❛s ♥♦s ❧✐✈r♦s ❍✐s✲ tór✐❛ ❞❛ ♠❛t❡♠át✐❝❛ ❞❡ ❈❛r❧ ❇❡♥❥❛♠✐♥ ❇♦②❡r✱ ✶✾✼✹ ❡ ■❞❡✐❛s ❣❡♥✐❛✐s ♥❛ ♠❛t❡♠át✐❝❛ ❞❡ ❙✉r❡♥❞r❛ ❱❡r♠❛✱ ✷✵✶✸✳
❊①✐st❡ ✉♠❛ s✉♣♦s✐çã♦ ❞❡ q✉❡ ♦ ❤♦♠❡♠ ♣r✐♠❡✐r♦ ❛♣r❡♥❞❡✉ ❛ ❝♦♥t❛r ♣❛r❛ ❞❡♣♦✐s ❢❛❧❛r✳ ❱✐❛✲s❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❡♥✉♠❡r❛r ♦✉ ❛❣r✉♣❛r ❝♦✐s❛s✱ ♦ q✉❡ ❣❡r♦✉ ❛ ❝♦♥t❛❣❡♠✳ ❊st❛ ❡r❛ ❢❡✐t❛ ♥❛ ❢♦r♠❛ ❞❡ ❛❣r✉♣❛♠❡♥t♦s ❡ ❡♥t❡♥❞❡✲s❡ q✉❡✱ ❛♣ós ❝❡rt❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♦❜❥❡t♦s✱ ❝♦✐s❛s ♦✉ ❛♥✐♠❛✐s✱ ❛ q✉❛♥t✐❞❛❞❡ ❡r❛ ✈✐st❛ ❝♦♠♦ ✧✉♠ ♠♦♥t❡✧✳
❈♦♠ ❛ ❡✈♦❧✉çã♦ ❞♦ ❤♦♠❡♠ q✉❡ ♣❛ss♦✉ ❞❡ ♥ô♠❛❞❡ ❛ ❛❞♦t❛r ❧♦❝❛✐s ✜①♦s ✭❡st❛♠♦s ♥♦s r❡❢❡r✐♥❞♦ ❤á ♣♦✉❝♦ ♠❛✐s ❞❡ ✶✵ ♠✐❧ ❛♥♦s ❛trás✮✱ ✈✐✈❡♥❞♦ ❡♠ ❣r✉t❛s ❡ ❝❛✈❡r♥❛s ♣❛r❛ ♣r♦t❡❣❡r✲s❡ ❞❛ ❝❤✉✈❛✱ ❞♦ ❢r✐♦ ❡ ❞❡ ❛♥✐♠❛✐s s❡❧✈❛❣❡♥s✱ ❤♦✉✈❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞♦ ❛r♠❛③❡♥❛♠❡♥t♦ ❡ ❡st♦❝❛❣❡♠ ❞❡ ❛❧✐♠❡♥t♦s✳ P❛ss♦✉✲s❡ ❛ ♣r❛t✐❝❛r ♥ã♦ s♦♠❡♥t❡ ❛ ❝❛ç❛ ❡ ❝♦❧❡t❛ ❞❡ ❢r✉t♦s✱ ♠❛s t❛♠❜é♠ ♦ ❝✉❧t✐✈♦ ❞❡ ♣❧❛♥t❛s ❡ ❛ ❝r✐❛çã♦ ❞❡ ❛♥✐♠❛✐s✳ ■ss♦ ✐♥✢✉❡♥❝✐♦✉ ❞✐r❡t❛♠❡♥t❡ ♥❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✉♠❛ ♥♦✈❛ ❢♦r♠❛ ❞❡ ❝♦♥t❛❣❡♠✱ ♣♦✐s ❤❛✈✐❛ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞♦ ❤♦♠❡♠ ❡♠ ❝♦♥tr♦❧❛r ♦ s❡✉ r❡❜❛♥❤♦✳ ❆❧❣✉♠❛ ♥♦✈✐❧❤❛ ♦✉ ♦✉tr♦ ❛♥✐♠❛❧ ♣♦❞❡r✐❛ ❢✉❣✐r✱ ❧♦❣♦✱ ❝♦♠♦ ♣❡r❝❡❜❡r s❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❛♥✐♠❛✐s ❞❡ ✧♦♥t❡♠ ♣❛r❛ ❤♦❥❡✧s❡ ♠❛♥t❡✈❡❄
❆ ✉t✐❧✐③❛çã♦ ❞❡ ♣❡❞r❛s ❝♦♠♦ ♦❜❥❡t♦ ❞❡ ❝♦♥t❛❣❡♠ ❢♦✐ ❛❞❡r✐❞❛ ❛tr❛✈és ❞❡ ✉♠❛ r❡❧❛çã♦ ❜✐❥❡t♦r❛ s✐♠♣❧❡s ✭r❡❧❛çã♦ ✉♠ ♣❛r❛ ✉♠✮✿ ❝❛❞❛ ❛♥✐♠❛❧ s❡r✐❛ r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ ♣❡❞r❛✳ ❖ ❛♥✐♠❛❧ s❛í❛ ♣❛r❛ ♣❛st❛r❀ ❧♦❣♦✱ ✉♠❛ ♣❡❞r❛ ❡r❛ ✐♥s❡r✐❞❛ ❡♠ ✉♠ s❛❝♦✳ ❆♦ ✜♥❛❧ ❞♦ ❞✐❛✱ à ♠❡❞✐❞❛ q✉❡ ♦ ❛♥✐♠❛❧ r❡t♦r♥❛r❛ ♣❛r❛ s❡✉ ❧♦❝❛❧ ❞❡ ♦r✐❣❡♠✱ ❛s ♣❡❞r❛s ✐❛♠ s❡♥❞♦ r❡t✐r❛❞❛s✳ ❉❡st❛ ❢♦r♠❛ ♦ ❝♦♥tr♦❧❡ ❡r❛ ♠❛♥t✐❞♦ ❡✱ ❝❛s♦ s♦❜r❛ss❡ ❛❧❣✉♠❛ ♣❡❞r❛✱ ♣❡r❝❡❜✐❛✲s❡ ❡♥tã♦ q✉❡ ❛❧❣✉♠ ❛♥✐♠❛❧ ♥ã♦ ❤❛✈✐❛ r❡t♦r♥❛❞♦✳ ❊st❡ ❢♦✐ ♦ ✐♥í❝✐♦ ❞❛ ❝r✐❛çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦✳
◆❡ss❛ é♣♦❝❛ ♦ ❤♦♠❡♠ ❝♦♠✐❛ ♦ q✉❡ ❧❤❡ ❡r❛ ♦❢❡r❡❝✐❞♦ ♣❡❧❛ ♥❛t✉r❡③❛✳ P❛r❛ r❡❣✐str❛r ♦s ❛♥✐♠❛✐s ♠♦rt♦s ♥✉♠❛ ❝❛ç❛❞❛✱ ♦s ❤♦♠❡♥s ❢❛③✐❛♠ ♠❛r❝❛s ❡♠ ✉♠❛ ✈❛r❛✳ ❯s❛✈❛✲s❡ t❛♠❜é♠ ♦ss♦s ❞❡ ❧♦❜♦s ♣❛r❛ t❛✐s ♠❛r❝❛çõ❡s✳ ◗✉❛♥❞♦ ❞❡s❝♦❜r✐✉ ♦ ❢♦❣♦✱ ❛♣r❡♥❞❡✉ ❛ ❝♦③✐♥❤❛r ♦s ❛❧✐♠❡♥t♦s ❡ ❛ ♣r♦t❡❣❡r✲s❡ ♠❡❧❤♦r ❝♦♥tr❛ ♦ ❢r✐♦✳ ❆ ❡s❝r✐t❛ ❛✐♥❞❛ ♥ã♦ t✐♥❤❛ s✐❞♦ ❝r✐❛❞❛✳ ■ss♦ é ✉♠ ❣r❛♥❞❡ ✐♥❞í❝✐♦ q✉❡ ♥♦s ❧❡✈❛ ❛ ❝r❡r q✉❡ ♦ ❤♦♠❡♠ r❡❛❧♠❡♥t❡ ❞❡s❡♥✈♦❧✈❡✉ ♣r✐♠❡✐r♦ ❛ ❝♦♥t❛❣❡♠ ❞♦ q✉❡ ❛ ❡s❝r✐t❛✳
❖✉tr❛ ♠❛♥❡✐r❛ ❛❞♦t❛❞❛ ❡r❛ r❡❧❛❝✐♦♥❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❛♥✐♠❛✐s ♦✉ ✐t❡♥s ✭❞✐✈✐sõ❡s ❡♠ ❣r✉♣♦s✮ ❝♦♠ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❞❡❞♦s ❞❛s ♠ã♦s✿ ❝✐♥❝♦ ❞❡❞♦s✱ ❝✐♥❝♦ ♣❡✐①❡s✱ ❝✐♥❝♦ ❜❛stõ❡s✱
✷✵ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛
❝✐♥❝♦ ❛♥✐♠❛✐s✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❆ ✐❞❡✐❛ ❞❡ ❝♦♥t❛❣❡♠ ❡st❛✈❛ r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ♦s ❞❡❞♦s ❞❛ ♠ã♦✳ ❆ss✐♠✱ ❛♦ ❝♦♥t❛r ❛s ♦✈❡❧❤❛s✱ ♦ ♣❛st♦r s❡♣❛r❛✈❛ ❛s ♣❡❞r❛s ❡♠ ❣r✉♣♦s ❞❡ ❝✐♥❝♦ ✭s❡♠ s❛❜❡r ♦ q✉❡ ❡r❛ ♦ ❝✐♥❝♦✱ ♣♦✐s ❛♣❡♥❛s ❡♥t❡♥❞✐❛ ❛ q✉❛♥t✐❞❛❞❡ ❞♦ ❣r✉♣♦✮✳ ❉♦ ♠❡s♠♦ ♠♦❞♦ ♦s ❝❛ç❛❞♦r❡s ❝♦♥t❛✈❛♠ ♦s ❛♥✐♠❛✐s ❛❜❛t✐❞♦s✱ tr❛ç❛♥❞♦ r✐s❝♦s ♥❛ ♠❛❞❡✐r❛ ♦✉ ❢❛③❡♥❞♦ ♥ós ❡♠ ✉♠❛ ❝♦r❞❛✱ t❛♠❜é♠ ❞❡ ❝✐♥❝♦ ❡♠ ❝✐♥❝♦✳
❉❡✈✐❞♦ ❛ ❡st❡ t✐♣♦ ❞❡ ✧❝♦♥t❛❣❡♠✧ s✉r❣✐✉ ❛ ♣❛❧❛✈r❛ ❝á❧❝✉❧♦✱ q✉❡✱ ❡♠ ❧❛t✐♠✱ s✐❣♥✐✜❝❛ ❝♦♥t❛s ❝♦♠ ♣❡❞r❛s✳
✷✳✶ ❖ s✉r❣✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ◆❛t✉r❛✐s
❆♦ ❧♦♥❣♦ ❞❛ ❤✐stór✐❛ ♦❜s❡r✈❛✲s❡ ♦ ❛✈❛♥ç♦ ❞❛ ♠❛t❡♠át✐❝❛ ❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❡①tr❡♠❛ ❞❡ ❝♦♥t❛r ♦✉ ♠❡s♠♦ ❞❡ r❡❧❛❝✐♦♥❛r q✉❛♥t✐❞❛❞❡s✳ ■ss♦ ✐♥st✐❣♦✉ ♦ ❤♦♠❡♠ ❛ ❞❡s❡♥✈♦❧✈❡r sí♠❜♦❧♦s ♥♦ ✐♥t✉✐t♦ ❞❡ ❡①♣r❡ss❛r ✐♥ú♠❡r❛s s✐t✉❛çõ❡s✳ ❆ ♦r✐❣❡♠ ❡ ❛ ❢♦r♠✉❧❛çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ◆ú♠❡r♦ ♦❝♦rr❡✉ ❝♦♠ ♦ ♣ró♣r✐♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♠❛t❡♠át✐❝❛✳
