P❘❖❋▼❆❚
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P❛r❛♥á ✽✶✺✸✶✲✾✾✵✱ ❈✉r✐t✐❜❛✱ P❘❇r❛③✐❧Pr❡♣r✐♥t P❘❖❋▼❆❚ ✶ ✭✷✵✶✹✮ ✶✷ ❞❡ ♥♦✈❡♠❜r♦✱ ✷✵✶✹
❉✐s♣♦♥í✈❡❧ ✈✐❛ ■◆❚❊❘◆❊❚✿ ❤tt♣✿✴✴✇✇✇✳♠❛t✳✉❢♣r✳❜r
❚ÓP■❈❖❙ ❉❊ ❆❘■❚▼➱❚■❈❆✿ ❖
❆▲●❖❘■❚▼❖ ❉❊ ❊❯❈▲■❉❊❙
♣♦r
❚ó♣✐❝♦s ❞❡ ❆r✐t♠ét✐❝❛✿ ♦ ❛❧❣♦r✐t♠♦ ❞❡
❊✉❝❧✐❞❡s
❘❛❢❛❡❧ ❞❛ ❙✐❧✈❛ ❈♦rt✐❛♥♦
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❯❋P❘
✵✶✾✵✽✶✲✾✾✵✱ ❈✉r✐t✐❜❛✱ P❘
❇r❛s✐❧
❡✲♠❛✐❧✿ r❛❢❛❡❧✳❝♦rt✐❛♥♦❅❣♠❛✐❧✳❝♦♠
✽ ❞❡ ❉❡③❡♠❜r♦ ❞❡ ✷✵✶✹
❘❡s✉♠♦
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ é ♣❛rt❡ ❞❡ ✉♠❛ ❝♦❧❡çã♦ ❞❡ ❛t✐✈✐❞❛❞❡s r❡s♦❧✈✐❞❛s✱ ❝♦♠❡♥✲ t❛❞❛s ❡ ❝♦♠♣❧❡♠❡♥t❛r❡s ❡♥✈♦❧✈❡♥❞♦ ❝♦♥❝❡✐t♦s ❞❡ ❛r✐t♠ét✐❝❛✳ ❆♣r❡s❡♥t❛♠♦s ✉♠❛ ♣r♦♣♦st❛ ❞❡ ♠❛t❡r✐❛❧ ♣❛r❛ s❡r ✉t✐❧✐③❛❞♦ ❡♠ ❝✉rs♦s ❡①tr❛❝✉rr✐❝✉❧❛r❡s✱ ❝✉❥♦ ♦❜❥❡t✐✈♦ é ♣♦t❡♥❝✐❛❧✐③❛r ♦ ✐♥t❡r❡ss❡ ♣❡❧♦ ❡st✉❞♦ ❞❛ ▼❛t❡♠át✐❝❛✳ P♦r ❡st❡ ♠♦t✐✈♦✱ ❜✉s❝❛♠♦s ❛t✐✈✐❞❛❞❡s q✉❡ ❞❡s♣❡rt❡♠ ❛ ♠♦t✐✈❛çã♦ ❞♦s ❛❧✉♥♦s ❡ ❛s ❞❡s❝r❡✈❡♠♦s ❞❡ ♠❛♥❡✐r❛ ❛ ❛✉①✐❧✐❛r ♦s ♣r♦❢❡ss♦r❡s ❡♠ s✉❛ ❛♣❧✐❝❛çã♦✱ ❝♦♠ r❡s♣♦st❛s✱ ❝♦♠❡♥tár✐♦s ❡ s✉❣❡stõ❡s✳ ❆ ❛❜♦r❞❛❣❡♠ ❞♦s t❡♠❛s s❡ ❢❛③ ❝♦♠ ♦ ✉s♦ r❡❝♦rr❡♥t❡ ❞❡ ❛s♣❡❝t♦s ❤✐stór✐❝♦s✱ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✱ ❥♦❣♦s ❡ ❛t✐✲ ✈✐❞❛❞❡s r❡❝r❡❛t✐✈❛s✳
P❛❧❛✈r❛s✲❈❤❛✈❡✿ ❆r✐t♠ét✐❝❛ ✲ ❘❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ✲ ❏♦❣♦s
✶ ■♥tr♦❞✉çã♦
❖ ♠❛t❡r✐❛❧ ❛ s❡❣✉✐r é ✉♠❛ ♣r♦♣♦st❛ ❞❡ ♣r♦❥❡t♦ ♣❛r❛ ✉♠ ❝✉rs♦ ❞❡ ♠❛t❡♠át✐❝❛ ❡①tr❛❝✉rr✐❝✉❧❛r ♣❛r❛ ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❙✉r❣✐✉ ❞❛ ♣❡r❝❡♣çã♦ ❞❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✉♠ ❛t❡♥❞✐♠❡♥t♦ ❞✐❢❡r❡♥❝✐❛❞♦ ❛ ❛❧✉♥♦s ❝♦♠ ❛❧❣✉♠ ✐♥t❡r❡ss❡ ♥❛ ❞✐s❝✐♣❧✐♥❛✱ ♣♦✐s ❡♠ ♥♦ss❛s ❡①♣❡r✐ê♥❝✐❛s ♣♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ ♥♦ ❞✐❛ ❛ ❞✐❛ ❞❡ ✉♠❛ s❛❧❛ ❞❡ ❛✉❧❛ ♥ã♦ é s❡♠♣r❡ ♣♦ssí✈❡❧ ❞❛r ❛t❡♥çã♦ ❛♦s ❛❧✉♥♦s ❝♦♠ ✉♠ ♠❛✐♦r ✐♥t❡r❡ss❡ ❡ ❤❛❜✐❧✐❞❛❞❡s ❡♠ ♠❛t❡♠át✐❝❛✳ ❆ ❝♦♥s❡q✉ê♥❝✐❛ ❞✐ss♦ é ✉♠ ❞❡s♣❡r❞í❝✐♦ ❞❡ ♣♦ssí✈❡✐s t❛❧❡♥t♦s q✉❡ sã♦ ❞❡✐①❛❞♦s ❛ ♠❛r❣❡♠✱ ❡♠ ✉♠ s✐st❡♠❛ ❞❡ ❡♥s✐♥♦ q✉❡ ❣❡r❛❧♠❡♥t❡ ♦❢❡r❡❝❡ ♣r♦❥❡t♦s ❡①tr❛❝❧❛ss❡ ❛♣❡♥❛s ❞❡ r❡❢♦rç♦ ❡s❝♦❧❛r ♣❛r❛ s❛♥❛r ❞✐✜❝✉❧❞❛❞❡s✱ ❥á q✉❡ ❞✐✜❝✐❧♠❡♥t❡ t❡♠♦s ❛❧❣♦ ♥♦ s❡♥t✐❞♦ ❞❡ ♣♦t❡♥❝✐❛❧✐③❛r ❤❛❜✐❧✐❞❛❞❡s✳ ❈❛❜❡ r❡ss❛❧t❛r q✉❡ ❛ ✐❞❡✐❛
♥ã♦ é s❡❧❡❝✐♦♥❛r ❛♣❡♥❛s ♦s ♠❡❧❤♦r❡s ❛❧✉♥♦s ♣❛r❛ ♣❛rt✐❝✐♣❛r✳ ❖ ✐❞❡❛❧ é q✉❡ s❡❥❛ ❛❜❡rt♦ ❛ t♦❞♦ ❛❧✉♥♦ s✐♠♣❛t✐③❛♥t❡ ❞❛ ❞✐s❝✐♣❧✐♥❛ ❡ q✉❡ ✈♦❧✉♥t❛r✐❛♠❡♥t❡ s❡ ✐♥s❝r❡✈❛ ♥♦ ❝✉rs♦✳
❖s tó♣✐❝♦s ❛❜♦r❞❛❞♦s sã♦✿ ♣❛r✐❞❛❞❡✱ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ❡ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐✲ ❞❡s✳ ❖s t❡♠❛s sã♦ ❞❡s❡♥✈♦❧✈✐❞♦s ✐♥✐❝✐❛❧♠❡♥t❡ ❝♦♠ ✉♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛ s♦❜r❡ ♦ ♣r♦t❛❣♦♥✐st❛ ♣r✐♥❝✐♣❛❧ ❞♦ ❛ss✉♥t♦✳ ❊♠ s❡❣✉✐❞❛ sã♦ ♣r♦♣♦st❛s ❛t✐✈✐❞❛❞❡s✱ q✉❡ r❡♠❡t❡♠ à r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✱ ♥♦s q✉❛✐s ♦ ❛❧✉♥♦ ✈❛✐ ❣r❛❞❛t✐✈❛♠❡♥t❡ ❝❤❡✲ ❣❛♥❞♦ às ❝♦♥❝❧✉sõ❡s✳ ◆♦ t❡①t♦✱ ❛s ❛t✐✈✐❞❛❞❡s ❡stã♦ ❝♦♠ r❡s♣♦st❛s ❡ ❝♦♠❡♥tár✐♦s✳ ❆ ✈❡rsã♦ ❞♦ ❛❧✉♥♦✱ s❡♠ r❡s♣♦st❛s✱ é ❡♥❝♦♥tr❛❞❛ ♥♦ ❛♥❡①♦✳ ❇✉s❝❛♠♦s ❛❜♦r❞❛r ♦s ❛ss✉♥t♦s ❞❡ ♠❛♥❡✐r❛ ❝✉r✐♦s❛ ❡ ❧ú❞✐❝❛ ❝♦♠ ❛ ✐♥t❡♥çã♦ ❞❡ ❞❡s♣❡rt❛r ✉♠❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦s tó♣✐❝♦s ♠❛✐s ❞❡❧✐❝❛❞♦s ❞♦ tr❛❜❛❧❤♦ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❝❛✉s❛r ✉♠❛ ✐♥q✉✐❡t❛✲ çã♦ ♥♦ s❡♥t✐❞♦ ❞❡ ❜✉s❝❛r ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ q✉❡ ✈á ❛❧é♠ ❞♦ q✉❡ ❛ ❡s❝♦❧❛ ♦❢❡r❡❝❡ às t✉r♠❛s r❡❣✉❧❛r❡s✳