❆ ♣❛rt✐r ❞❛s ♥❡❝❡ss✐❞❛❞❡s ❞✐ár✐❛s ❞♦ ❤♦♠❡♠✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦ ♥❛t✉r❛❧ s✉r❣✐✉ ♣♦r ♠❡✐♦ ❞❛ ❝♦♥t❛❣❡♠ ❞❡ ♦❜❥❡t♦s✳ ❆ss✐♠✱ ❡st❡ ❝♦♥❝❡✐t♦ ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ♣❡❧❛s ♥❛çõ❡s✱ ❡♠ ❝♦♥❥✉♥t♦ ❝♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ s✉❛s ❢♦r♠❛s ♣ró♣r✐❛s ❞❡ ❡s❝r✐t❛✱ ❝r✐❛♥❞♦ ♦ s✐st❡♠❛ ❞❡ ❝♦♥t❛❣❡♠✳
❉✐✈❡rs♦s s✐st❡♠❛s ❞❡ ♥✉♠❡r❛çã♦ ❢♦r❛♠ ❝r✐❛❞♦s ❡♠ t♦❞♦ ♦ ♠✉♥❞♦ ♥♦ ❞❡❝♦rr❡r ❞♦s t❡♠♣♦s✱ s❡♥❞♦ ♦s ♠❛✐s ❛♥t✐❣♦s ♦r✐❣✐♥ár✐♦s ❞♦ ❊❣✐t♦✱ ❙✉♠ér✐❛ ❡ ❇❛❜✐❧ô♥✐❛✳ ❖✉tr♦s s✐st❡✲ ♠❛s ❞❡ ♥✉♠❡r❛çã♦ ❜❛st❛♥t❡ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❈❤✐♥ês✱ ▼❛✐❛s✱ ●r❡❣♦✱ ❘♦♠❛♥♦✱ ■♥❞✐❛♥♦ ❡ ❆rá❜✐❝♦✳
❈♦♠ ♦ ♣❛ss❛r ❞♦s t❡♠♣♦s✱ ♦ ❤♦♠❡♠ ♣❛ss♦✉ ❛ ❜✉s❝❛r ❛❧❣♦ q✉❡ r❡♣r❡s❡♥t❛ss❡ ❞❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s t❛✐s s✐t✉❛çõ❡s✳ ❆♣ós ♦ s✉r❣✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❤♦✉✈❡ ✉♠❛ r❡✈♦❧✉çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ❝♦♥t❛❣❡♠✱ ♣♦✐s✱ ❛ ♣❛rt✐r ❞❡ ❡♥tã♦✱ ♣❛ss♦✉✲s❡ ❛ r❡❧❛❝✐♦♥❛r sí♠❜♦❧♦s ❛ ❞❡t❡r♠✐♥❛❞❛s q✉❛♥t✐❞❛❞❡s✳ ❈❛❞❛ q✉❛♥t✐❞❛❞❡ ♣♦❞❡r✐❛ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠ sí♠❜♦❧♦ ♥✉♠ér✐❝♦✳
▼❡s♠♦ ❝♦♠ ♦ ♣❛ss❛r ❞❡ t❛♥t♦s ❛♥♦s✱ ❛✐♥❞❛ ❤♦❥❡ ❡①✐st❡♠ ❛✉t♦r❡s ❡✴♦✉ ❡st✉❞✐♦s♦s q✉❡ ❛r❣✉♠❡♥t❛♠ s❡ ♦ ❛❧❣❛r✐s♠♦ ✧③❡r♦✧ é ♦✉ ♥ã♦ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ❆ ✐♥tr♦❞✉çã♦ ❞♦ ③❡r♦ ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s é r❡❧❛t✐✈❛♠❡♥t❡ ♥♦✈❛ ♥❛ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛✳ ❖s ❣r❡❣♦s✱ r♦♠❛♥♦s✱ ❡❣í♣❝✐♦s ❡ ❜❛❜✐❧ô♥✐♦s ♥ã♦ ❞❡✐①❛r❛♠ ❡✈✐❞ê♥❝✐❛s ❝❧❛r❛s ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ sí♠❜♦❧♦ ♣❛r❛ ❞❡s✐❣♥❛r ♦ ③❡r♦ ♥♦s s❡✉s s✐st❡♠❛s ♥✉♠ér✐❝♦s✳
❚❛✐s ❞✐s❝✉ssõ❡s s✉r❣❡♠ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s sã♦ ♥ú♠❡r♦s ❞❛ ♥❛t✉r❡③❛ ❡ ♥ã♦ é ♥❛t✉r❛❧ ✉t✐❧✐③❛r ③❡r♦ ❝♦♠♦ ❝♦♥t❛❣❡♠✳ ◆ã♦ s❡ ❞✐③ q✉❡ ❡♠ ❞❡t❡r♠✐♥❛❞♦ ❧♦❝❛❧ ❡①✐st❡♠ ✧③❡r♦ ❝❛r♥❡✐r♦s✧✱ ♣♦r ❡①❡♠♣❧♦✳ ❏á ♦✉tr♦s ❛✜r♠❛♠ s❡r ♥❛t✉r❛❧ ♣♦✐s ③❡r♦ s❡r✐❛ ♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ ❡ ♦ ❞✐✈✐s♦r ❝❡♥tr❛❧ ❡♥tr❡ ♦s ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s ❡ ♣♦s✐t✐✈♦s✳ ❙❡ ③❡r♦ é ♦✉ ♥ã♦ ♥❛t✉r❛❧✱ ✐ss♦ ♥✉♥❝❛ ❢♦✐ ✉♠ ♣r♦❜❧❡♠❛✱ ✈✐st♦ q✉❡ ❡♠ q✉❛❧q✉❡r ❞❡♠♦♥str❛çã♦ ♠❛t❡♠át✐❝❛ q✉❡ ♥❡❝❡ss✐t❡ ❞♦ ③❡r♦✱ ❛❝r❡s❝❡♥t❛✲s❡ ❡st❡ ❛ ❛❧❣✉♠ ❝♦♥❥✉♥t♦ ♥✉♠ér✐❝♦ ♦✉ ♥ã♦✳
❖ s✉r❣✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ◆❡❣❛t✐✈♦s ✷✶
s❡❣✉✐♥t❡ ❢♦r♠❛✿ N ❂ ④✵✱ ✶✱ ✷✱ ✸✱ ✹✱ ✳✳✳⑥✳
✷✳✷ ❖ s✉r❣✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ◆❡❣❛t✐✈♦s
❆♣❡s❛r ❞♦s ❡❣í♣❝✐♦s ❛✐♥❞❛ ♥ã♦ ❝♦♥❤❡❝❡r❡♠ ♦s ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s✱ ❥á ❡✈✐❞❡♥❝✐❛✈❛♠ s✉❛ ❡①✐stê♥❝✐❛ ♣♦r ♠❡✐♦ ❞❛s ❝♦♥str✉çõ❡s ❞❡ ✉♠❛ ♠❛❧❤❛ q✉❛❞r✐❝✉❧❛❞❛ ❡ ❞❡ ♣✐râ♠✐❞❡s✳ ❊❧❡s t♦♠❛✈❛♠ ✉♠❛ ❧✐♥❤❛ r❡t✐❧í♥❡❛ r❡♥t❡ ❛♦ ❝❤ã♦ ❝♦♠♦ ✧❧✐♥❤❛ ③❡r♦✧ ❡ ❞❡♥♦♠✐♥❛✈❛♠ ❛s ❧✐♥❤❛s ❛❝✐♠❛ ❝♦♠♦ s❡♥❞♦ ❝ú❜✐❝♦s ❛❝✐♠❛ ❞♦ ③❡r♦ ❡ ❛s ❧✐♥❤❛s ❛❜❛✐①♦ ❞♦ ❝❤ã♦ ♣♦r ❝ú❜✐❝♦s ❛❜❛✐①♦ ❞❡ ③❡r♦✳
❖s ❝❤✐♥❡s❡s t✐♥❤❛♠ ❛❧❣✉♥s ♠ét♦❞♦s ♣❛r❛ tr❛❜❛❧❤❛r ❝♦♠ ♦s ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s ❛✐♥❞❛ ✧❞❡s❝♦♥❤❡❝✐❞♦s✧✳ ❊st❡s ❡r❛♠ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ♦♣♦st♦s ❝♦♠♣❧❡♠❡♥t❛r❡s✱ ♣♦ré♠ ♥ã♦ ❡r❛♠ ❝♦♥s✐❞❡r❛❞♦s ❝♦♠♦ ❡♥t❡s ♠❛t❡♠át✐❝♦s ❡ s✐♠ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ r❡s♦❧✈❡r ❛❧❣✉♥s ♣r♦❜❧❡♠❛s✱ t❛✐s ❝♦♠♦ ❛ ❡①tr❛çã♦ ❞❛s r❛í③❡s ❞❡ ❞❡t❡r♠✐♥❛❞❛s ❡q✉❛çõ❡s✳ P❛r❛ ♦s ❍✐♥❞✉s✱ ❛ ♠❛t❡♠át✐❝❛ ❡r❛ ♠✉✐t♦ ✈♦❧t❛❞❛ ♣❛r❛ ♣rát✐❝❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❛ ❛str♦♥♦♠✐❛✱ ♣♦ré♠ t❛♠❜é♠ ❤♦✉✈❡ ❞❡❞✐❝❛çã♦ ♥♦s ❡st✉❞♦s ❞❡ ❡q✉❛çõ❡s q✉❛❞rát✐❝❛s ❡ ❧✐✲ ♥❡❛r❡s ❞✐♦❢❛♥t✐♥❛s✱ ♦♥❞❡ ✜❝♦✉ ❡✈✐❞❡♥t❡ ❛ ❛❝❡✐t❛çã♦ ❞♦s ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s ❝♦♠♦ ♣♦ssí✈❡❧ s♦❧✉çã♦ ♣❛r❛ ❡ss❛s ❡q✉❛çõ❡s✱ ♠❡s♠♦ ❞❡s❝♦♥❤❡❝❡♥❞♦✲s❡ t❛✐s ♥ú♠❡r♦s ❝♦♠♦ ❝♦♥❤❡❝❡♠♦s ❤♦❥❡✳
❖ s✉r❣✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s✱ ♣❛r❛ ♦s ❣r❡❣♦s✱ ❝♦♥t♦✉ ❝♦♠ ✐♠♣♦rt❛♥t❡s ❛❧✐❛✲ ❞♦s✿ ❚❛❧❡s ❞❡ ▼✐❧❡t♦ ✭✻✷✹ ✲ ✺✹✽ ❛✳❈✮ ❡ P✐tá❣♦r❛s ❞❡ ❙❛♠♦s ✭✺✼✵ ✲ ✹✾✵ ❛✳❈✮✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛s ❡s❝♦❧❛s ❏ô♥✐❝❛ ❡ P✐t❛❣ór✐❝❛✳ P♦r ✐ss♦✱ ♥❛ ❝✐✈✐❧✐③❛çã♦ ❣r❡❣❛✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú✲ ♠❡r♦ ♥❡❣❛t✐✈♦ s❡ ❞❡✉ ❞❡ ❢♦r♠❛ ♠❛✐s ❡✈✐❞❡♥t❡✱ ✈✐st♦ q✉❡ ❉✐♦❢❛♥t♦ ❥á ♦s ✉t✐❧✐③❛✈❛ ♥❛s r❡s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s✳ P♦ré♠ ♦ ♥ú♠❡r♦ ♥ã♦ ❡r❛ ✈✐st♦ ❝♦♠♦ ♥❡❣❛t✐✈♦✱ ♠❛s s✐♠ ❝♦♠♦ ✉♠ ♥ú♠❡r♦ ❝♦♠ ✉♥✐❞❛❞❡ ♥❡❣❛t✐✈❛✳ ❊①✳✿ ✲✶✵ ❂ ✶✵ ✉♥✐❞❛❞❡s ♥❡❣❛t✐✈❛s✳
❈♦♠ ♦ ❝r❡s❝✐♠❡♥t♦ ❞♦ ✐♠♣ér✐♦ ➪r❛❜❡✱ ❇❛❣❞á s❡ ✜r♠❛r❛ ❝♦♠♦ ✉♠ ❝❡♥tr♦ ❞❡ ♣r♦❞✉çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ ❡r❛ ♠✉✐t♦ ✐♥✢✉❡♥❝✐❛❞♦ ♣❡❧♦ ❝♦♠ér❝✐♦✳ ❊st❡✱ ♣♦r s✉❛ ✈❡③✱ ❢♦r❛ ✉♠ ❢❛t♦r ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛❝❡✐t❛çã♦ ❞♦s ♥ú♠❡r♦s ■♥t❡✐r♦s ◆❡❣❛t✐✈♦s✱ ✉♠❛ ✈❡③ q✉❡ ❡st❡s ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s r❡♣r❡s❡♥t❛✈❛♠ ♦ ♣r❡❥✉í③♦ ♥♦ ❝♦♠ér❝✐♦✳
❖s ✐♥❣❧❡s❡s ♣✉❜❧✐❝❛r❛♠ ❧✐✈r♦s ❛❝❡r❝❛ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥❡❣❛t✐✈♦s ❡♠ q✉❡ ❛❧❣✉♥s ❛✉t♦r❡s ❞❡❢❡♥❞✐❛♠ ❛ s✉❛ ✉t✐❧✐③❛çã♦ ❡ ♦✉tr♦s ❡r❛♠ t♦t❛❧♠❡♥t❡ ❝♦♥tr❛ ♦ s❡✉ ✉s♦✱ ♥ã♦ ❛❝❡✐t❛♥❞♦✲♦s ❝♦♠♦ ❡♥t❡s ♠❛t❡♠át✐❝♦s✳ ◆❡ss❛ ♣❡rs♣❡❝t✐✈❛✱ ♦ ❛ss✉♥t♦ ❢♦r❛ ♠✉✐t♦ ❞✐s❝✉✲ t✐❞♦ ❡ ❞❡s❡♥❝❛❞❡♦✉ ✉♠ ❝r❡s❝✐♠❡♥t♦ ❞❡ ❡st✉❞♦s ♠❛t❡♠át✐❝♦s ❛❝❡r❝❛ ❞♦ t❡♠❛✳
❉✐❛♥t❡ ❞❛s ♣r♦❞✉çõ❡s ♠❛t❡♠át✐❝❛s ❞♦s ✐♥❣❧❡s❡s✱ ❞❛ ❢♦rt❡ ✐♥✢✉ê♥❝✐❛ ❞♦ ❝♦♠ér❝✐♦ ❡ ❞♦s ❡st✉❞♦s ❞❡ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s✱ ♦s ◆ú♠❡r♦s ■♥t❡✐r♦s ◆❡❣❛t✐✈♦s ♣❛ss❛r❛♠ ❛ s❡r ✈✐st♦s ❝♦♠♦ ❡♥t❡s ♠❛t❡♠át✐❝♦s ❡♠ ♠❡❛❞♦s ❞♦ sé❝✉❧♦ ❳❱■■■✱ ❢❛③❡♥❞♦ ♣❛rt❡ ❞❡ ♥♦ss♦s ❡st✉❞♦s ❡ ❞❡ ♥♦ss♦ ❞✐❛✲❛✲❞✐❛✳
✷✷ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛
r❡♣r❡s❡♥t❛r t❡♠♣❡r❛t✉r❛s ❛❝✐♠❛ ❡ ❛❜❛✐①♦ ❞❡ ✵♦ ❈✱ ♣♦r ❡①❡♠♣❧♦✳ ❆strô♥♦♠♦s ❡ ❢ís✐❝♦s
♣r♦❝✉r❛✈❛♠ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ♣❛r❛ ❡①♣r❡ss❛r ❛ ❛tr❛çã♦ ❡♥tr❡ ❞♦✐s ❝♦r♣♦s ❡ s❡♥t✐❞♦s ❞❡ ♠♦✈✐♠❡♥t♦s✳
◆♦ ❝❛s♦ ❞❛s ♦✉tr❛s ❝✐ê♥❝✐❛s✱ q✉❛♥❞♦ ✉♠ ❝♦r♣♦ ❛❣❡ ❝♦♠ ✉♠❛ ❢♦rç❛ s♦❜r❡ ♦✉tr♦✱ ❡st❡ s❡❣✉♥❞♦ r❡❛❣❡ ❝♦♠ ❢♦rç❛ ❞❡ ♠❡s♠❛ ✐♥t❡♥s✐❞❛❞❡✱ ♣♦ré♠✱ ❝♦♠ s❡♥t✐❞♦ ❝♦♥trár✐♦✳ ❆ t❛r❡❢❛ ♥ã♦ ❡st❛✈❛ s♦♠❡♥t❡ ❡♠ ❝r✐❛r ❡st❡s ♥♦✈♦s ♥ú♠❡r♦s❀ ❡r❛ ♣r❡❝✐s♦ ❡♥❝♦♥tr❛r sí♠❜♦❧♦s q✉❡ ♣❡r♠✐t✐ss❡♠ ♦♣❡r❛çõ❡s ♠❛t❡♠át✐❝❛s ❝♦♠ ✜♥s ❞❡ ✈❡r❛❝✐❞❛❞❡ ❝♦♠ t❛✐s ♥ú♠❡r♦s✱ ❞❡ ♠♦❞♦ ♣rát✐❝♦ ❡ ❡✜❝✐❡♥t❡✳
❯♠❛ ♥♦✈❛ ♥♦t❛çã♦ ♥✉♠ér✐❝❛ ♣r❡❝✐s❛r✐❛ s❡r ✐♥st✐t✉í❞❛✳ ❆ ♠❛♥❡✐r❛ q✉❡ ❢♦✐ ❡♥❝♦♥tr❛❞❛ ♣❛r❛ r❡s♦❧✈❡r t❛✐s ♣r♦❜❧❡♠❛s ❝♦♥s✐st✐❛ ♥♦ ✉s♦ ❞♦s sí♠❜♦❧♦s ✰ ❡ ✲✳ ❊s❝r❡✈✐❛✲s❡ ❛❧❣♦ ❞♦ t✐♣♦ ♥✰ ✭♦ s✐♥❛❧ ✰ à ❢r❡♥t❡ ❞♦ ♥✉♠❡r❛❧ ✐♥❞✐❝❛✈❛ ✉♠❛ ❝♦♠♣r❛✱ t❡♠♣❡r❛t✉r❛s ❛❝✐♠❛ ❞❡ ③❡r♦ ❡ s❡♥t✐❞♦ ❢❛✈♦rá✈❡❧ ❛♦ ❞♦ ♠♦✈✐♠❡♥t♦✮ ♦✉ ♥✲ ✭♦ s✐♥❛❧ ✲ à ❞✐r❡✐t❛ ❞♦ ♥✉♠❡r❛❧ ✐♥❞✐❝❛✈❛ ✉♠❛ ✈❡♥❞❛✱ t❡♠♣❡r❛t✉r❛s ❛❜❛✐①♦ ❞❡ ③❡r♦ ❡ s❡♥t✐❞♦ ❝♦♥trár✐♦ ❛♦ ❞♦ ♠♦✈✐♠❡♥t♦✮✳ P♦st❡r✐♦r♠❡♥t❡ ❢♦r❛♠ ❞❡♥♦♠✐♥❛❞♦s ♥ú♠❡r♦s ♣♦s✐t✐✈♦s ❡ ♥❡❣❛t✐✈♦s✳
❈♦♠ ❡ss❛ ♥♦✈❛ s✐♠❜♦❧♦❣✐❛ ❢♦r❛ ✐♥st✐t✉í❞❛ ✉♠❛ ♥♦✈❛ ♠❛♥❡✐r❛ ❞❡ ✈❡r ♦s ♥ú♠❡r♦s✳ ❖s ♠❛t❡♠át✐❝♦s ❞❛ é♣♦❝❛ ❞❡s❡♥✈♦❧✈❡r❛♠ té❝♥✐❝❛s ♦♣❡r❛tór✐❛s ❝❛♣❛③❡s ❞❡ ❡①♣r❡ss❛r q✉❛❧✲ q✉❡r s✐t✉❛çã♦ ❡♥✈♦❧✈❡♥❞♦ t❛✐s ♥ú♠❡r♦s✳ ❙✉r❣✐r✐❛ ❡♥tã♦ ✉♠ ♥♦✈♦ ❝♦♥❥✉♥t♦ ♥✉♠ér✐❝♦ ♣♦st❡r✐♦r♠❡♥t❡ r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❧❡tr❛ ❩ ❞❡ ❩❛❤❧ ✭♥ú♠❡r♦ ❡♠ ❛❧❡♠ã♦✮✱ s❡♥❞♦ ❢♦r♠❛❞♦ ♣❡❧♦s ♥ú♠❡r♦s ♣♦s✐t✐✈♦s ✭◆❛t✉r❛✐s✮ ❡ s❡✉s r❡s♣❡❝t✐✈♦s ♦♣♦st♦s✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ♥ú✲ ♠❡r♦ ♥❡✉tr♦ ♣♦❞❡♥❞♦ s❡r ❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ Z ❂ ④✳✳✳✱ ✲✸✱ ✲✷✱ ✲✶✱ ✵✱ ✶✱ ✷✱ ✸✱✳✳✳⑥✳