❊♠ ✈ár✐♦s ♠♦♠❡♥t♦s ♥♦ t❡①t♦✱ ♦ ♣r♦❢❡ss♦r ❡♥❝♦♥tr❛rá s✉❣❡stõ❡s ❞❡ ❝♦♠♦ ♦r✲ ❣❛♥✐③❛r ❛ ❛♣❧✐❝❛çã♦ ❞❛s ❛t✐✈✐❞❛❞❡s ❡♠ s❛❧❛✳ P♦ré♠✱ é ✐♠♣♦rt❛♥t❡ q✉❡ ❤❛❥❛ ✉♠❛ ♣r❡♣❛r❛çã♦ ✐♥✐❝✐❛❧ ♥♦ s❡♥t✐❞♦ ❞❡ ♠❡s❝❧❛r ❛ ♣r♦♣♦st❛ ❞♦ t❡①t♦ ❝♦♠ ❛ ❤❛❜✐❧✐❞❛❞❡ ❡ ❡①♣❡r✐ê♥❝✐❛ ❞♦ ❛♣❧✐❝❛❞♦r✳ ■❣✉❛❧♠❡♥t❡ ✐♠♣♦rt❛♥t❡ é ❛ r❡❛❧✐③❛çã♦ ❞❡ ✉♠❛ ❧❡✐t✉r❛ ✐♥✐❝✐❛❧ ❞♦ ♥í✈❡❧ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❞❛ t✉r♠❛✳ ◆❡st❡ s❡♥t✐❞♦ ❡ ❝❛s♦ s❡❥❛ ♥❡❝❡ssár✐♦✱ ❞❡✈❡ s❡r ♣❡♥s❛❞♦ ✉♠ ♣ré✲❝✉rs♦ ♥♦ q✉❛❧ s❡rã♦ tr❛❜❛❧❤❛❞❛s ❛s ❛r❡st❛s ✐❞❡♥t✐✜❝❛❞❛s✳
✷ ▼❡t♦❞♦❧♦❣✐❛
Pr♦❝✉r❛♠♦s ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ❞❛ ♠❛t❡♠át✐❝❛ ❞❡ ❢♦r♠❛ ❛ ✈❛❧♦r✐③❛r ❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ❞✐❢❡r❡♥t❡ ❞❛ ❢♦r♠❛ ❝❧áss✐❝❛ ♥❛ q✉❛❧ s❡ ❞❡✜♥❡ ♥♦✈♦s ❝♦♥t❡ú❞♦s ❝♦♠ ❛✉①í❧✐♦ ❞♦s ❛ss✉♥t♦s ❛♥t❡r✐♦r❡s✱ ❝♦♠♣❧❡t❛♥❞♦ ❝♦♠ ❡①❡♠♣❧♦s ❡ ♣r♦❜❧❡♠❛s q✉❡ ✉t✐❧✐③❛♠ ❡ss❡s s❛❜❡r❡s✱ s❡❣✉♥❞♦ ❇r♦✉ss❡❛✉ ✭✶✾✾✻✱ ♣❣ ✸✻✮ s♦❜r❡ ❛ ♠❛♥❡✐r❛ ❞❡ ❡♥s✐♥♦ ❝❧áss✐❝❛
✧❛♣❛❣❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ❛ ❤✐stór✐❛ ❞❡st❡s s❛❜❡r❡s✱ ✐st♦ é✱ ❛ s✉❝❡s✲ sã♦ ❞❛s ❞✐✜❝✉❧❞❛❞❡s ❡ ❞❛s q✉❡stõ❡s q✉❡ ♣r♦✈♦❝❛r❛♠ ♦ ❛♣❛r❡❝✐♠❡♥t♦ ❞♦s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s✱ ❛ s✉❛ ✉t✐❧✐③❛çã♦ ♣❛r❛ ❛ ❝♦❧♦❝❛çã♦ ❞❡ ♥♦✲ ✈♦s ♣r♦❜❧❡♠❛s✱ ❛ ✐♥tr✉sã♦ ❞❡ té❝♥✐❝❛s ❡ ❞❡ q✉❡stõ❡s r❡s✉❧t❛♥t❡s ❞♦s ♣r♦❣r❡ss♦s ❞♦s ♦✉tr♦s s❡❝t♦r❡s✱ ❛ r❡❥❡✐çã♦ ❞❡ ❞❡t❡r♠✐♥❛❞♦s ♣♦♥t♦s ❞❡ ✈✐st❛✱ ❝♦♥s✐❞❡r❛♥❞♦ ❢❛❧s♦s ♦✉ ❞❡s❛❞❡q✉❛❞♦s✱ ❡ ❛s ♥✉♠❡r♦s❛s q✉❡r❡❧❛s ❛ s❡✉ r❡s♣❡✐t♦✳✧
❇✉s❝❛♠♦s ❝♦❧❡t❛r ♣r♦❜❧❡♠❛s ❡ ❥♦❣♦s q✉❡ ❞❡s❡♥✈♦❧✈❡ss❡♠ ♦ ❛s♣❡❝t♦ ❝✐❡♥tí✜❝♦ ♥♦s ❛❧✉♥♦s✱ s✐t✉❛çõ❡s q✉❡ ❡♥✈♦❧✈❡♠ ❤❛❜✐❧✐❞❛❞❡s ❞❡ ❧❡✈❛♥t❛r ❤✐♣ót❡s❡s✱ ❝♦♥❥❡❝t✉r❛r✱ ✐♥t✉✐r✱ t❡st❛r ❡ ✈❛❧✐❞❛r r❡s✉❧t❛❞♦s✱ ❝♦♠♦ ❇r♦✉ss❡❛✉ ✭✶✾✾✻✱ ♣❣ ✸✽✮ ❞❡s❝r❡✈❡
✧✉♠❛ ❜♦❛ r❡♣r♦❞✉çã♦ ♣❡❧♦ ❛❧✉♥♦ ❞❡ ✉♠❛ ❛❝t✐✈✐❞❛❞❡ ❝✐❡♥tí✜❝❛ ❡①✐❣❡ q✉❡ ❡❧❡ ❛❥❛✱ ❢♦r♠✉❧❡✱ ♣r♦✈❡✱ ❝♦♥str✉❛ ♠♦❞❡❧♦s✱ ❧✐♥❣✉❛❣❡♥s✱ ❝♦♥❝❡✐t♦s✱
t❡♦r✐❛s✱ ♦s tr♦q✉❡ ❝♦♠ ♦✉tr♦s✱ r❡❝♦♥❤❡ç❛ ❛q✉❡❧❡s q✉❡ sã♦ ❝♦♥❢♦r♠❡s à ❝✉❧t✉r❛✱ r❡t✐r❡ ❞❡st❛ ❛q✉❡❧❡s q✉❡ ❧❤❡ sã♦ út❡✐s✱ ❡t❝✳✧
❆ r❡s♣❡✐t♦ ❞❛ ❡s❝♦❧❤❛ ❞❛ ❛r✐t♠ét✐❝❛✱ ❛❝r❡❞✐t❛♠♦s q✉❡ ❤á ❝♦♥t❡ú❞♦ q✉❡ ♥ã♦ ❡stã♦ ❡✈✐❞❡♥❝✐❛❞♦s ♥♦s ♣❛râ♠❡tr♦s ❝✉rr✐❝✉❧❛r❡s✱ ♣♦ré♠ ♣♦❞❡♠ s❡r ❛❜♦r❞❛❞♦s ❡ r❡❧❛❝✐✲ ♦♥❛❞♦s ❝♦♠ ♦s q✉❡ ❧á ❡stã♦✳ ❆ ❛r✐t♠ét✐❝❛ t❛♠❜é♠ é ✐♠♣♦rt❛♥t❡✱ ♣♦✐s s❡❣✉♥❞♦ ●♦♠❡s✱ ✭✷✵✵✽✱ ♣❣ ✶✻✵✮
✧❆ ❛r✐t♠ét✐❝❛ é ✐♥✈♦❝❛❞❛ ❝♦♠♦ ♦ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦ s❡♥sí✈❡❧ ❞❛ ♥❡❝❡ss✐❞❛❞❡ ❞♦s s✐❣♥♦s✿ ❝♦♥❢♦r♠❡ ❈♦♥❞✐❧❧❛❝✱ ♥ã♦ ♣♦❞❡rí❛♠♦s ❢❛③❡r ♣r♦❣r❡ss♦s ❛❧❣✉♠ ♥♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s✱ s❡ ♥ã♦ ✐♠❛❣✐♥áss❡✲ ♠♦s ♥♦♠❡s ♣❛r❛ t♦❞❛s ❛s ✐❞é✐❛s q✉❡ ❢♦r♠❛♠♦s ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛ ✐❞é✐❛ ❞❡ ✉♥✐❞❛❞❡✱ q✉❡ ❡❧❡ s✉♣õ❡ ❥á t❡r r❡❝❡♥✐❞♦ ✉♠ ♥♦♠❡✳✧
❊s❝♦❧❤❡♠♦s ✉t✐❧✐③❛r ❞❡ ❛❧❣✉♥s ❥♦❣♦s✱ ♣♦✐s ❡st❡s ❛✉①✐❧✐❛♠ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❤❛❜✐❧✐❞❛❞❡s ❞❡ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦✱ ❞❡s♣❡rt❛ ✉♠ ♠❛✐♦r ✐♥t❡r❡ss❡✱ ♣♦❞❡✲s❡ ❛♥❛❧✐s❛r ♦s ❡rr♦s ❡ ❡❧❛❜♦r❛r ❡str❛té❣✐❛s ♣❛r❛ ✈❡♥❝❡r✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❊♠❡r✐q✉❡ ✭✶✾✾✾✱ ♣❣ ✶✽✽✮
✧❝♦❧♦❝❛r ♦ ❛❧✉♥♦ ❞✐❛♥t❡ ❞❡ s✐t✉❛çõ❡s ❞❡ ❥♦❣♦ ♣♦❞❡ s❡r ✉♠❛ ❜♦❛ ❡str❛té❣✐❛ ♣❛r❛ ❛♣r♦①✐♠á✲❧♦ ❞♦s ❝♦♥t❡ú❞♦s ❝✉❧t✉r❛✐s ❛ s❡r❡♠ ✈❡✐❝✉❧❛✲ ❞♦s ♥❛ ❡s❝♦❧❛✱ ❛❧é♠ ❞❡ ♣♦❞❡r ❡st❛r ♣r♦♠♦✈❡♥❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♥♦✈❛s ❡str✉t✉r❛s ❝♦❣♥✐t✐✈❛s✧
❚❛♠❜é♠ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❛s♣❡❝t♦ ❝♦❣♥✐t✐✈♦ ❞♦ ❥♦❣♦ P❛✉❧♦ ✭✶✾✾✾✱ ♣❣ ✶✾✵✮ ❞✐③ ✧♥❡❝❡ss✐❞❛❞❡ ❡ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❝♦♥str✉çã♦ ❞❡ ♥♦✈♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❡ ♣r♦❝❡❞✐♠❡♥t♦s✱ ❞❡ ❞❡s❝♦❜r✐r ❡rr♦s ❡ ❞❡ ✐♠❛❣✐♥❛r ❢♦r♠❛s ❞❡ s✉♣❡rá✲❧♦s✱ ❞❡♥tr❡ ♦✉tr♦s ❞❡s❛✜♦s✳✧