✷✳✸ ❆ ❞❡s❝♦❜❡rt❛ ❞♦s ♥ú♠❡r♦s ❘❛❝✐♦♥❛✐s
◆♦ ❊❣✐t♦ ❆♥t✐❣♦✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❤á ✸✵✵✵ ❛♥t❡s ❞❡ ❈r✐st♦✱ ♦❝♦rr✐❛ ✉♠ ❢❡♥ô♠❡♥♦ ✐♥t❡r❡ss❛♥t❡✿ ❛s ❞✐✈✐sõ❡s ❞❡ t❡rr❛ ❡r❛♠ ❢❡✐t❛s ❛tr❛✈és ❞❡ ♠❛r❝❛çõ❡s ♥♦ ♣ró♣r✐♦ s♦❧♦ ♣❡✲ ❧♦s ❣❡ô♠❡tr❛s ❞♦s ❢❛r❛ós✱ ♣❛r❛ ❛ s✉❛ ♣♦♣✉❧❛çã♦✳ ◆❛ é♣♦❝❛ ❞❛ ❝❤❡✐❛ ❞♦ ❘✐♦ ◆✐❧♦✱ ♥♦ ♣❡rí♦❞♦ ❞❡ ❥✉♥❤♦ ❛ s❡t❡♠❜r♦✱ ❛ á❣✉❛ s✉❜✐❛ ❡ ❝♦❜r✐❛ t♦❞❛s ❛s ♠❛r❝❛çõ❡s ❞❡ t❡rr❛✳ ❊r❛♠ ♠❛r❝❛çõ❡s ❝♦♠ ❝♦r❞❛s✱ q✉❡ s❡r✐❛ ✉♠❛ ❡s♣é❝✐❡ ❞❡ ♠❡❞✐❞❛✱ ❞❡♥♦♠✐♥❛❞❛ ❡st✐r❛❞♦r❡s ❞❡ ❝♦r❞❛s✳ ❉❡✈✐❞♦ ❛ ❝❤❡✐❛✱ ♦s ❞♦♥♦s ❞❛s t❡rr❛s ❡r❛♠ ♦❜r✐❣❛❞♦s r❡♠❛r❝❛r❡♠ ♥♦✈❛♠❡♥t❡ ❡st❛s ❞✐✈✐sõ❡s✳ ◆♦ ❡♥t❛♥t♦✱ ♣♦r ♠❛✐s ❡✜❝❛③❡s q✉❡ t❡♥t❛ss❡♠ s❡r✱ ♥ã♦ ❡♥❝♦♥tr❛✈❛♠ ✉♠❛ ♠❛♥❡✐r❛ s✐♠♣❧❡s ♣❛r❛ t❛❧ ✉t✐❧✐③❛♥❞♦ ❛♣❡♥❛s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ♦ q✉❡ ♦s ❧❡✈♦✉ ❛ ♣❛rt✐❝✐✲ ♦♥❛r ✈❛❧♦r❡s ✐♥t❡✐r♦s ✉t✐❧✐③❛♥❞♦ ❢r❛çõ❡s✳ ❆s ♣❡ss♦❛s ✉t✐❧✐③❛✈❛♠ ❛s ❝♦r❞❛s✱ ❡st✐❝❛♥❞♦✲❛s ❡ ❛ss✐♠ ✈❡r✐✜❝❛✈❛♠ q✉❛♥t❛s ✈❡③❡s ❞❡t❡r♠✐♥❛❞❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ❡st❛✈❛ ❝♦♥t✐❞❛ ♥♦s ❧❛❞♦s ❞❛ ár❡❛ ❞❡♠❛r❝❛❞❛✱ ♠❛s r❛r❛♠❡♥t❡ ❛ ♠❡❞✐❞❛ ❞❛✈❛ ❝♦rr❡t❛ ♥♦ t❡rr❡♥♦ ♣♦✐s ♥ã♦ ❝❛❜✐❛ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❞❡ ✈❡③❡s ♥♦s ❧❛❞♦s ❞♦ t❡rr❡♥♦❀ ❝♦♠ ✐ss♦ ♣❡r❝❡❜❡✉✲s❡ ❛ ♥❡❝❡ss✐✲ ❞❛❞❡ ❞❡ ❝r✐❛r ✉♠ ♥♦✈♦ t✐♣♦ ❞❡ ♥ú♠❡r♦ ✲ ♦ ♥ú♠❡r♦ ❢r❛❝✐♦♥ár✐♦✱ ♦✉✱ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t♦✱ ❛s ❢r❛çõ❡s✳
❖ s✉r❣✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ■rr❛❝✐♦♥❛✐s ✷✸
❆ss✐♠✱ ♦ ❝♦♥❥✉♥t♦ r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❧❡tr❛Qq✉❡ ❞❡r✐✈❛❞❛ ❞❛ ♣❛❧❛✈r❛ ❧❛t✐♥❛ q✉♦t✐✠❡♥s✱
❝✉❥♦ s✐❣♥✐✜❝❛❞♦ é q✉❛♥t❛s ✈❡③❡s ❡ ❝♦♥té♠ t♦❞♦s ♦s ♥ú♠❡r♦s ✧①✧❞❛ ❢♦r♠❛
x= a
b, a, b∈Z, b6= 0. ✭✷✳✶✮
❍á q✉❛tr♦ ❢♦r♠❛s ❞❡ s❡ ❛♣r❡s❡♥t❛r ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✿
• ❋r❛çõ❡s ✭♣ró♣r✐❛s ♦✉ ✐♠♣ró♣r✐❛s✮❀
• ◆ú♠❡r♦s ♠✐st♦s ✭q✉❡ é ✉♠❛ ✈❛r✐❛çã♦ ❞❛s ❢r❛çõ❡s ✐♠♣ró♣r✐❛s✮❀ • ◆ú♠❡r♦s ❞❡❝✐♠❛✐s ❞❡ ❡s❝r✐t❛ ✜♥✐t❛ ♦✉ ❞í③✐♠❛ ✜♥✐t❛❀
• ❉í③✐♠❛s ✐♥✜♥✐t❛s✱ q✉❡ sã♦ ♥ú♠❡r♦s ❞❡❝✐♠❛✐s ❝✉❥❛ ❡s❝r✐t❛ ❛♣❛r❡❝❡♠ ♣❡rí♦❞♦s ♥✉♠ér✐❝♦s ✐♥✜♥✐t♦s✳
✷✳✹ ❖ s✉r❣✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ■rr❛❝✐♦♥❛✐s
P✐tá❣♦r❛s✱ ✜❧ós♦❢♦ ❡ ♠❛t❡♠át✐❝♦ ❣r❡❣♦✱ ♣♦r ✈♦❧t❛ ❞♦ sé❝✉❧♦ ❱■ ❛✳❈✳✱ ❢♦r♠♦✉ ❡ ❢✉♥❞♦✉ ❛ s♦❝✐❡❞❛❞❡ s❡❝r❡t❛ ❡ ♠íst✐❝❛ ❝❤❛♠❛❞❛ ❞❡ ✧♣✐t❛❣ór✐❝♦s✧✳ ❖s ♠❡♠❜r♦s ❞❡ t❛❧ s♦❝✐❡❞❛❞❡ ❞❡❞✐❝❛✈❛♠✲s❡ ❛♦ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s ✲ ❝♦♥s✐❞❡r❛♥❞♦✲♦s ❝♦♠♦ ❡ssê♥❝✐❛s ❞❛s ❝♦✐s❛s ✲ ♣♦r ❛❝r❡❞✐t❛r❡♠ q✉❡ ❉❡✉s ❤❛✈✐❛ s❡❣✉✐❞♦ ♣❛❞rõ❡s ♥✉♠ér✐❝♦s ♥❛ ❝r✐❛çã♦ ❞♦ ❯♥✐✈❡rs♦ ❡ q✉❡ t♦❞❛s ❛s ❝♦✐s❛s ♣♦❞❡r✐❛♠ s❡r ❡①♣❧✐❝❛❞❛s ♣♦r ♠❡✐♦ ❞❡ ♥ú♠❡r♦s✳
❆ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s s✉r❣✐✉ ❞❡s❞❡ ❡♥tã♦ ❝♦♠ P✐tá❣♦r❛s ❡ s❡✉s ❞✐s❝í♣✉❧♦s ❡st✉❞❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❊st❡s ❡ ❛s ❢r❛çõ❡s ❛té ❡♥tã♦ ❥á ❡r❛♠ ❝♦♥❤❡❝✐✲ ❞♦s✱ ❛♣❡s❛r ❞❛s ❢r❛çõ❡s ♥ã♦ s❡r❡♠ ❝♦♥s✐❞❡r❛❞❛s ❝♦♠♦ ❢r❛çõ❡s✱ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t❛s✱ ♠❛s r❡♣r❡s❡♥t❛✈❛♠ ❝♦♠♣❛r❛çõ❡s ❡♥tr❡ ❞✉❛s ❣r❛♥❞❡③❛s ❞❡ ♠❡s♠❛ ❡s♣é❝✐❡✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ r❡❧❛çã♦ ✧♣❛r❛ t♦❞♦ ❡ q✉❛❧q✉❡r tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❛ ár❡❛ ❞♦ q✉❛✲ ❞r❛❞♦ ❝♦♥str✉í❞♦ s♦❜r❡ ♦ s❡✉ ♠❛✐♦r ❧❛❞♦ é ✐❣✉❛❧ à s♦♠❛ ❞❛s ár❡❛s ❞♦s q✉❛❞r❛❞♦s ❝♦♥s✲ tr✉í❞♦s s♦❜r❡ ♦s ♦✉tr♦s ❞♦✐s ❧❛❞♦s✧✱ q✉❡ ❤♦❥❡ ❝♦♥❤❡❝❡♠♦s ❝♦♠♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ s✉r❣✐✉ ✉♠ ♣r♦❜❧❡♠❛✿ ♦ ❝á❧❝✉❧♦ ❞❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❝✉❥♦ ❧❛❞♦ ♠❡❞❡ ✶ ✉♥✐❞❛❞❡✳ ❈♦♠♦ ❡❧❡s ❛♣❡♥❛s ❝♦♥❤❡❝✐❛♠ ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❢♦✐ ❝♦♠ ❣r❛♥❞❡ s✉r♣r❡s❛ ❡ ❝❤♦q✉❡ ❛♦ ❞❡s❝♦❜r✐r❡♠ q✉❡ ❤❛✈✐❛♠ s❡❣♠❡♥t♦s ❞❡ r❡t❛ ❝✉❥❛s ♠❡❞✐❞❛s ♥ã♦ ♣♦❞✐❛♠ s❡r ❡①♣r❡ss❛s ♣♦r ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✳ ❊st❛ ♣r✐♠❡✐r❛ ❞❡s❝♦❜❡rt❛ ❞❡ ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ é ❛tr✐❜✉í❞❛ ❛ ❍✐♣❛s♦ ❞❡ ▼❡t❛♣♦♥t♦✱ ❞✐s❝í♣✉❧♦ ❞❡ P✐tá❣♦r❛s✳
❖s P✐t❛❣ór✐❝♦s ❝♦♥s✐❞❡r❛r❛♠ q✉❡❜r❛❞❛ ❛ ❤❛r♠♦♥✐❛ ❞♦ ✉♥✐✈❡rs♦✱ ❥á q✉❡ ♥ã♦ ♣♦❞✐❛♠ ❛❝❡✐t❛r ❛ r❛✐③ q✉❛❞r❛❞❛ ❞❡ ❞♦✐s ❝♦♠♦ ✉♠ ♥ú♠❡r♦✱ ♠❛s ♥ã♦ ♣♦❞✐❛♠ ♥❡❣❛r q✉❡ ❡st❛ r❛✐③ ❡r❛ ❛ ♠❡❞✐❞❛ ❞❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦ ✉♥✐tár✐♦✳
✷✹ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛
s♦♠❡♥t❡ ❡♠ ✶✽✼✷ q✉❡ ♦ ♠❛t❡♠át✐❝♦ ❛❧❡♠ã♦ ❉❡❞❡❦✐♥❞ ❢❡③ ❡♥tr❛r ♥❛ ❛r✐t♠ét✐❝❛✱ ❡♠ t❡r♠♦s r✐❣♦r♦s♦s✱ ♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s q✉❡ ❛ ❣❡♦♠❡tr✐❛ ❤❛✈✐❛ s✉❣❡r✐❞♦ ❤á ♠❛✐s ❞❡ ✈✐♥t❡ sé❝✉❧♦s✳
❍♦❥❡ ❡st❡ ♥ú♠❡r♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ √2 ❡ ✈❡r❡♠♦s ❛❜❛✐①♦ ✉♠❛ ♣♦ssí✈❡❧ ❞❡♠♦♥str❛✲
çã♦ ❞❡ t❛❧ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡✿
❙✉♣♦♥❤❛♠♦s ❡♥tã♦ q✉❡ √2 s❡❥❛ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r q✉❡
√
2 ❂ a
b✱ ❝♦♠ ❛✱ ❜ ∈ ❩ ❡ ❜ 6= ✵✳ ❙✉♣♦♥❤❛♠♦s q✉❡ a
b ❥á s❡❥❛ ✉♠❛ ❢r❛çã♦ ✐rr❡❞✉tí✈❡❧✱ ♦✉ s❡❥❛✱ ♠❞❝ ✭❛✱ ❜✮ ❂ ✶✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ ❡❧❡✈❛♥❞♦✲s❡ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❛♦ q✉❛❞r❛❞♦ t❡r❡♠♦s q✉❡
√
2 = a
b =⇒(
√
2)2 =a
b 2
=⇒2 = a
2
b2 =⇒2b
2 =a2. ✭✷✳✷✮
P♦❞❡♠♦s ❡♥tã♦ ♦❜s❡r✈❛r q✉❡ a2 é ✉♠ ♥ú♠❡r♦ ♣❛r✱ ❛ss✐♠ ❝♦♠♦2b2✳ ❉❡ss❛ ♠❛♥❡✐r❛✱
❛✜r♠❛♠♦s q✉❡ ❛ ❂ ✷❦✱ ❝♦♠ ❦ ∈ ❩✳
❙✉❜st✐t✉✐♥❞♦ ❛ ❂ ✷❦ ❡♠ ✭✶✳✷✮✱ t❡r❡♠♦s q✉❡
2b2 = (2k)2 =⇒4k2 = 2b2 =⇒2k2 =b2. ✭✷✳✸✮ ◆♦t❡ q✉❡ b2 t❛♠❜é♠ é ✉♠ ♥ú♠❡r♦ ♣❛r✳ ❉❡ss❛ ❢♦r♠❛✱ ❝♦♠♦ a2 é ✉♠ ♥ú♠❡r♦ ♣❛r ❡
b2 t❛♠❜é♠ é✱ s❡❣✉❡ q✉❡ ❛ ❡ ❜ sã♦ ♥ú♠❡r♦s ♣❛r❡s✱ ♣♦✐s s❡ ♦ q✉❛❞r❛❞♦ ❞❡ ✉♠ ♥ú♠❡r♦ é
♣❛r✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ ♦ ♥ú♠❡r♦ t❛♠❜é♠ é✳ P♦ré♠✱ ❛ ❡ ❜ s❡♥❞♦ ♣❛r❡s✱ ♠❞❝✭❛✱❜✮ 6= ✶✳
❆❜s✉r❞♦✳
P♦rt❛♥t♦✱ √2 ♥ã♦ é ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧❀ ❧♦❣♦✱ é ✐rr❛❝✐♦♥❛❧✳
✷✳✺ ❆ ❢✉♥❞❛♠❡♥t❛çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❘❡❛✐s
❆ ✉♥✐ã♦ ❞❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s ❛té ❡♥tã♦ ❝♦♥❤❡❝✐❞♦s ♦r✐❣✐♥♦✉ ❛ ❝r✐❛çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ r❡s♣♦♥sá✈❡❧ ♣♦r r❡♣r❡s❡♥t❛r ❡ ♦r❣❛♥✐③❛r ♦s ♥ú♠❡r♦s ❡♠ ✉♠ ú♥✐❝♦ ❝♦♥❥✉♥t♦✳
❉❡s❞❡ ✶✾✺✽✱ ❉❡❞❡❦✐♥❞✱ q✉❛♥❞♦ ❞❛✈❛ ❛✉❧❛s ❞❡ ❝á❧❝✉❧♦✱ s❡ ♠♦str♦✉ ✐♥t❡r❡ss❛❞♦ ❡♠ ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈✐❛♠ ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✳ P❛r❛ ❡❧❡✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡ ❞❡✈❡r✐❛ s❡r ❞❡s❡♥✈♦❧✈✐❞♦ ❛tr❛✈és ❞❡ ✉♠❛ ❛r✐t♠ét✐❝❛ ♣✉r❛ ✐s❡♥t❛ ❞♦ ✉s♦ ❞❡ ❣❡♦♠❡tr✐❛✳
❖s ♥ú♠❡r♦s ❈♦♠♣❧❡①♦s ✷✺
❖ ❞♦♠í♥✐♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ♣♦❞❡ s❡r ❡st❡♥❞✐❞♦ ❞❡ ♠♦❞♦ ❛ ❢♦r♠❛r ✉♠ ❝♦♥✲ t✐♥✉✉♠ ❞❡ ♥ú♠❡r♦s r❡❛✐s s❡ s✉♣✉s❡r♠♦s ♦ q✉❡ ❛❣♦r❛ s❡ ❝❤❛♠❛ ♦ ❛①✐♦♠❛ ❞❡ ❈❛♥t♦r✲ ❉❡❞❡❦✐♥❞ ✲ q✉❡ ♦s ♣♦♥t♦s s♦❜r❡ ❛ r❡t❛ ♣♦❞❡♠ s❡r ♣♦st♦s ❡♠ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❝♦♠ ♦s ♥ú♠❡r♦s r❡❛✐s✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ ♣❛r❛ t♦❞❛ ❞✐✈✐sã♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❡♠ ❞✉❛s ❝❧❛ss❡s ❆ ❡ ❇ t❛✐s q✉❡ t♦❞♦ ♥ú♠❡r♦ ❞❛ ♣r✐♠❡✐r❛ ❝❧❛ss❡ ❆ é ♠❡♥♦r q✉❡ t♦❞♦ ♥ú♠❡r♦ ❞❛ s❡❣✉♥❞❛ ❝❧❛ss❡ ❇✱ ❡①✐st❡ ✉♠ ❡ s♦♠❡♥t❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ q✉❡ ♣r♦❞✉③ ❡ss❛ ❝❧❛ss✐✜❝❛✲ çã♦✳ ❙❡ ❆ t❡♠ ✉♠ ♠á①✐♠♦ ♦✉ s❡ ❇ t❡♠ ♠í♥✐♠♦✱ ♦ ❝♦rt❡ ❞❡✜♥❡ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧❀ ♠❛s s❡ ❆ ♥ã♦ t❡♠ ♠á①✐♠♦ ❡ ❇ ♥ã♦ t❡♠ ♠í♥✐♠♦✱ ❡♥tã♦ ♦ ❝♦rt❡ ❞❡✜♥❡ ✉♠ ♥ú♠❡r♦ ✐r✲ r❛❝✐♦♥❛❧✳ ❊❧❡ ♦❜s❡r✈♦✉ q✉❡ ♦s t❡♦r❡♠❛s ❢✉♥❞❛♠❡♥t❛✐s s♦❜r❡ ❧✐♠✐t❡s ♣♦❞❡ s❡r ♣r♦✈❛❞♦s r✐❣♦r♦s❛♠❡♥t❡ s❡♠ ❛♣❡❧♦ à ❣❡♦♠❡tr✐❛✳ ❋♦✐ ❛ ❣❡♦♠❡tr✐❛ q✉❡ ✐♥✐❝✐♦✉ ♦ ❝❛♠✐♥❤♦ ♣❛r❛ ✉♠❛ ❞❡✜♥✐çã♦ ❝♦♥✈❡♥✐❡♥t❡ ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡✱ ♠❛s ♥♦ ✜♠ ❢♦✐ ❡①❝❧✉í❞❛ ❞❛ ❞❡✜♥✐çã♦ ❛r✐t♠ét✐❝❛ ❢♦r♠❛❧ ❞♦ ❝♦♥❝❡✐t♦✳