❯t✐❧✐③❛♠♦s t❛♠❜é♠ ❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✱ ✉♠❛ ✈❡③ q✉❡ ♣r♦♣✐❝✐❛ ✉♠ ❛♠❜✐❡♥t❡ ❡♠ q✉❡ ♦ ❛❧✉♥♦ é s❡r ❛t✐✈♦ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠ ❡ s❡❣✉♥❞♦ ❖♥✉❝❤✐❝ ✭✶✾✾✾✱ ♣❣ ✷✵✹✮
✧❛ t❡♥❞ê♥❝✐❛ é ❝❛r❛❝t❡r✐③❛r ❡ss❡ tr❛❜❛❧❤♦ ❝♦♥s✐❞❡r❛♥❞♦ ♦s ❡st✉❞❛♥t❡s ❝♦♠♦ ♣❛rt✐❝✐♣❛♥t❡s ❛t✐✈♦s✱ ♦s ♣r♦❜❧❡♠❛s ❝♦♠♦ ✐♥str✉♠❡♥t♦s ♣r❡❝✐s♦s ❡ ❜❡♠ ❞❡✜♥✐❞♦s ❡ ❛ ❛t✐✈✐❞❛❞❡ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❝♦♠♦ ✉♠❛ ❝♦♦r❞❡♥❛çã♦ ❝♦♠♣❧❡①❛ s✐♠✉♥t❛♥❡❛ ❞❡ ✈ár✐♦s ♥í✈❡✐s ❞❡ ❛t✐✈✐❞❛❞❡✳✧
✸ ❖ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s
✸✳✶ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛✿ ❊✉❝❧✐❞❡s
❊♠306 ❛✳❈✳ ♦ ❝♦♥tr♦❧❡ ❞❛ ♣❛rt❡ ❡❣í♣❝✐❛ ❞♦ ✐♠♣ér✐♦ ❞❡ ❆❧❡①❛♥❞r❡✱ ❖ ●r❛♥❞❡✱ ❡s✲ t❛✈❛ ♥❛s ♠ã♦s ❞❡ Pt♦❧♦♠❡✉ ■✳ ❊ss❡ ❣♦✈❡r♥❛❞♦r ✈♦❧t♦✉ ❛ s✉❛ ❛t❡♥çã♦ ♣❛r❛ ❡s❢♦rç♦s ❝♦♥str✉t✐✈♦s✳ ❯♠ ❞❡ s❡✉s ♣r✐♠❡✐r♦s ❛t♦s ❢♦✐ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ✐♥st✐t✉t♦ ❝♦♥❤❡✲ ❝✐❞♦ ❝♦♠♦ ♠✉s❡✉✳ ❈♦♠♦ ♣r♦❢❡ss♦r❡s✱ ❡❧❡ ❝❤❛♠♦✉ ✉♠ ❣r✉♣♦ ❞❡ sá❜✐♦s ❞❡ ♣r✐♠❡✐r❛
❧✐♥❤❛✱ ❡♥tr❡ ❡❧❡s ❊✉❝❧✐❞❡s✱ ❞❡ ❝✉❥❛ ✈✐❞❛ s❛❜❡✲s❡ ♣♦✉❝♦✳ ❚ã♦ ♦❜s❝✉r❛ ✜❝♦✉ s✉❛ ✈✐❞❛ q✉❡ ♥❡♥❤✉♠ ❧✉❣❛r ❞❡ ♥❛s❝✐♠❡♥t♦ é ❛ss♦❝✐❛❞♦ ❛♦ s❡✉ ♥♦♠❡✳ ❊♠❜♦r❛ ❡❞✐çõ❡s ❞❡ s✉❛ ♣r✐♥❝✐♣❛❧ ♦❜r❛✱ ❖s ❊❧❡♠❡♥t♦s✱ ❢r❡q✉❡♥t❡♠❡♥t❡ ♦ ✐❞❡♥t✐✜❝❛ss❡♠ ❝♦♠♦ ❊✉❝❧✐❞❡s ❞❡ ▼❡❣❛r❛✱ ❡ ✉♠ r❡tr❛t♦ ❞❡ ❢r❡q✉❡♥t❡♠❡♥t❡ ❛♣❛r❡ç❛ ❡♠ ❤✐stór✐❛s ❞❛ ▼❛t❡♠át✐❝❛✱ tr❛t❛✲s❡ ❞❡ ✉♠ ❡rr♦ ❞❡ ✐❞❡♥t✐❞❛❞❡✳ ❖ ✈❡r❞❛❞❡✐r♦ ❊✉❝❧✐❞❡s ❞❡ ▼❡❣❛r❛ ❡r❛ ✉♠ ❞✐s❝í♣✉❧♦ ❞❡ ❙ó❝r❛t❡s ❡✱ ❡♠❜♦r❛ s❡ ♣r❡♦❝✉♣❛ss❡ ❝♦♠ ❧ó❣✐❝❛✱ ♥ã♦ s❡ s❡♥t✐❛ ♠❛✐s ❛tr❛í❞♦ ♣❡❧❛ ▼❛t❡♠át✐❝❛ q✉❡ s❡✉ ♠❡str❡✳
◆♦ss♦ ❊✉❝❧✐❞❡s✱ ❡♠ ❝♦♥tr❛st❡✱ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛✱ ♣♦r✲ q✉❡ ❢♦✐ ❝❤❛♠❛❞♦ ❧á ♣❛r❛ ❡♥s✐♥❛r ▼❛t❡♠át✐❝❛✳ ❉❛ ♥❛t✉r❡③❛ ❞❡ s❡✉ tr❛❜❛❧❤♦ ♣♦❞❡✲ s❡ ♣r❡s✉♠✐r q✉❡ t✐✈❡ss❡ ❡st✉❞❛❞♦ ❝♦♠ ❞✐s❝í♣✉❧♦s ❞❡ P❧❛tã♦✱ s❡ ♥ã♦ ♥❛ ♣ró♣r✐❛ ❆❝❛❞❡♠✐❛✳ ▲❡♥❞❛s ❛ss♦❝✐❛❞❛s ❛ ❊✉❝❧✐❞❡s ♦ ♣✐♥t❛✈❛♠ ❝♦♠♦ ✉♠ ❜♦♥❞♦s♦ ✈❡❧❤♦✳ Pt♦❧♦♠❡✉ ✉♠❛ ✈❡③ ♣❡r❣✉♥t♦✉✲❧❤❡ s❡ ❤❛✈✐❛ ✉♠ ❝❛♠✐♥❤♦ ♠❛✐s ❝✉rt♦✱ ♣❛r❛ ❛ ❣❡♦✲ ♠❡tr✐❛✱ q✉❡ ♦ ❡st✉❞♦ ❞❡ ❖s ❊❧❡♠❡♥t♦s✱ ❡ ❊✉❝❧✐❞❡s ❧❤❡ r❡s♣♦♥❞❡✉ q✉❡ ♥ã♦ ❤❛✈✐❛ ❡str❛❞❛ r❡❛❧ ♣❛r❛ ❛ ❣❡♦♠❡tr✐❛✳
❊✈✐❞❡♥t❡♠❡♥t❡ ❊✉❝❧✐❞❡s ♥ã♦ ❞❛✈❛ ê♥❢❛s❡ ❛ ❛s♣❡❝t♦s ♣rát✐❝♦s ❞♦ ❛ss✉♥t♦✳ ❍á ✉♠❛ ❡stór✐❛ ❝♦♥t❛❞❛ s♦❜r❡ ❡❧❡ q✉❡ ❞✐③ q✉❡ q✉❛♥❞♦ ✉♠ ❡st✉❞❛♥t❡ ♣❡r❣✉♥t♦✉ ♣❛r❛ q✉❡ s❡r✈✐❛ ♦ ❡st✉❞♦ ❞❛ ❣❡♦♠❡tr✐❛✱ ❊✉❝❧✐❞❡s ❞✐ss❡ ❛ s❡✉ ❡s❝r❛✈♦ q✉❡ ❞❡ss❡ três ♠♦❡❞❛s ❛♦ ❡st✉❞❛♥t❡ ✧♣♦✐s ❡❧❡ ♣r❡❝✐s❛ t❡r ❧✉❝r♦ ❝♦♠ ♦ q✉❡ ❛♣r❡♥❞❡✧✳ ❊♠❜♦r❛ s❡ t❡♥❤❛♠ ♣❡r❞✐❞♦ ♠❛✐s ❞❛ ♠❡t❛❞❡ ❞♦s s❡✉s ❧✐✈r♦s✱ ❛✐♥❞❛ r❡st❛r❛♠✱ ♣❛r❛ ❢❡❧✐❝✐✲ ❞❛❞❡ ❞♦s sé❝✉❧♦s ✈✐♥❞♦✉r♦s✱ ♦s tr❡③❡ ❢❛♠♦s♦s ❧✐✈r♦s q✉❡ ❝♦♥st✐t✉❡♠ ♦s ❊❧❡♠❡♥t♦s ✭❙t♦✐❝❤❡✐❛✮✳ P✉❜❧✐❝❛❞♦s ♣♦r ✈♦❧t❛ ❞❡ 300 ❛✳❈✳ ❛í ❡stá ❝♦♥t❡♠♣❧❛❞❛ ❛ ❛r✐t♠ét✐❝❛✱ ❛ ❣❡♦♠❡tr✐❛ ❡ ❛ á❧❣❡❜r❛✳ ▼✉✐t♦s ♦✉tr♦s t❡①t♦s ❧❤❡ sã♦ ❛tr✐❜✉í❞♦s✳ ❖ tr❛❜❛❧❤♦ ❞❡ ❊✉❝❧✐❞❡s é tã♦ ✈❛st♦ q✉❡ ❛❧❣✉♥s ❤✐st♦r✐❛❞♦r❡s ♥ã♦ ❛❝r❡❞✐t❛✈❛♠ q✉❡ ❢♦ss❡ ♦❜r❛ ❞❡ ✉♠ só ❤♦♠❡♠✳
❖s tr❛❜❛❧❤♦s ♠❛t❡♠át✐❝♦s q✉❡ ❝❤❡❣❛r❛♠ ❛té ♥ós ❢♦r❛♠ ✐♥✐❝✐❛❧♠❡♥t❡ tr❛❞✉✲ ③✐❞♦s ♣❛r❛ ár❛❜❡✱ ❞❡♣♦✐s ♣❛r❛ ❧❛t✐♠✱ ❡ ❛ ♣❛rt✐r ❞❡st❡s ❞♦✐s ✐❞✐♦♠❛s ♣❛r❛ ♦✉tr❛s ❧í♥❣✉❛s ❡✉r♦♣❡✐❛s✳ ❊♠❜♦r❛ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❥á ❢♦ss❡♠ ❝♦♥❤❡❝✐❞♦s ❛♥t❡r✐♦r♠❡♥t❡ ❛ s✉❛ é♣♦❝❛✱ ♦ q✉❡ ✐♠♣♦ss✐❜✐❧✐t❛ ✉♠❛ ❛♥á❧✐s❡ ❝♦♠♣❧❡t❛ ❞❛ s✉❛ ♦r✐❣✐♥❛❧✐❞❛❞❡✱ ♣♦❞❡✲s❡ ❝♦♥s✐❞❡r❛r ♦ s❡✉ tr❛❜❛❧❤♦ ❣❡♥✐❛❧✳ ❆♦ r❡❝♦❧❤❡r t✉❞♦ ♦ q✉❡ ❡♥tã♦ ❝♦♥❤❡❝✐❛✱ s✐st❡✲ ♠❛t✐③❛ ♦s ❞❛❞♦s ❞❛ ✐♥t✉✐çã♦ ❡ s✉❜st✐t✉✐ ✐♠❛❣❡♥s ❝♦♥❝r❡t❛s ♣♦r ♥♦çõ❡s ❛❜str❛t❛s✱ ♣❛r❛ ♣♦❞❡r r❛❝✐♦❝✐♥❛r s❡♠ q✉❛❧q✉❡r ❛♣♦✐♦ ✐♥t✉✐t✐✈♦✳
✸✳✷ ❆t✐✈✐❞❛❞❡s
❆t✐✈✐❞❛❞❡ ✸✳✶ P❛r ♦✉ í♠♣❛r ♠❛❧✉❝♦