❆ ♥♦çã♦ ❞❡ ❝♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✱ ♥♦ s✐st❡♠❛ ❞❡ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ ♦✉ ✉♠❛ ❝♦♥str✉✲ çã♦ ❡q✉✐✈❛❧❡♥t❡ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ t✐♥❤❛ ❛❣♦r❛ s✉❜st✐t✉í❞♦ ❛ ❣r❛♥❞❡③❛ ❣❡♦♠étr✐❝❛ ♥❛ ❛♥á❧✐s❡✳
✷✳✻ ❖s ♥ú♠❡r♦s ❈♦♠♣❧❡①♦s
❆té ❝❡r❝❛ ❞❡ ✶✻✺✵✱ ❛s ú♥✐❝❛s r❛í③❡s ♣❛r❛ ❡q✉❛çõ❡s ❝♦♥s✐❞❡r❛❞❛s ❝♦♠♦ ✈❡r❞❛❞❡✐r❛s ❡r❛♠ ❛s q✉❡ ❝♦rr❡s♣♦♥❞❡ss❡♠ à ❣r❛♥❞❡③❛s ❝♦♠♦ ❝♦♠♣r✐♠❡♥t♦s✱ ár❡❛s✱ ✈♦❧✉♠❡s✱ ♠❛s✲ s❛s✱ ❡t❝✳ P♦ré♠✱ ♥❛ r❡s♦❧✉çã♦ ❞❡ ❛❧❣✉♠❛s ❡q✉❛çõ❡s q✉❛❞rát✐❝❛s ❡♥❝♦♥tr❛✈❛♠✲s❡ r❛í③❡s q✉❡ ❡r❛♠ ❞✐✈✐❞✐❞❛s ❡♠ ✈❡r❞❛❞❡✐r❛s ✭❝♦rr❡s♣♦♥❞✐❛♠ ❛♦s r❡❛✐s ♣♦s✐t✐✈♦s✮ ❡ ❢❛❧s❛s ✭q✉❡ ❝♦rr❡s♣♦♥❞✐❛♠ ❛♦s r❡❛✐s ♥❡❣❛t✐✈♦s ❡ ♥ã♦ ❡r❛♠ ❝♦♥s✐❞❡r❛❞❛s ❝♦♠♦ ❧❡❣ít✐♠❛s✮✳ ❆s ú♥✐✲ ❝❛s ❡ r❛r❛s ♦❝♦rrê♥❝✐❛s ❞❡ r❛í③❡s ♥❡❣❛t✐✈❛s ♥❡ss❡ ♣❡rí♦❞♦ s✉r❣✐❛♠ ❡♠ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥t❛❜✐❧✐❞❛❞❡✱ ♦♥❞❡ ❡r❛♠ ✐♥t❡r♣r❡t❛❞❛s ❝♦♠♦ ❞í✈✐❞❛s✳
❊♠ ✶✺✹✺✱ ❈❛r❞❛♥♦✱ ❛♦ t❡♥t❛r r❡s♦❧✈❡r ❛ ❡q✉❛çã♦x3 ❂ ✹ ✰ ✶✺①✱ ❝♦♥st❛t♦✉ q✉❡✱ ♠❡s♠♦
s❛❜❡♥❞♦ q✉❡ ① ❂ ✹ ❡r❛ ✉♠❛ r❛✐③ ✈❡r❞❛❞❡✐r❛ ❡ ❧❡❣ít✐♠❛✱ ❛ ❡q✉❛çã♦ ① ❂ q3
(2 +√−121) +q3
(2−√−121)
❡①♣✉♥❤❛ ✉♠ ✈❛❧♦r ❡str❛♥❤♦✳
❊st❛ r❡❧❛çã♦ é ♦❜t✐❞❛ ❝♦♠ ♦ ✉s♦ ❞❛ r❡❣r❛ ❞❡ ❞❛❧ ❋❡rr♦✲❚❛rt❛❣❧✐❛✱ q✉❡ é ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ❡q✉❛çõ❡s ❞♦ t❡r❝❡✐r♦ ❣r❛✉✳
✯ ❯♠❛ ❜r❡✈❡ ❞❡❞✉çã♦ ❞❛ ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦
❙❡❥❛ ✉♠❛ ❡q✉❛çã♦ ❝♦♥t❡♥❞♦ ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ ✸ ❞❛ ❢♦r♠❛ay3 ✰by2 ✰ ❝② ✰ ❞
❂ ✵ ❝♦♠ ❛ ❂ ✶✱ ♦✉ s❡❥❛✱ y3 ✰ by2 ✰ ❝② ✰ ❞ ❂ ✵✳ ❚♦♠❡ ❛ s✉❜st✐t✉✐çã♦ ② ❂ ① ✲ b
3 ❞❡
❢♦r♠❛ ❛ tr❛♥s❢♦r♠❛r ❡st❛ ❡q✉❛çã♦ ❡♠ ✉♠❛ ❝ú❜✐❝❛ s❡♠ ♦ t❡r♠♦ ❡♠ x2✳
✷✻ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛
x− b
3
3
+b
x− b
3
2
+c
x− b
3
+d= 0 =⇒ ✭✷✳✹✮
x3− 3x
2b
3 + 3xb2
9 −
b3
27+b
x2− 2xb
3 +
b2
9
+cx−bc
3 +d= 0 =⇒ ✭✷✳✺✮
x3+x
−b2
3 +c
+2b
3
27 −
bc
3 +d= 0. ✭✷✳✻✮
❊st❛ ❡q✉❛çã♦ ♦❜t✐❞❛ ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛ x3 ✰ P① ✰ ❈ ❂ ✵✳ ➱ ❝♦♥s✐❞❡r❛❞❛ q✉❡
❛ r❛✐③ ❞❡st❛ ❡q✉❛çã♦ s❡❥❛ ❞❛❞❛ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞♦ t✐♣♦ ① ❂ ✉ ✰ ✈✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ t❡r❡♠♦s q✉❡
(u+v)3+P(u+v) +C = 0 =⇒ ✭✷✳✼✮
u3+ 3u2v+ 3uv2+v3+uP +vP +C = 0 =⇒ ✭✷✳✽✮ u3+v3+ (3uv+P)(u+v) +C = 0. ✭✷✳✾✮ ❈♦♠♦ ✸✉✈ ✰ P ❂ ✵✱ ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ s❡rá ❞❛❞❛ ♣♦ru3 ✰v3 ❂ ✲❈✳ ❉❡st❛ ❢♦r♠❛✱
t❡r❡♠♦s ♦ s❡❣✉✐♥t❡ s✐st❡♠❛✿
❉❡ss❛ ♠❛♥❡✐r❛ t❡♠♦s q✉❡u3 ❡v3 sã♦ ❛ s♦♠❛ ❡ ♦ ♣r♦❞✉t♦ ❞❛s r❛í③❡s ❞❡ ✉♠❛ ❡q✉❛çã♦
❞♦ s❡❣✉♥❞♦ ❣r❛✉y2 ✰ ❈② ✲ P3✳✷✼−1 ❂ ✵✳ ❆ss✐♠✱ s✉❜st✐t✉✐♥❞♦ t❛✐s s♦❧✉çõ❡s ♥♦ ♠ét♦❞♦
❞❡ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞❡ s❡❣✉♥❞♦ ❣r❛✉✱ t❡r❡♠♦s q✉❡
y′ =u3 = −C+
r
C2+ 4P
3
27 2 e y
′′=v3 = −C−
r
C2+ 4P
3
27
2 . ✭✷✳✶✵✮
❉❡ss❛ ♠❛♥❡✐r❛✱ t❡r❡♠♦s q✉❡
u= 3 v u u u
t−C+
r
C2+4P
3
27
2 e v = 3
v u u u
t−C−
r
C2+4P
3
27
2 . ✭✷✳✶✶✮
❋♦✐ ❝♦♥s✐❞❡r❛❞♦ q✉❡ ❛ r❛✐③ ❞❡st❛ ❡q✉❛çã♦ é ❞♦ t✐♣♦ ① ❂ ✉ ✰ ✈✳ P♦rt❛♥t♦
x=u+v =
3
v u u u
t−C+
r
C2+4P
3 27 2 + 3 v u u u
t−C−
r
C2+4P
3
27
❖s ♥ú♠❡r♦s ❈♦♠♣❧❡①♦s ✷✼
x= 3
s −C 2 + r C2 4 + P3 27 + 3 s −C 2 − r C2 4 + P3
27. ✭✷✳✶✸✮
❊♥✜♠✱ ❡st❛ ✜❝♦✉ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❋Ó❘▼❯▲❆ ❉❊ ❈❆❘❉❆◆❖✳ ❊st❛ ❢ór♠✉❧❛✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ♠ét♦❞♦ ❞❡ r❡s♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡ q✉❛rt♦ ❣r❛✉✱ ❢♦✐ ❛♣r❡s❡♥t❛❞❛ ❛ ♣ú❜❧✐❝♦ ❡♠ ✶✺✹✺✱ ♥♦ ❆rs ▼❛❣♥❛ ✭❞♦ ❧❛t✐♠✿ ❆ ●r❛♥❞❡ ❆rt❡✮✱ ❞❡ ❈❛r❞❛♥♦✱ q✉❡ ❢♦✐ ♦ ♣r✐♠❡✐r♦ ❧✐✈r♦ ❞❡ ➪❧❣❡❜r❛ ❞❛ ❘❡♥❛s❝❡♥ç❛ ❛ ✐r ❛❧é♠ ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣❡❧♦s ♠❛t❡♠át✐❝♦s ❞❛ ❛♥t✐❣✉✐❞❛❞❡ ❡ ♣❡❧♦s ♠❛t❡♠át✐❝♦s ár❛❜❡s✳ ❆ ♣✉❜❧✐❝❛çã♦ ❞❡st❛ ♦❜r❛ ❞❡✉ ✉♠ ♥♦✈♦ ✐♠♣✉❧s♦ ❛♦ ❡st✉❞♦ ❞❛ á❧❣❡❜r❛✳
❱♦❧t❛♥❞♦ ❛♦ ♥♦ss♦ ♣r♦❜❧❡♠❛ ✐♥✐❝✐❛❧✱ ♦✉ s❡❥❛✱ ♦ ❞❡ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦x3 ❂ ✹ ✰ ✶✺①✱
♣❡❧❛ ❢ór♠✉❧❛✱ t❡r❡♠♦s q✉❡
x= 3
s
−(−4) 2 +
r
(−4)2
4 +
(−15)3
27 + 3
s
−(−4) 2 −
r
(−4)2
4 +
(−15)3
27 =⇒ ✭✷✳✶✹✮
x= 3
s
2 +
r
4 + (−1)
3.53.33
27 + 3
s
2− r
4 + (−1)
3.53.33
27 =⇒ ✭✷✳✶✺✮
x= 3
q
2 +√4 +−1.53+ 3
q
2−√4 +−1.53 =⇒ ✭✷✳✶✻✮
x= 3
q
2 +√−121 + 3
q
2−√−121. ✭✷✳✶✼✮
❈❛r❞❛♥♦ s❡ ❞❡♣❛r♦✉ ❝♦♠ ✉♠❛ r❛✐③ q✉❛❞r❛❞❛ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❡❣❛t✐✈♦✱ ♦ q✉❡ ♦ ✐♠♣❡❞✐✉ ❞❡ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ ♣❛r❛ ❝❤❡❣❛r q✉❡ ① ❂ ✹✳ ❉❛í✱ ♣❡♥s♦✉✲s❡ ❡♠ ❡s❝r❡✈❡rp3
2±√−121
❞❡ ✉♠❛ ♦✉tr❛ ♠❛♥❡✐r❛✱ ❛❧❣♦ ❞♦ t✐♣♦
a+b√−1 = 3
q
2 +√−121 ✭✷✳✶✽✮
❡ q✉❡
a−b√−1 = 3
q
2−√−121, com a, b ∈R. ✭✷✳✶✾✮
❊❧❡✈❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✶✽✮ ❛♦ ❝✉❜♦ ❞❡ ❛♠❜♦s ♦s ❧❛❞♦s✱ ♦❜t❡r❡♠♦s q✉❡
(a+b√−1)3 = 2 +√−121 =⇒ ✭✷✳✷✵✮ a3+ 3a2b√−1 + 3ab2(−1) +b3(−1)√−1 = 2 +√−121 =⇒ ✭✷✳✷✶✮ 3a2b√−1−b3√−1−11√−1 = 2 + 3ab2−a3 =⇒ ✭✷✳✷✷✮
√
−1 = 2 + 3ab
2−a3
✷✽ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛
−1 =
2 + 3ab2 −a3
3a2b−b3−11
2
. ✭✷✳✷✹✮
❆♥❛❧♦❣❛♠❡♥t❡✱ ❡❧❡✈❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✶✾✮ ❛♦ ❝✉❜♦ ❞❡ ❛♠❜♦s ♦s ❧❛❞♦s✱ ♦❜t❡r❡♠♦s q✉❡
a−b√−1 = (2−√−121)13 =⇒ ✭✷✳✷✺✮
−1 =
2 + 3ab2 −a3
−3a2b+b3+ 11
2
. ✭✷✳✷✻✮
❈♦♠♦ ❛s ❡q✉❛çõ❡s ✭✷✳✷✹✮ ❡ ✭✷✳✷✻✮ sã♦ ❛♠❜❛s ✐❣✉❛✐s ❛ ✲✶✱ t❡r❡♠♦s q✉❡
2 + 3ab2−a3
3a2b−b3−11
2
=
2 + 3ab2−a3
−3a2b+b3 + 11
2 =⇒ ✭✷✳✷✼✮
2 + 3ab2−a3
3a2b−b3−11
=
2 + 3ab2−a3
−3a2b+b3+ 11
. ✭✷✳✷✽✮
❉❡ss❛ ♠❛♥❡✐r❛✱ t❡r❡♠♦s ❞✉❛s s✐t✉❛çõ❡s✿
2 + 3ab2−a3
3a2b−b3−11 =
2 + 3ab2−a3
−3a2b+b3+ 11 ✭✷✳✷✾✮
♦✉
2 + 3ab2 −a3
3a2b−b3−11 =
2 + 3ab2−a3
3a2b−b3−11. ✭✷✳✸✵✮
P❡r❝❡❜❡✲s❡ q✉❡ ❡♠ ✭✷✳✸✵✮ ❛ ✐❣✉❛❧❞❛❞❡ é ✈á❧✐❞❛ s❡♠♣r❡✱ r❡s♣❡✐t❛❞❛s ❛s ❝♦♥❞✐çõ❡s ❞❡ ❡①✐stê♥❝✐❛✱ ✈✐st♦ q✉❡ ❛♠❜❛s ❛s ❡①♣r❡ssõ❡s sã♦ ❡①❛t❛♠❡♥t❡ ✐❞ê♥t✐❝❛s✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ ✐r❡♠♦s r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ ✭✶✳✷✾✮✳
2 + 3ab2−a3
3a2b−b3−11 =
−2−3ab2+a3
3a2b−b3−11 =⇒ ✭✷✳✸✶✮
2 + 3ab2−a3 =−2−3ab2+a3 =⇒ ✭✷✳✸✷✮ 4 + 6ab2−2a3 = 0 =⇒ ✭✷✳✸✸✮
2 + 3ab2−a3 = 0. ✭✷✳✸✹✮ ❖ ❝❛s♦ ❡♠ q✉❡ C2
22 ✰
p3
33 ❃ ✵ ✱ ❡r❛ ❝❤❛♠❛❞♦ ♥❛ é♣♦❝❛ ❞❡ ❝❛s✉s ✐rr❡❞✉❝✐❜✐❧✐s ♣♦rq✉❡
❖s ♥ú♠❡r♦s ❈♦♠♣❧❡①♦s ✷✾
❋♦r❛♠ ♥❡❝❡ssár✐♦s ♠❛✐s ❞❡ ✷✺ ❛♥♦s ♣❛r❛ q✉❡ ❇♦♠❜❡❧❧✐✱ ❡♠ ✶✺✼✷✱ r❡s♦❧✈❡ss❡ ❡st❡ ♣r♦❜❧❡♠❛✳
❇♦♠❜❡❧❧✐ t❡✈❡ ✉♠❛ ✐❞❡✐❛ ❜❛st❛♥t❡ ♦✉s❛❞❛ ♣❛r❛ ❛ é♣♦❝❛ ❞❡ r❡❛❧✐③❛r ♦♣❡r❛çõ❡s ✉t✐✲ ❧✐③❛♥❞♦ ❛s ♠❡s♠❛s ❝♦♥❞✐çõ❡s ❡st❛❜❡❧❡❝✐❞❛s ♣❛r❛ ♦s ♥ú♠❡r♦s r❡❛✐s✱ ❝♦♠ ❛ r❡ss❛❧✈❛ ❞❡ q✉❡ ✭√−1)2❂✲✶✱ ♦✉ s❡❥❛✱ ❛❞✐❝✐♦♥♦✉✲s❡ ✉♠❛ ♥♦✈❛ ♣r♦♣r✐❡❞❛❞❡ q✉❡ ♥ã♦ ❢❛③✐❛ ♣❛rt❡ ❞♦
❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ❇♦♠❜❡❧❧✐ ❛❞♠✐t✐✉ q✉❡ p3
2 +√−121 ♣✉❞❡ss❡ s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛ m+n√−1 ❡ q✉❡ 3
p
2−√−121 ❞❛ ❢♦r♠❛ m−n√−1✳ ❋❡✐t♦ ✐ss♦✱ ❛ ❡q✉❛çã♦ ♣♦❞❡ s❡r r❡s♦❧✈✐❞❛✱ ♣♦✐s✱ ❞❡
✭✷✳✶✾✮✱ ① ❂ m+n√−1✰ m−n√−1❂ ✷♠✱ ❝♦♠ ♠ ❂ ✷✱ ♣♦✐s ① ❂ ✹✳
2 +√−121 = 2 +√−1.121 = 2 +√−1.112 = 2 + 11√−1 = ✭✷✳✸✺✮
8−6 + (12−1)√−1 = 23+ 3.2.(√−1)2+ 22.3√−1−√−1 = ✭✷✳✸✻✮
23+ 22.3√−1 + 3.2.(√−1)2+ (√−1)3 = (2 +√−1)3. ✭✷✳✸✼✮ ▲♦❣♦✱ t❡♠✲s❡ q✉❡
2 +√−121 = (2 +√−1)3. ✭✷✳✸✽✮ ❆♥❛❧♦❣❛♠❡♥t❡✱ t❡♠✲s❡ q✉❡
2−√−121 = (2−√−1)3. ✭✷✳✸✾✮ ❉❡ ❢❛t♦✱ ♣♦✐s