❊ss❛ ❛t✐✈✐❞❛❞❡ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r♦♣✐❝✐❛r ❛♦ ♣r♦❢❡ss♦r ✉♠ ♠ét♦❞♦ ❧ú❞✐❝♦ ♣❛r❛ ✐♥tr♦❞✉③✐r✱ ❡♠ s❛❧❛✱ ❛ ♣❛r✐❞❛❞❡ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ s✉❛ ❛♥á❧✐s❡ ♥❛s ♦♣❡r❛çõ❡s ❜ás✐❝❛s✿ ❛❞✐çã♦✱ s✉❜tr❛çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦✳ ❖ ❥♦❣♦ ❛❜♦r❞❛ ❛✐♥❞❛ ✉♠ tr❛❜❛❧❤♦ ❡s♣❡❝í✜❝♦ ❝♦♠ r❡st♦s ❞❡ ❞✐✈✐sõ❡s ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ ❡ss❡ t❡♠❛ ♣♦❞❡ s❡r ❞❡s❡♥✈♦❧✈✐❞♦✱ s❡♠ t❛♥t♦ r✐❣♦r✱ ❥✉♥t♦ ❛♦s ❛❧✉♥♦s✳ ❈♦♠ ❡ss❛ ❛t✐✈✐❞❛❞❡ ❡s♣❡r❛✲s❡ q✉❡ ♦ ❛❧✉♥♦ ❞❡s❡♥✈♦❧✈❛ ✉♠❛ ❛♥á❧✐s❡ ♠❛✐s ❢♦r♠❛❧ s♦❜r❡ ❝♦♠♦ ♣r♦❝❡❞❡r
✉♠❛ ❞❡♠♦♥str❛çã♦ ♦♥❞❡ ❛ ❛♥á❧✐s❡ ❞❡ t♦❞♦s ♦s ❝❛s♦s ♣♦ssí✈❡✐s é ✉t✐❧✐③❛❞❛✳ P❛r❛ ❛ ❛t✐✈✐❞❛❞❡ ✈❛♠♦s ♣r❡❝✐s❛r ❞❛ ❢♦❧❤❛ ❞❡ q✉❡stõ❡s q✉❡ ❡♥❝♦♥tr❛✲s❡ ♥♦ ❛♥❡①♦✳
❉❡s❝r✐çã♦ ❞❛ ❛t✐✈✐❞❛❞❡✿ ❖ ♣r♦❢❡ss♦r ❞❡✈❡ s❡♣❛r❛r ♦s ❛❧✉♥♦s ❡♠ tr✐♦s✱ ♦♥❞❡ ✉♠ ❞♦s ♣❛rt✐❝✐♣❛♥t❡s ✐♥✐❝✐❛rá ♦ ❥♦❣♦ ❝♦♠♦ ♦ ♠❡❞✐❛❞♦r✳ ❊♠ s❡❣✉✐❞❛✱ ♦ ♣r♦❢❡ss♦r ❡①♣❧✐❝❛rá ♦ ❥♦❣♦✿ ❈❛❞❛ ✉♠ ❞♦s ❥♦❣❛❞♦r❡s ❡s❝♦❧❤❡rá ♣❛r ♦✉ í♠♣❛r ❡ ❛♣ós✱ ❛♥♦t❛rá✱ s❡❝r❡t❛♠❡♥t❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❛ s✉❛ ❡s❝♦❧❤❛✱ ❡♠ ✉♠❛ ❢♦❧❤❛ ❞❡ ♣❛♣❡❧ q✉❡ s❡rá ❡♥tr❡❣✉❡ ❛♦ ♠❡❞✐❛❞♦r✳ ❖ ♠❡❞✐❛❞♦r s❡ ❡♥❝❛rr❡❣❛rá ❞❡ ♠✉❧t✐♣❧✐❝❛r ♦s ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s ❡ s❡ ♦ r❡s✉❧t❛❞♦ ❢♦r ♣❛r✱ ❣❛♥❤❛ ♦ ❥♦❣❛❞♦r q✉❡ ❡s❝♦❧❤❡✉ ❡ss❛ ♦♣çã♦✳ ❈❛s♦ ❝♦♥trár✐♦ ❣❛♥❤❛rá ♦ ❥♦❣❛❞♦r q✉❡ ❡s❝♦❧❤❡✉ ❛ ♦♣çã♦ í♠♣❛r✳ ❆♣ós ❛❧❣✉♠❛s r♦❞❛❞❛s é ✐♥t❡r❡ss❛♥t❡ ♦❜s❡r✈❛r ❥✉♥t♦ ❛ t✉r♠❛ ♦ q✉❡ s❡r✐❛♠ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ❡ q✉❛❧ ♦ ♠♦t✐✈♦ ❞❛ ❛✉sê♥❝✐❛ ❞♦ ♥ú♠❡r♦ ③❡r♦✳ ❉❡♣♦✐s ❞❡ ❥♦❣❛r❡♠ ♣♦r ♠❛✐s ✉♠ t❡♠♣♦ ♦ ♣r♦❢❡ss♦r ❞❡✈❡rá ♣❡❞✐r ❛♦s ❛❧✉♥♦s q✉❡ r❡s♣♦♥❞❛♠ à q✉❡stã♦ ✉♠ ❞❛ ❢♦❧❤❛ ❞❡ ❛t✐✈✐❞❛❞❡s ❞❛ ♣á❣✐♥❛ ✷✶✳
◗✉❡stã♦ ✸✳✶✳✶ ❊①✐st❡ ✉♠❛ ❡str❛té❣✐❛ q✉❡ ♣♦ss✐❜✐❧✐t❡ ✈❡♥❝❡r s❡♠♣r❡❄ ❈❛s♦ ❡①✐st❛✱ q✉❛❧ s❡r✐❛❄
❘❡s♣♦st❛ ❝♦♠❡♥t❛❞❛✿ ❙✐♠✱ ❜❛st❛ q✉❡ ♦ ❛❧✉♥♦ ❡s❝♦❧❤❛ ❛ ♦♣çã♦ ♣❛r ❡ ❛♥♦t❡ ✉♠ ♥ú♠❡r♦ ♣❛r ❡♠ s✉❛ ✜❝❤❛✳ ❊s♣❡r❛✲s❡ q✉❡ ♦s ❛❧✉♥♦s ♣❡r❝❡❜❛♠ q✉❡ s❡ ❡s❝♦❧❤❡✲ r❡♠ ❛ ♦♣çã♦ ♣❛r ❡ ❛♥♦t❛r ❡♠ s❡✉ ♣❛♣❡❧ ✉♠ ♥ú♠❡r♦ ♣❛r ❡❧❡ s❡♠♣r❡ ✈❡♥❝❡rá✱ ♦✉ s❡❥❛✱ ♦ ❥♦❣♦ ♥ã♦ é ❥✉st♦✱ ♣♦✐s ♦s ♣❛rt✐❝✐♣❛♥t❡s ♥ã♦ ♣♦ss✉❡♠ ❛s ♠❡s♠❛s ❝❤❛♥❝❡ ❞❡ ❣❛♥❤❛r✳ ❖❜s❡r✈❛❞♦ ❡ss❡ ❢❛t♦ ♦ ♣r♦❢❡ss♦r ❞❡✈❡rá s♦❧✐❝✐t❛r ❛♦s ❛❧✉♥♦s q✉❡ r❡s♣♦♥✲ ❞❛♠ à q✉❡stã♦ ❞♦✐s ❞❛ ❢♦❧❤❛ ❞❡ ❛t✐✈✐❞❛❞❡s ❞❛ ♣á❣✐♥❛ ✷✶ ♣♦✐s ♥❡❧❛ ✐rã♦ ✈❡r✐✜❝❛r ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♣♦r q✉❡ ✐st♦ ❛❝♦♥t❡❝❡✳
◗✉❡stã♦ ✸✳✶✳✷ ◆♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✱ ✉♠ ♥ú♠❡r♦ ♣❛r✱ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ 2n✱ ❝♦♠ n∈N∗✳ ❏á ✉♠ ♥ú♠❡r♦ í♠♣❛r ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦
❝♦♠♦2k−1✱ ❝♦♠k ∈N∗✳ ❈♦♠♦ ♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r ❡s❝♦❧❤❡✉ ❛ ♦♣çã♦ ♣❛r ❡ ❛♥♦t♦✉
❡♠ s✉❛ ✜❝❤❛ ✉♠ ♥ú♠❡r♦ ♣❛r✱ t❡♠♦s q✉❡ ♦ ♦✉tr♦ ❥♦❣❛❞♦r só ♣♦❞❡rá ❡s❝♦❧❤❡r ✉♠ ♥ú♠❡r♦ ♣❛r ♦✉ ✉♠ ♥ú♠❡r♦ í♠♣❛r✳ ❋❡✐t❛s ❡ss❛s ❝♦♥s✐❞❡r❛çõ❡s✱ ♣❡❞❡✲s❡✿
❛✳ ♦ ♣r♦❞✉t♦ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s ♣❛r❡s✱ ♦✉ s❡❥❛ 2n.2k✱ ❝♦♠ n, k ∈N∗✳ ❆♥❛❧✐s❡ ❛
♣❛r✐❞❛❞❡✳
❘❡s♣♦st❛ ❝♦♠❡♥t❛❞❛✿ 2n.2k = 4nk = 2.(2nk)✱ ❝♦♠♦ 2nk ∈ N∗✱ t❡♠♦s
q✉❡ 2nk = r✳ P♦rt❛♥t♦ 2n.2k = 2.(2nk) = 2r✱ ♦✉ s❡❥❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡
❞♦✐s ♥ú♠❡r♦s ♣❛r❡s r❡s✉❧t❛rá ❡♠ ✉♠ ♥ú♠❡r♦ ♣❛r✳
❜✳ ♦ ♣r♦❞✉t♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♣❛r ♣♦r ✉♠ ♥ú♠❡r♦ í♠♣❛r✱ ♦✉ s❡❥❛2n.(2k−1)❝♦♠
n, k ∈N∗✳ ❆♥❛❧✐s❡ ❛ ♣❛r✐❞❛❞❡✳
❘❡s♣♦st❛ ❝♦♠❡♥t❛❞❛✿ 2n.(2k −1) = 4nk − 2n = 2.(2nk −n)✱ ❝♦♠♦ (2nk−n)∈N∗✱ t❡♠♦s q✉❡ (2nk−n) =r✳ P♦rt❛♥t♦ 2n.(2k−1) = 2.(2nk−
n) = 2r✱ ♦✉ s❡❥❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ r❡s✉❧t❛ ❡♠ ✉♠ ♥ú♠❡r♦ ♣❛r✳
❝✳ ❈♦♠ ♦s r❡s✉❧t❛❞♦s ❞♦s ✐t❡♥s ❛ ❡ ❜ ♦ q✉❡ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r❄
❘❡s♣♦st❛ ❝♦♠❡♥t❛❞❛✿ P♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♥❡st❡s ❝❛s♦s ♦ r❡s✉❧t❛❞♦ s❡rá s❡♠♣r❡ ✉♠ ♥ú♠❡r♦ ♣❛r✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦s ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s✳ ❆ss✐♠ t❡♠♦s q✉❡ s❡ ♦ ❥♦❣❛❞♦r ❡s❝♦❧❤❡r ♣❛r ❡ ❛♥♦t❛r ❡♠ s✉❛ ❢♦❧❤❛ ✉♠ ♥ú♠❡r♦ ♣❛r✱ s❡rá ♦ ✈❡♥❝❡❞♦r✳
❆t✐✈✐❞❛❞❡ ✸✳✷ ▼✉❞❛♥❞♦ ✉♠ ♣♦✉❝♦ ❛s r❡❣r❛s
❖s ❥♦❣❛❞♦r❡s ❡s❝♦❧❤❡♠ ❛ ♦♣çã♦ ♣❛r ♦✉ í♠♣❛r✱ ♠❛s ❛♦ ✐♥✈és ❞❡ ❡s❝r❡✈❡r q✉❛❧q✉❡r ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ♣❛r❛ ❥♦❣❛r✱ ❡❧❡s ❞❡✈❡♠ ❡s❝♦❧❤❡r s♦♠❡♥t❡ ♦s ♥ú♠❡r♦ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s q✉❡ ♥ã♦ sã♦ ❞✐✈✐sí✈❡✐s ♣♦r três✱ ♣♦r ❡①❡♠♣❧♦ 2,4,5,7, . . .✱ ♦✉ s❡❥❛✱ ♦s
♥ú♠❡r♦s q✉❡ ♥ã♦ sã♦ ♠ú❧t✐♣❧♦s ❞❡ três✳ ❖ ♠♦❞♦ ❞❡ s❡ ♦❜t❡r ♦ r❡s✉❧t❛❞♦ t❛♠❜é♠ ♠✉❞❛✱ ♣♦✐s ❛♣ós ♦ ♠❡❞✐❛❞♦r ♠✉❧t✐♣❧✐❝❛r ♦s ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s ❡❧❡ ❞✐✈✐❞✐rá ❡ss❡ r❡s✉❧t❛❞♦ ♣♦r três ❡ ❛ ❛♥á❧✐s❡ ❞❛ ♣❛r✐❞❛❞❡ s❡rá ❢❡✐t❛ ♥♦ r❡st♦ ❞❡st❛ ❞✐✈✐sã♦✳ ❖✉ s❡❥❛✱ q✉❛♥❞♦ ♦ r❡st♦ ❢♦r ♣❛r ❣❛♥❤❛ ♦ ❥♦❣❛❞♦r q✉❡ ❡s❝♦❧❤❡✉ ❛ ♦♣çã♦ ♣❛r✱ ❝❛s♦ ❝♦♥trár✐♦ ❣❛♥❤❛ ♦ ❥♦❣❛❞♦r q✉❡ ❡s❝♦❧❤❡✉ í♠♣❛r✳ ❆♦ ❡①♣❧✐❝❛r ❡ss❛ ♠✉❞❛♥ç❛✱ ♦ ♣r♦❢❡ss♦r ♣♦❞❡rá ❛❜♦r❞❛r ❛ q✉❡stã♦ ❞❡ ♠ú❧t✐♣❧♦s ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❡ ❛ ❞✐✈✐sã♦ ♣♦r três✱ ❡♠ ❡s♣❡❝✐❛❧ ✉t✐❧✐③❛♥❞♦ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✳ ❆♣ós ❛❧❣✉♠❛s ♣❛rt✐❞❛s ♦ ♣r♦❢❡ss♦r ❞❡✈❡rá s♦❧✐❝✐t❛r q✉❡ ♦s ❛❧✉♥♦s r❡s♣♦♥❞❛♠ à q✉❡stã♦ ✉♠ ❞❛ ❢♦❧❤❛ ❞❡ ❛t✐✈✐❞❛❞❡s ❞❛ ♣á❣✐♥❛ ✷✶✳
◗✉❡stã♦ ✸✳✷✳✶ ❊①✐st❡ ❛❧❣✉♠❛ ❡str❛té❣✐❛ q✉❡ ♣❡r♠✐t❛ ✈❡♥❝❡r s❡♠♣r❡❄ ❈❛s♦ ❡①✐st❛✱ q✉❛❧ é❄
❘❡s♣♦st❛ ❝♦♠❡♥t❛❞❛✿ ❊s♣❡r❛✲s❡ q✉❡ ♦s ❛❧✉♥♦s ✜q✉❡♠ ✐♥q✉✐❡t♦s ❝♦♠ r❡✲ ❧❛çã♦ à ♦❜t❡♥çã♦ ❞❡ss❛ ❡str❛té❣✐❛✱ ♣♦✐s ❞❡ ❢❛t♦ ❡❧❛ ♥ã♦ ❡①✐st❡✱ ♠❛s ♦ ♣r♦❢❡ss♦r ❞❡✈❡rá ❡①♣❧♦r❛r ❡ss❡ ❢❛t♦ ♦✉✈✐♥❞♦ ❡ t❡st❛♥❞♦ ♣♦ssí✈❡✐s ♠❛♥❡✐r❛s s✉❣❡r✐❞❛s✳ ❉✐s✲ ❝✉t✐❞❛s ❡ss❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❡s♣❡r❛✲s❡ q✉❡ ❛ t✉r♠❛ ❡st❡❥❛ ♠♦t✐✈❛❞❛ ❛ ❛❝♦♠♣❛♥❤❛r ❛ ❥✉st✐✜❝❛t✐✈❛ ❢♦r♠❛❧ ❞❛ ❛✉sê♥❝✐❛ ❞❛ ❡str❛té❣✐❛✳ P❛r❛ r❡s♦❧✈❡r ❡ss❛ ♥♦✈❛ s✐t✉❛çã♦ t❡♠♦s q✉❡ ❛♥❛❧✐s❛r ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♣♦r três✳ ◆❡st❡ ✐♥st❛♥t❡ é ♣♦ssí✈❡❧ ♣❛r❛ ♦ ♣r♦❢❡ss♦r r❡❛❧✐③❛r ❛❧❣✉♠❛s ❞✐✈✐sõ❡s ♣♦r três ❝♦♠ ✐♥t✉✐t♦ ❞❡ ❛♥❛❧✐s❛r ♦ r❡st♦ ❞❡ss❛s ❞✐✈✐sõ❡s ❡ ❝♦♥❝❧✉✐r q✉❡ ♣❛r❛ ❡ss❡ r❡st♦ só ❡①✐st❡♠ três ♦♣çõ❡s✱ ♦ ③❡r♦✱ ♦ ✉♠ ♦✉ ♦ ❞♦✐s✳ ❆ss✐♠ ♣♦❞❡✲s❡ r❡♣r❡s❡♥t❛r ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❞✐✈✐❞✐❞♦ ♣♦r três ♥❛ ❢♦r♠❛ 3n+r✱ ❝♦♠ n, r ∈ N ❡ 0 ≤ r ≤ 2✳ ❙✉♣♦♥❤❛ q✉❡ ♦
♥ú♠❡r♦ ❡s❝♦❧❤✐❞♦ ♣❡❧♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r s❡❥❛ ❞❛ ❢♦r♠❛3n+r0✱ ❝♦♠ n, r0 ∈N∗ ❡
0≤r0 ≤2✱ ♦♥❞❡ r0 é ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ♣♦r três ❡ q✉❡ ♦ ♦✉tr♦ ♥ú♠❡r♦ ❡s❝♦❧❤✐❞♦
s❡❥❛ ❞❛ ❢♦r♠❛3k+r1✱ ❝♦♠k, r1 ∈N∗ ❡0≤r1 ≤2♦♥❞❡r1 é ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ♣♦r
três✳ ❈♦♠♦ ❛ s♦❧✉çã♦ é ♦❜t✐❞❛ ♣❡❧❛ ❛♥á❧✐s❡ ❞♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦s ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s ♣♦r três✱ t❡r❡♠♦s q✉❡ ❡❢❡t✉❛r ❡ss❛ ♠✉❧t✐♣❧✐❝❛çã♦✳ ❆ss✐♠ t❡♠♦s✱
(3n+r0).(3k+r1) = 9nk+ 3nr1+ 3kr0 +r1r0 = 3(3nk+nr1 +nr0) +r1r0
❡ ❝♦♠♦3nk+nr1+nr0 ∈N∗✱ t❡♠♦s q✉❡ 3nk+nr1+nr0 =u✳ P♦rt❛♥t♦
(3n+r0).(3k+r1) = 3(3nk+nr1+nr0) +r1r0 = 3u+r1r0
♦✉ s❡❥❛ r1r0 é ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦s ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s ♣♦r
três✳ ❆ss✐♠ ❜❛st❛ ❛♥❛❧✐s❛r ♦s ♣♦ssí✈❡✐s r❡st♦s ❞❛s ❞✐✈✐sõ❡s ♣♦r três ❞❡ ❝❛❞❛ ✉♠ ❞♦s ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s ❡ ♠✉❧t✐♣❧✐❝❛r ✉♠ ♣❡❧♦ ♦✉tr♦ ♣❛r❛ ♦❜t❡r ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦✱ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ss❡s ♥ú♠❡r♦s✱ ♣♦r três✳ ▲♦❣♦ ✐r❡♠♦s ❛♥❛❧✐s❛r t♦❞♦s ♦s ❝❛s♦s ♣♦ssí✈❡✐s ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ t❛❜❡❧❛✳ ▲❡♠❜r❛♥❞♦ q✉❡ ♥ã♦ é ♣♦ss✐✈❡❧ ♣❛r❛ ♥❡♥❤✉♠ ❞♦s ❥♦❣❛❞♦r❡s ❡s❝♦❧❤❡r ♥ú♠❡r♦s ❞✐✈✐sí✈❡✐s ♣♦r três✱ ♦✉ s❡❥❛✱ ♥ã♦ t❡r❡♠♦s ♦ r❡st♦ ✐❣✉❛❧ ❛ ③❡r♦ ♣❛r❛ ♥❡♥❤✉♠ ❞♦s ❥♦❣❛❞♦r❡s✳
❘❡st♦s ❞♦1◦ ❥♦❣❛❞♦r ❘❡st♦s ❞♦2◦ ❥♦❣❛❞♦r ▼✉❧t✐♣❧✐❝❛çã♦ ❞♦s r❡st♦s P❛r✐❞❛❞❡
✶ ✶ ✶✳✶ ❂ ✶ í♠♣❛r
✶ ✷ ✶✳✷ ❂ ✷ ♣❛r
✷ ✶ ✷✳✶ ❂ ✷ ♣❛r
✷ ✷ ✷✳✷ ❂ ✹ ❂ ✶ í♠♣❛r
◆❡st❡ ♠♦♠❡♥t♦ ❝❛❜❡ ❛♦ ♣r♦❢❡ss♦r ❡①❡♠♣❧✐✜❝❛r ❡ss❛ s✐t✉❛çã♦ ❝♦♠ ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s ♣❡❧♦s ❛❧✉♥♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛ ú❧t✐♠❛ ❧✐♥❤❛ ❞❛ t❛❜❡❧❛ ♦♥❞❡ ❛ ♠ú❧t✐✲ ♣❧✐❝❛çã♦ ❞♦s r❡st♦s r❡s✉❧t❛ ❡♠ q✉❛tr♦✳ ❆ ú❧t✐♠❛ ❧✐♥❤❛ ❞❛ t❛❜❡❧❛ ♠❡r❡❝❡ ✉♠❛ ❛t❡♥çã♦ ❡s♣❡❝✐❛❧✱ ♣♦✐s ♦ ♥ú♠❡r♦ r❡s✉❧t❛♥t❡ ✭r❡st♦✮ s❡rá ♠❛✐♦r ❞♦ q✉❡ ♦ ❞✐✈✐s♦r✳ ❯♠❛ ❡str❛té❣✐❛ ❝♦♥s✐st❡ ❡♠ ✈♦❧t❛r à ❝♦♥❝❧✉sã♦✱ ❢❡✐t❛ ❛❝✐♠❛✱ q✉❡ ♥♦s ♣❡r♠✐t✐✉ ❛ ❝♦♥str✉çã♦ ❞❛ t❛❜❡❧❛ ❡ ❧❡♠❜r❛r q✉❡ ❝❛s♦ r1.r0 ❢♦r ♠❛✐♦r q✉❡ três✱ ❡♥tã♦ ❡❧❡ ♣♦❞❡
s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ três✱ ♦✉ s❡❥❛ ♥❛ ❢♦r♠❛3y+r✱ ❝♦♠y∈N∗✳ ❆ss✐♠
❜❛st❛ ❞✐✈✐❞✐r ♦ ♥ú♠❡r♦ ♦❜t✐❞♦ ♣♦r três ❡ ❛♥❛❧✐s❛r ♦ r❡st♦✳ ◆♦ ❝❛s♦ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡4 ♣♦r3 é ♦ ♥ú♠❡r♦ 1✳
❖✉tr❛ ✐❞❡✐❛ é r❡t✐r❛r q✉❛♥t♦s ♥ú♠❡r♦s três ❢♦r❡♠ ♣♦ssí✈❡✐s ❞♦ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ ❡ ❛♥❛❧✐s❛r ♦ q✉❡ ✧s♦❜r❛✧✳ ❉❡ ❢❛t♦ 4−1.3 = 1✳ ❆ss✐♠ ❢♦✐ ♣♦ssí✈❡❧ r❡t✐r❛r ✉♠❛ ✈❡③ ♦ ♥ú♠❡r♦ três ❡ ❛✐♥❞❛ s♦❜r♦✉ ✉♠✳ ❆♥❛❧✐s❛♥❞♦ ❛ t❛❜❡❧❛ ♣♦❞❡✲s❡ ❝♦♥❝❧✉✐r q✉❡ ♥ã♦ ❡①✐st❡ ✉♠❛ ❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛✱ ♣♦✐s ♦s ♣❛rt✐❝✐♣❛♥t❡s ♥ã♦ ♣♦ss✉❡♠ ❛❝❡ss♦ ❛♦ ♥ú♠❡r♦ ❞❡ s❡✉ ❛❞✈❡rsár✐♦✳
P♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ ❥♦❣♦ é ❥✉st♦✱ ♣♦rq✉❡ ❡①✐st❡♠ ❛s ♠❡s♠❛s ❝❤❛♥❝❡s ❞❡ ✈❡♥✲ ❝❡r ♣❛r❛ ❝❛❞❛ ❡s❝♦❧❤❛✳ ➱ ✐♠♣♦rt❛♥t❡ q✉❡ ♦ ♣r♦❢❡ss♦r t❡st❡ ❡ ✐♥st✐❣✉❡ ♦s ❛❧✉♥♦s ❛ ♠♦❞✐✜❝❛r ♦ ❥♦❣♦ ❞❡ ♠❛✐s ♠❛♥❡✐r❛s ♦✉ ❛té ❛ ❝r✐❛r s❡✉s ♣ró♣r✐♦s ❥♦❣♦s ✉t✐❧✐③❛♥❞♦ r❡st♦s✱ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡ ♣❛r✐❞❛❞❡✳ P♦r ❡①❡♠♣❧♦✱ s❡rá q✉❡ ♦ ❥♦❣♦✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ❞✐✈✐s♦r ♥❛t✉r❛❧ ❡s❝♦❧❤✐❞♦✱ é s❡♠♣r❡ ❥✉st♦❄ ❖ ♣r♦❢❡ss♦r ❞❡✈❡ ❡st❛r ❝✐❡♥t❡ ❞❡ q✉❡ ❛ s♦❧✉çã♦ ♣r♦♣♦st❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛♣ós ❛ ♠♦❞✐✜❝❛çã♦ ❞♦ ❥♦❣♦✱ ❝♦♥st✐t✉✐ ❡♠ ✉♠❛ ❛♥á❧✐s❡ ❞❡ ❝❛s♦s ♣❛r❛ ❞❡t❡r♠✐♥❛r t♦❞♦s ♦s r❡st♦s ♣♦ssí✈❡✐s ❞❛ ❞✐✈✐sã♦ ❛♥❛❧✐s❛❞❛✳ ❊ss❡ ♠ét♦❞♦ é ♠✉✐t♦ ✐♠♣♦rt❛♥t❡✱ ♣♦✐s ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ❡♠ ♦✉tr❛s ár❡❛s ❛❧é♠ ❞❛ ❛r✐t♠ét✐❝❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❡♠ ❛♥á❧✐s❡ ❝♦♠❜✐♥❛tór✐❛✳ P♦r ❡ss❡ ♠♦t✐✈♦ é ✐♠✲ ♣♦rt❛♥t❡ q✉❡ ♦s ❛❧✉♥♦s ❡♥t❡♥❞❛♠ q✉❡ ❡ss❛ s♦❧✉çã♦ é s❛t✐s❢❛tór✐❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✳ ❆t✐✈✐❞❛❞❡ ✸✳✸ ❙❡①t❛✲❢❡✐r❛ 13
❊ss❛ ❛t✐✈✐❞❛❞❡ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♠♦str❛r ❞❡ ❢♦r♠❛ ❧ú❞✐❝❛ ❝♦♠♦ ❛ ❛♥á❧✐s❡ ❞❡ r❡st♦s ❞❡ ❞✐✈✐sõ❡s ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛ ♥❛s ♠❛✐s
❞✐✈❡rs❛s s✐t✉❛çõ❡s✳ P❛r❛ ❡st❛ ❛t✐✈✐❞❛❞❡ ✈❛♠♦s ♣r❡❝✐s❛r ❞❛ ❢♦❧❤❛ ❞❡ q✉❡stõ❡s q✉❡ ❡♥❝♦♥tr❛✲s❡ ♥♦ ❛♥❡①♦ ❡ ✉♠ ❝❛❧❡♥❞ár✐♦ ♣♦r ❡q✉✐♣❡✳
❉❡s❝r✐çã♦ ❞❛ ❛t✐✈✐❞❛❞❡✿ ❖ ♣r♦❜❧❡♠❛ ❛ s❡❣✉✐r é ✉♠ t❛♥t♦ ❝✉r✐♦s♦ ❡ ♣♦❞❡ s❡r ♣r♦♣♦st♦ ❛♦s ❛❧✉♥♦s ❝♦♠♦ ✉♠ ❞❡s❛✜♦✳ ❖ ♣r♦❢❡ss♦r ❞❡✈❡rá ❞✐✈✐❞✐r ❛ s❛❧❛ ❡♠ ❣r✉♣♦s ❞❡ q✉❛tr♦ ♣❡ss♦❛s ❡ ♣❛r❛ ❝❛❞❛ ❣r✉♣♦ ❞❡✈❡rá s❡r ❡♥tr❡❣✉❡ ✉♠ ❝❛❧❡♥❞ár✐♦✱ ♣r❡❢❡r❡♥❝✐❛❧♠❡♥t❡ ♦s ❝❛❧❡♥❞ár✐♦s ❞❡✈❡♠ s❡r ❞❡ ❛♥♦s ❞✐❢❡r❡♥t❡s✳
◗✉❡stã♦ ✸✳✸✳✶ ◗✉❛❧ é ♦ ♥ú♠❡r♦ ♠á①✐♠♦ ❞❡ s❡①t❛✲❢❡✐r❛s tr❡③❡ q✉❡ ♣♦❞❡♠ ♦❝♦rr❡r ♥✉♠ ❛♥♦ q✉❡ ♥ã♦ é ❜✐ss❡①t♦❄ ◆❡st❡ ❝❛s♦✱ ❡♠ q✉❡ ❞✐❛ ❞❛ s❡♠❛♥❛ ❝❛✐ ♦ ❞é❝✐♠♦ ❞✐❛ ❞♦ ❛♥♦❄
➱ ✐♠♣♦rt❛♥t❡ q✉❡ ♦ ♣r♦❢❡ss♦r r❡ss❛❧t❡ q✉❡ ♦ ♣r♦❜❧❡♠❛ ♥ã♦ ❞❡✜♥❡ ✉♠ ❛♥♦ ❡s♣❡❝í✜❝♦✱ ♠❛s s✐♠ q✉❡ s❡❥❛ ✉♠ ❛♥♦ q✉❡ ♥ã♦ é ❜✐ss❡①t♦✳ ❆ s❡❣✉✐r ❡st✉❞❛r❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳
❘❡s✉❧t❛❞♦ ✸✳✶ ▼♦♥t❡ ✉♠❛ t❛❜❡❧❛ ❡s♣❡❝✐✜❝❛♥❞♦ q✉❛♥t♦s ❞✐❛s ♣♦ss✉❡♠ ❝❛❞❛ ♠ês ❞♦ ❛♥♦✳ ❊♠ s❡❣✉✐❞❛ ❞❡t❡r♠✐♥❡ q✉❛♥t♦s ❞✐❛s ❞♦ ♠ês ❞❡ ♠❛rç♦ s❡ ♣❛ss❛r❛♠ ❛té ♦ ❞✐❛11 ❞❡ ♠❛rç♦ ❡ q✉❛♥t♦s ❞✐❛s ❢❛❧t❛♠ ♣❛r❛ t❡r♠✐♥❛r ♦ ♠ês ❞❡ ♠❛rç♦ t❡♥❞♦ ❝♦♠♦ ❜❛s❡ ♦ ❞✐❛ 11 ❞❡ ♠❛rç♦✳
❚❛❜❡❧❛ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❞✐❛s ❞❡ ❝❛❞❛ ♠ês
▼ês ❚♦t❛❧ ❞❡ ❞✐❛s
❏❛♥❡✐r♦ ✸✶
❋❡✈❡r❡✐r♦ ✷✽ ♦✉ ✷✾ s❡ ❢♦r ❛♥♦ ❜✐ss❡①t♦
▼❛rç♦ ✸✶
❆❜r✐❧ ✸✵
▼❛✐♦ ✸✶
❏✉♥❤♦ ✸✵
❏✉❧❤♦ ✸✶
❆❣♦st♦ ✸✶
❙❡t❡♠❜r♦ ✸✵
❖✉t✉❜r♦ ✸✶
◆♦✈❡♠❜r♦ ✸✵
❉❡③❡♠❜r♦ ✸✶
❚❡♠♦s 11❞✐❛s ❞♦ ♠ês ❞❡ ♠❛rç♦ ❛té ♦ ❞✐❛ 11 ❞❡ss❡ ♠ês ❡ ❢❛❧t❛♠ 20❞✐❛s ♣❛r❛ ❛❝❛❜❛r ♦ ♠❡s♠♦ ♠ês t❡♥❞♦ ❝♦♠♦ ❜❛s❡ ♦ ❞✐❛11 ❞❡ ♠❛rç♦✳
❈♦♠❡♥tár✐♦ ❘❡s✉❧t❛❞♦ ✸✳✶✿ ➱ ♣r❡❝✐s♦ s❛❜❡r q✉❛♥t♦s ❞✐❛s ♣♦ss✉❡♠ ❝❛❞❛ ♠ês ♥♦ ❛♥♦ ♣❛r❛ q✉❡ ♦ ♣r♦❜❧❡♠❛ s❡❥❛ r❡s♣♦♥❞✐❞♦✳ ❆❧é♠ ❞✐ss♦✱ ♦s ❛❧✉♥♦s ❞❡✈❡♠ ♣❡r❝❡❜❡r q✉❡ s❡ ❢❛❧❛r♠♦s ❞♦ ❞✐❛ 11 ❞❡ ♠❛rç♦✱ ✐ss♦ s✐❣♥✐✜❝❛ q✉❡ s❡ ♣❛ss❛r❛♠ 11 ❞✐❛s ❞♦ ♠ês ❞❡ ♠❛rç♦ ❡ ❢❛❧t❛♠ 20 ❞✐❛s ♣❛r❛ ❛❝❛❜❛r ❡ss❡ ♠ês✱ ♣♦✐s ♠❛rç♦ ♣♦ss✉✐ 31❞✐❛s✳ ❆♣ós t❡r❡♠ ❞❡❜❛t✐❞♦ ❡ s♦❧✉❝✐♦♥❛❞♦ ♦ ♣r♦❜❧❡♠❛ ♦ ♣r♦❢❡ss♦r ❞❡✈❡ ♣r♦♣♦r ❛♦s ❛❧✉♥♦s q✉❡ r❡s♣♦♥❞❛♠ ❛♦ ❘❡s✉❧t❛❞♦ ❞♦✐s ❞❛ ❢♦❧❤❛ ❞❡ ❛t✐✈✐❞❛❞❡s✳
❘❡s✉❧t❛❞♦ ✸✳✷ ❙❛❜❡♥❞♦ q✉❡ ♦ ❞✐❛2❞❡ ♠❛rç♦ é ✉♠❛ q✉✐♥t❛✲❢❡✐r❛ ❞❡t❡r♠✐♥❡ q✉❛✐s ♦✉tr♦s ❞✐❛s ❞♦ ♠ês ❞❡ ♠❛rç♦ ♦❝♦rr❡♠ ❡♠ ✉♠❛ q✉✐♥t❛✲❢❡✐r❛✳ ❊♠ s❡❣✉✐❞❛ ❞❡t❡r♠✐♥❡ ✉♠❛ ❡str❛té❣✐❛ q✉❡ ♣❡r♠✐t❛ ❞❡t❡r♠✐♥❛r ❡sss❛ s✐t✉❛çã♦ s❡♠ ✉s❛r ♦ ❝❛❧❡♥❞ár✐♦✳
❈♦♠❡♥tár✐♦ ❘❡s✉❧t❛❞♦ ✸✳✷✿ ◆♦t❡ q✉❡✱ s❡ ♦ ❞✐❛ 02❞❡ ♠❛rç♦ é ✉♠❛ q✉✐♥t❛✲ ❢❡✐r❛✱ ❡♥tã♦ ❞✐❛ 09 ❞❡ ♠❛rç♦✱ ❞✐❛ 16 ❞❡ ♠❛rç♦✱ ❞✐❛ 23 ❞❡ ♠❛rç♦ ❡ 30 ❞❡ ♠❛rç♦ s❡rã♦ t♦❞♦s ♥✉♠❛ q✉✐♥t❛✲❢❡✐r❛✳ ◆♦t❡ q✉❡ 9 = 2 + 7✱ 16 = 9 + 7✱ 23 = 16 + 7✱ 30 = 23+7✳ P♦rt❛♥t♦✱ ❜❛st❛ s♦♠❛r♠♦s s❡t❡ ❛♦ ❞✐❛ ❡s❝♦❧❤✐❞♦ ♣❛r❛ ♦❜t❡r ♦ ♣ró①✐♠♦ ❞✐❛ q✉❡ s❡rá✱ ♣♦r ❡①❡♠♣❧♦✱ ✉♠❛ q✉✐♥t❛✲❢❡✐r❛✳ ❊ss❡ ♣r♦❝❡ss♦ é ✈á❧✐❞♦ ♣❛r❛ q✉❛❧q✉❡r ❞✐❛ ❞❛ s❡♠❛♥❛✳ ❆♣ós✱ ♦s ❛❧✉♥♦s ❞❡✈❡♠ r❡s♣♦♥❞❡r ❛♦ r❡s✉❧t❛❞♦ três ❞❛ ❢♦❧❤❛ ❞❡ ❛t✐✈✐❞❛❞❡s✳
❘❡s✉❧t❛❞♦ ✸✳✸ ❉❡t❡r♠✐♥❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ❞✐❛ 11 ❞❡ ♠❛rç♦ ❡ ♦ ❞✐❛ 11 ❞❡ ❛❜r✐❧✱ ♦✉ s❡❥❛✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❞✐❛s q✉❡ ❡①✐st❡ ❡♥tr❡ 11 ❞❡ ♠❛rç♦ ❡ 11 ❞❡ ❛❜r✐❧ ✐♥❝❧✉s✐✈❡✳
❈♦♠❡♥tár✐♦ ❞❛ ❘❡s✉❧t❛❞♦ ✸✳✸✿ ❊♥tr❡ ♦s ❞✐❛s 11 ❞❡ ♠❛rç♦ ❡ 11 ❞❡ ❛❜r✐❧ ✐♥❝❧✉s✐✈❡✱ t❡♠♦s 31❞✐❛s✳
Pr✐♠❡✐r♦✿ ❊s❝♦❧❤✐❞♦ ♦ ❞✐❛ ❞♦ ♠ês✱ ✈❛♠♦s ❞❡t❡r♠✐♥❛r ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡ss❡ ❞✐❛ ❡ ♦ ú❧t✐♠♦ ❞✐❛ ❞♦ ♠ês✳ P♦r ❡①❡♠♣❧♦✱ ❡s❝♦❧❤✐❞♦ ♦ ❞✐❛11 ❞❡ ♠❛rç♦ t❡♠♦s q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡ss❛ ❞❛t❛ ❡ ♦ ú❧t✐♠♦ ❞✐❛ ❞♦ ♠ês ❞❡ ♠❛rç♦ é ✐❣✉❛❧ ❛ 20✱ ♣♦✐s 31−11 = 20✳
❙❡❣✉♥❞♦✿ ❙♦♠❛♠♦s ❛ ❞✐❢❡r❡♥ç❛ ♦❜t✐❞❛ ❛♥t❡r✐♦r♠❡♥t❡ ❛♦ ❞✐❛ ❡s❝♦❧❤✐❞♦ ❡ ❛ss✐♠ ♦❜t❡r❡♠♦s ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❞✐❛ ❞❡ ✉♠ ♠ês ❡ ❡ss❡ ♠❡s♠♦ ❞✐❛ ❞♦ ♠ês s❡❣✉✐♥t❡ ✐♥❝❧✉s✐✈❡✳ ❘❡t♦♠❛♥❞♦✱ t❡♠♦s q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ 11 ❞❡ ♠❛rç♦ ❡ 11❞❡ ❛❜r✐❧ é 31✱ ♣♦✐s20 + 11 = 31✳
❆♣ós ♦s r❡s✉❧t❛❞♦s t❡r❡♠ s✐❞♦ ❛♣r❡s❡♥t❛❞♦s ❡ ❞✐s❝✉t✐❞♦s ❡♠ s❛❧❛ ♦ ♣r♦❢❡ss♦r ♣♦❞❡ ♦r✐❡♥t❛r ♦s ❛❧✉♥♦s ❛ r❡s♣♦♥❞❡r❡♠ à q✉❡stã♦ ❞♦✐s ❞❛ ❢♦❧❤❛ ❞❡ ❛t✐✈✐❞❛❞❡s✳ ◗✉❡stã♦ ✸✳✸✳✷ ❈♦♠ ❜❛s❡ ♥♦ r❡s✉❧t❛❞♦ três✱ ❝♦♥str✉❛ ✉♠❛ t❛❜❡❧❛ q✉❡ ✐♥❞✐q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ❞✐❛s 13 ❞❡ ❝❛❞❛ ♠ês✳
❘❡s♣♦st❛ ❝♦♠❡♥t❛❞❛✿ ❈♦♥❢♦r♠❡ ♦ r❡s✉❧t❛❞♦ três✱ t❡♠♦s ❛ t❛❜❡❧❛ ❛❜❛✐①♦ ❝♦♠ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ❞✐❛s13❞❡ ❝❛❞❛ ♠ês✳
✶✸✴✵✶ ✶✸✴✵✷ ✶✸✴✵✸ ✶✸✴✵✹ ✶✸✴✵✺ ✶✸✴✵✻ ✶✸✴✵✼ ✶✸✴✵✽ ✶✸✴✵✾ ✶✸✴✶✵ ✶✸✴✶✶
✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲
✶✸✴✵✷ ✶✸✴✵✸ ✶✸✴✵✹ ✶✸✴✵✺ ✶✸✴✵✻ ✶✸✴✵✼ ✶✸✴✵✽ ✶✸✴✵✾ ✶✸✴✶✵ ✶✸✴✶✶ ✶✸✴✶✷
✸✶ ✷✽ ✸✶ ✸✵ ✸✶ ✸✵ ✸✶ ✸✶ ✸✵ ✸✶ ✸✵
❆♣ós ❛ ❝♦♥str✉çã♦ ❞❛ t❛❜❡❧❛ ♦ ♣r♦❢❡ss♦r ❞❡✈❡ ♦r✐❡♥t❛r ♦s ❛❧✉♥♦s ❛ r❡s♣♦♥❞❡r❡♠ ❛ q✉❡stã♦ ❞♦✐s ❞❛ ❢♦❧❤❛ ❞❡ ❛t✐✈✐❞❛❞❡s✳
◗✉❡stã♦ ✸✳✸✳✸ ❯t✐❧✐③❛♥❞♦ ♦ r❡s✉❧t❛❞♦ 3.2 ❞❡t❡r♠✐♥❡ q✉❛✐s ❞✐❛s tr❡③❡ ♦❝♦rr❡♠ ♥✉♠ ♠❡s♠♦ ❞✐❛ ❞❛ s❡♠❛♥❛✳
❘❡s♣♦st❛ ❝♦♠❡♥t❛❞❛✿ ❆♣❧✐❝❛♥❞♦ ♦ ❘❡s✉❧t❛❞♦ 3.2♥❛ t❛❜❡❧❛ ✉♠✱ t❡♠♦s q✉❡ s❡ ❞✐✈✐❞✐r♠♦s ❛ ❞✐❢❡r❡♥ç❛ ♦❜t✐❞❛ ♣♦r7✱ t❡r❡♠♦s ❝♦♠♦ s❛❜❡r s❡ ♦s ❞✐❛s13♦❝♦rr❡♠ ♥♦ ♠❡s♠♦ ❞✐❛ ❞❛ s❡♠❛♥❛✳ ❆ss✐♠ t❡♠♦s ❛ t❛❜❡❧❛✳
✶✸✴✵✶ ✶✸✴✵✷ ✶✸✴✵✸ ✶✸✴✵✹ ✶✸✴✵✺ ✶✸✴✵✻ ✶✸✴✵✼ ✶✸✴✵✽ ✶✸✴✵✾ ✶✸✴✶✵ ✶✸✴✶✶
✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲
✶✸✴✵✷ ✶✸✴✵✸ ✶✸✴✵✹ ✶✸✴✵✺ ✶✸✴✵✻ ✶✸✴✵✼ ✶✸✴✵✽ ✶✸✴✵✾ ✶✸✴✶✵ ✶✸✴✶✶ ✶✸✴✶✷
✸✶ ✷✽ ✸✶ ✸✵ ✸✶ ✸✵ ✸✶ ✸✶ ✸✵ ✸✶ ✸✵
✸ ✵ ✸ ✷ ✸ ✷ ✸ ✸ ✷ ✸ ✷
❆♥❛❧✐s❛♥❞♦ ♦ r❡st♦ ❞❛s ❞✐✈✐sõ❡s✱ t❡♠♦s q✉❡ ❞✐❛13❞❡ ❢❡✈❡r❡✐r♦ ❡13❞❡ ♠❛rç♦ ♦❝♦rr❡♠
♥♦ ♠❡s♠♦ ❞✐❛ ❞❛ s❡♠❛♥❛✳ ➱ ❝♦♥✈❡♥✐❡♥t❡ q✉❡ ♦s ❞✐❛s q✉❡ ♦❝♦rr❡r❛♠ ♥✉♠ ♠❡s♠♦ ❞✐❛ ❞❛ s❡♠❛♥❛ s❡❥❛♠ ♥❛ s❡①t❛✲❢❡✐r❛✳ ❆ss✐♠ só r❡st❛ s❛❜❡r s❡ ❡①✐st❡ ♠❛✐s ❛❧❣✉♠ ♠ês q✉❡ ♣♦ss✉❛ ♦ ❞✐❛13 ♥❛ s❡①t❛✲❢❡✐r❛✳ P❛r❛ ✐ss♦ ♦ ♣r♦❢❡ss♦r ❞❡✈❡ ♦r✐❡♥t❛r ♦s ❛❧✉♥♦s ❛ r❡s♣♦♥❞❡r❡♠ ❛
q✉❡stã♦ três✳
◗✉❡stã♦ ✸✳✸✳✹ ❈♦♥str✉❛ ✉♠❛ t❛❜❡❧❛ ✐♥❞✐❝❛♥❞♦ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ❞✐❛s tr❡③❡✱ t❡♥❞♦ ❝♦♠♦ ❜❛s❡ ♦ ❞✐❛13 ❞❡ ❢❡✈❡r❡✐r♦✳ ❆♣ós✱ ❛♣❧✐q✉❡ ♦ r❡s✉❧t❛❞♦ 3.2❡ ✈❡r✐✜q✉❡ q✉❛✐s ❞✐❛s 13
❝❛❡♠ ♥♦ ♠❡s♠♦ ❞✐❛ ❞❛ s❡♠❛♥❛ q✉❡ ♦ ❞✐❛ 13 ❞❡ ❢❡✈❡r❡✐r♦✳
❊①❡♠♣❧♦ ✸✳✶ ❇❛st❛ ❛♥❛❧✐s❛r ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ❞✐❛13❞❡ ❢❡✈❡r❡✐r♦
❡ ♦ ❞✐❛ 13 ❞♦ ♠ês q✉❡ q✉❡r❡♠♦s✳ ❆ss✐♠✱ s❡ q✉❡r❡♠♦s ❛♥❛❧✐s❛r ♦ ❞✐❛ 13 ❞❡ ♠❛✐♦✱ ❜❛st❛
s♦♠❛r28+31+30 = 89❡ ✈❡r✐✜❝❛r ♦ r❡st♦ ❞♦ ♥ú♠❡r♦ ♦❜t✐❞♦ ♥❛ ❞✐✈✐sã♦ ♣♦r7✉t✐❧✐③❛♥❞♦✲
s❡ ❞♦ r❡s✉❧t❛❞♦3.2✳
✶✸✴✵✶ ✶✸✴✵✷ ✶✸✴✵✸ ✶✸✴✵✹ ✶✸✴✵✺ ✶✸✴✵✻ ✶✸✴✵✼ ✶✸✴✵✽ ✶✸✴✵✾ ✶✸✴✶✵ ✶✸✴✶✶
✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲
✶✸✴✵✷ ✶✸✴✵✸ ✶✸✴✵✹ ✶✸✴✵✺ ✶✸✴✵✻ ✶✸✴✵✼ ✶✸✴✵✽ ✶✸✴✵✾ ✶✸✴✶✵ ✶✸✴✶✶ ✶✸✴✶✷
✲ ✷✽ ✺✾ ✽✾ ✶✷✵ ✶✺✵ ✶✽✶ ✷✶✷ ✷✹✷ ✷✼✸ ✸✵✸
✲ ✵ ✸ ✺ ✶ ✸ ✻ ✷ ✹ ✵ ✷
P❡❧❛ t❛❜❡❧❛ ❛❝✐♠❛ ❝❤❡❣❛♠♦s ❛ ❝♦♥❝❧✉sã♦ ❞❡ q✉❡ ❡①✐st❡ ♠❛✐s ✉♠❛ s❡①t❛✲❢❡✐r❛13✱ q✉❡
♦❝♦rr❡ ♥♦ ❞✐❛ 13 ❞❡ ♥♦✈❡♠❜r♦✳ ▲♦❣♦ t❡♠♦s ♥✉♠ ♠❡s♠♦ ❛♥♦✱ ♥ã♦ ❜✐ss❡①t♦✱ ✉♠ t♦t❛❧
❞❡✱ ♥♦ ♠á①✐♠♦✱ três s❡①t❛s✲❢❡✐r❛s13✳ ❙❛❜❡♥❞♦ q✉❡ ♦ ❞✐❛13 ❞❡ ❢❡✈❡r❡✐r♦ ♦❝♦rr❡ ❡♠ ✉♠❛
s❡①t❛✲❢❡✐r❛✱ t❡♠♦s ❝♦♥❞✐çõ❡s ❞❡ ❞❡s❝♦❜r✐r ❡♠ q✉❛❧ ❞✐❛ ❞❛ s❡♠❛♥❛ ♦ ❞é❝✐♠♦ ❞✐❛ ❞♦ ❛♥♦ ♦❝♦rr❡✳ P❛r❛ ✐ss♦ ♦ ❣r✉♣♦ ❞❡✈❡rá r❡s♣♦♥❞❡r à q✉❡stã♦ q✉❛tr♦✳
◗✉❡stã♦ ✸✳✸✳✺ ❙❡ ♦ ❞✐❛ 13 ❞❡ ❢❡✈❡r❡✐r♦ ♦❝♦rr❡ ❡♠ ✉♠❛ s❡①t❛✲❢❡✐r❛✱ ❡♠ q✉❡ ❞✐❛ ❞❛
s❡♠❛♥❛ ♦❝♦rr❡ ♦ ❞✐❛ 10 ❞❡ ❥❛♥❡✐r♦❄
❘❡s♣♦st❛ ❝♦♠❡♥t❛❞❛✿ ❊♠ ✉♠ sá❜❛❞♦✳ ❇❛st❛ ✈♦❧t❛r♠♦s à t❛❜❡❧❛ ❞❛ ♣á❣✐♥❛ ✶✵ ❡ ♦❧❤❛r♠♦s ❛ ♣r✐♠❡✐r❛ ❝♦❧✉♥❛✳ ❙❡ ❞✐❛ 13 ♦❝♦rr❡ ♥✉♠❛ s❡①t❛ ❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❛
❞✐❢❡r❡♥ç❛ ❡♥tr❡13❞❡ ❥❛♥❡✐r♦ ❡ 13 ❢❡✈❡r❡✐r♦ é ✐❣✉❛❧ ❛3✱ t❡♠♦s q✉❡ ♦ ❞✐❛13 ❞❡ ❢❡✈❡r❡✐r♦✱
♣❛ss♦✉ três ❞✐❛s ❞❛ s❡♠❛♥❛ ❞♦ ❞✐❛ 13 ❞❡ ❥❛♥❡✐r♦✱ ♦✉ s❡❥❛✱ ♦ ❞✐❛ 13 ❞❡ ❥❛♥❡✐r♦ ♦❝♦rr❡
♥✉♠❛ t❡rç❛✲❢❡✐r❛✳ ▲♦❣♦✱ ♦ ❞✐❛10❞❡ ❥❛♥❡✐r♦✱ ♦ ❞é❝✐♠♦ ❞✐❛ ❞♦ ❛♥♦✱ ♦❝♦rr❡ ❡♠ ✉♠ sá❜❛❞♦✳
❆t✐✈✐❞❛❞❡ ✸✳✹ ❖ ❥♦❣♦ ❞❡ ❊✉❝❧✐❞❡s
❉❡s❝r✐çã♦ ❞❛ ❛t✐✈✐❞❛❞❡✿ ❙ã♦ ❞♦✐s ❥♦❣❛❞♦r❡s✱ ❝❛❞❛ ✉♠ ❡s❝♦❧❤❡✱ s❡❝r