(2−√−1)3 = 23−3.22√−1 + 3.2.(√−1)2−(√−1)3 = ✭✷✳✹✵✮
8−12√−1−6 +√−1 = 2−11√−1. ✭✷✳✹✶✮ P♦rt❛♥t♦✱ ❛ ❂ ✷ ❡ ❜ ❂ ✶ ❡♠ ✭✷✳✶✽✮ ❡ ✭✷✳✶✾✮ ❡✱ ❞❡ ✭✷✱✶✼✮✱ ❡♥❝♦♥tr❛✲s❡ q✉❡
x= 3
q
2 +√−121 + 3
q
2−√−121 = q3
(2 +√−1)3+q3
(2−√−1)3 = ✭✷✳✹✷✮
2 +√−1 + 2−√−1 = 2 + 2 = 4. ✭✷✳✹✸✮ ❖ ♣ró♣r✐♦ ❇♦♠❜❡❧❧✐ ♥ã♦ ❡st❛✈❛ ♠✉✐t♦ s❡❣✉r♦ ❞❡ s✉❛ ❝r✐❛çã♦✱ ❛té ♣♦rq✉❡✱ ♣❛r❛ ♦s ❞❡♠❛✐s ♠❛t❡♠át✐❝♦s ❞❛ é♣♦❝❛✱ ❡st❡s ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠❛ r❛✐③ q✉❛✲ ❞r❛❞❛ ❡r❛♠ ✈✐st♦s ❞❡ ♠❛♥❡✐r❛ s✉s♣❡✐t❛✱ ♦✉ s❡❥❛✱ ❡r❛♠ t♦❧❡r❛❞♦s ♥❛ ❢❛❧t❛ ❞❡ ❝♦✐s❛ ♠❡❧❤♦r✳
✸✵ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛
❆❧❜❡rt ●✐r❛r❞✱ ❡♠ ✶✻✷✾✱ ✉t✐❧✐③♦✉✱ ❡❢❡t✐✈❛♠❡♥t❡✱ ♦ sí♠❜♦❧♦ √−1q✉❛♥❞♦ ❡♥✉♥❝✐♦✉ ❛s
r❡❧❛çõ❡s ❡♥tr❡ r❛í③❡s ❡ ❝♦❡✜❝✐❡♥t❡s ❞❡ ✉♠❛ ❡q✉❛çã♦✳ ❊♠ ✶✼✾✼✱ ❈❛s♣❛r ❲❡ss❡❧ ✭❞✐♥❛✲ ♠❛rq✉ês✮ r❡♣r❡s❡♥t♦✉✱ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③✱ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❞❡ ❢♦r♠❛ ❣❡♦♠étr✐❝❛✱ ❡st❛❜❡❧❡❝❡♥❞♦ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ♦❜❥❡t✐✈❛ ❡♥tr❡ ❡st❡s ❡ ♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦✳
❖ sí♠❜♦❧♦ ✐✱ ♣❛r❛ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ √−1✱ ❢♦✐ ❝r✐❛❞♦ ♣♦r ▲❡♦♥❛r❞ ❊✉❧❡r✱ ♠❛s✱
✸ ❖ ■♥✜♥✐t♦
✸✳✶ P❛r❛❞♦①♦s✿ ✉♠❛ ♠❛♥❡✐r❛ ❝♦♥tr❛❞✐tór✐❛ ❞❡ ❡♥①❡r✲
❣❛r ♦ ✐♥✜♥✐t♦
◆♦ ♥♦ss♦ ❡♥t❡♥❞✐♠❡♥t♦✱ ♦ ✐♥✜♥✐t♦✱ ♠✉✐t❛s ✈❡③❡s✱ ❛❝❛❜❛ s❡♥❞♦ ✉♠ ♦❜stá❝✉❧♦ ♣❛r❛ ❛❧❣✉♥s ❡st✉❞✐♦s♦s✱ ♣❛r❛ ♦✉tr♦s✱ ❛ s♦❧✉çã♦ ❡✱ ♣❛r❛ ❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞❛s ♣❡ss♦❛s✱ ✉♠ ❞❡s❝♦♥❤❡❝✐❞♦✳ ❊①✐st❡♠ ❡st✉❞♦s ❛ r❡s♣❡✐t♦ ❞♦ ✐♥✜♥✐t♦ q✉❡ ❛❜♦r❞❛♠ ❝♦♥❝❡✐t♦s ❜❡♠ ❛✈❛♥✲ ç❛❞♦s✳ ❆❝r❡❞✐t♦ q✉❡ ♥❛ ♠❛t❡♠át✐❝❛ ♦ ✐♥✜♥✐t♦ s❡❥❛ ✈✐st♦ ❞❡ ♠❛♥❡✐r❛ ❡①ót✐❝❛✳ ❊ss❡ ♣♦❞❡ s❡r ♥✉♠ér✐❝♦ ♦✉ ❣❡♦♠étr✐❝♦✱ ♣♦✐s ♣♦❞❡✲s❡ ✐♠❛❣✐♥❛r ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠ ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♦✉ ♠❡s♠♦ ✉♠❛ r❡t❛ ✐♥✜♥✐t❛✱ ❝♦♥t❡♥❞♦ ❛ss✐♠✱ ✐♥✜♥✐t♦s ♣♦♥t♦s✳
◆♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ❳■❳ ❡ ♦ ✐♥í❝✐♦ ❞♦ ❳❳✱ ❛♣❛r❡❝❡r❛♠ ♥❛ ♠❛t❡♠át✐❝❛ ✈ár✐♦s r❡s✉❧t❛❞♦s ❡str❛♥❤♦s ❡✴ ♦✉ ♥❛❞❛ ✐♥t✉✐t✐✈♦s✳ ❆❧❣✉♥s ❞❡st❡s r❡s✉❧t❛❞♦s ❣❡r❛r❛♠ ❣r❛♥❞❡s ❝♦♥tr❛❞✐✲ çõ❡s✱ ❞❡♥♦♠✐♥❛❞♦s ❣❡♥❡r✐❝❛♠❡♥t❡ ❞❡ ♣❛r❛❞♦①♦s✱ q✉❡ sã♦ ♣❡♥s❛♠❡♥t♦s✱ ♣r♦♣♦s✐çõ❡s ♦✉ ❛r❣✉♠❡♥t♦s q✉❡ ❝♦♥tr❛r✐❛♠ ♦s ♣r✐♥❝í♣✐♦s ❜ás✐❝♦s ❡ ❣❡r❛✐s q✉❡ ❝♦st✉♠❛♠ ♦r✐❡♥t❛r ♦ ♣❡♥s❛♠❡♥t♦ ❤✉♠❛♥♦✳
❆❧❣✉♥s ♣❛r❛❞♦①♦s sã♦ ❜❡♠ ❢❛♠♦s♦s t❛✐s ❝♦♠♦ ♦ P❛r❛❞♦①♦ ❞♦ ❍♦t❡❧ ■♥✜♥✐t♦ ❞❡ ❍✐❧❜❡rt q✉❡ ❞✐③ q✉❡ ✉♠ ❤♦t❡❧ ❝♦♠ ✐♥✜♥✐t♦s q✉❛rt♦s s❡♠♣r❡ ♣♦❞❡rá r❡❝❡❜❡r ♠❛✐s ❣❡♥t❡✳ ❙❡ ♦s ✐♥✜♥✐t♦s q✉❛rt♦s ❥á ❡st✐✈❡r❡♠ ❝❤❡✐♦s ❝♦♠ ✐♥✜♥✐t♦s ❤ós♣❡❞❡s ❡ ✉♠ ♥♦✈♦ ❝❧✐❡♥t❡ ❛❞❡♥tr❛r ❛♦ ❤♦t❡❧✱ ♣❡❞❡✲s❡ ❛♦s ❤ós♣❡❞❡s q✉❡ s❡ ♠✉❞❡♠ ♣❛r❛ ♦ q✉❛rt♦ s❡❣✉✐♥t❡✱ ♦✉ s❡❥❛✱ s❡ ✉♠ ❤ós♣❡❞❡ ❡st✐✈❡r ♥♦ q✉❛rt♦ ❞❡ ♥ú♠❡r♦ ①✱ ❡st❡ s❡ ♠✉❞❛rá ♣❛r❛ ♦ q✉❛rt♦ ❞❡ ♥ú♠❡r♦ ① ✰ ✶✱ ❝♦♠ ① ∈N✳ ❉❡st❛ ♠❛♥❡✐r❛✱ ♦ ❤ós♣❡❞❡ ❞♦ q✉❛rt♦ ✶ ✈❛✐ ♣❛r❛ ♦ q✉❛rt♦ ✷ ❡ ❛ss✐♠
s✉❝❡ss✐✈❛♠❡♥t❡✳ ■st♦ s❡rá ♣♦ssí✈❡❧ ♣❛r❛ t♦❞♦s ❥á q✉❡ ❡①✐st❡♠ ✐♥✜♥✐t♦s q✉❛rt♦s ❡ ❞❡ss❛ ❢♦r♠❛ ♦ ✶◦ q✉❛rt♦ ✜❝❛rá ✈❛③✐♦ ♣❛r❛ ♦ ♥♦✈♦ ❤ós♣❡❞❡✳ ❈❛s♦ ❡st❡ ❝❤❡❣✉❡ ✉♠ ♥ú♠❡r♦
✐♥✜♥✐t♦✱ ♣❡❞✐r✲s❡✲á ❛♦s q✉❡ ❡stã♦ ❛❧♦❝❛❞♦s tr♦❝❛r❡♠ ❞❡ q✉❛rt♦✱ ♣♦ré♠✱ ❞❡st❛ ✈❡③✱ ✐♥❞♦ ♣❛r❛ ♦ q✉❛rt♦ ❝✉❥♦ ♥ú♠❡r♦ é ♦ ✷①✳ ❉❡ss❛ ❢♦r♠❛ ✜❝❛rã♦ q✉❛rt♦s s✉✜❝✐❡♥t❡s ✭t♦❞♦s ♦s q✉❡ tê♠ ♥ú♠❡r♦ í♠♣❛r✮ ♣❛r❛ ♦s ✐♥✜♥✐t♦s ♥♦✈♦s ❤ós♣❡❞❡s✳
❚❡♠♦s t❛♠❜é♠ ♦ P❛r❛❞♦①♦ ❞♦ ♠❡♥t✐r♦s♦ q✉❡ ❞✐③ ♦ s❡❣✉✐♥t❡✿ ❊♣✐♠é♥✐❞❡s é ❝r❡✲ t❡♥s❡ ✭♥❛s❝✐❞♦ ♥❛ ✐❧❤❛ ❞❡ ❈r❡t❛✮ ❡ ❛✜r♠❛ q✉❡ t♦❞♦s ♦s ❝r❡t❡♥s❡s ♠❡♥t❡♠✳ ❙❡ ❊♣✐♠é♥✐❞❡s ❢♦r ❝r❡t❡♥s❡ ❡ s❡ t♦❞♦s ♦s ❝r❡t❡♥s❡s ♠❡♥t❡♠ ❡♥tã♦✱ q✉❛♥❞♦ ❊♣✐♠é♥✐❞❡s ❛✜r♠❛ q✉❡ t♦❞♦s ♦s ❝r❡t❡♥s❡s ♠❡♥t❡♠ ❡❧❛ ❡stá ❞✐③❡♥❞♦ ❛ ✈❡r❞❛❞❡✱ ♠❛s✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ❊♣✐♠é♥✐❞❡s é ❝r❡t❡♥s❡ ❡ ♣♦r ✐ss♦ t❛♠❜é♠ ♠❡♥t❡✦ ❧♦❣♦✱ ❡❧❛ ♠❡♥t❡ ❡ ❞✐③ ❛ ✈❡r❞❛❞❡✳
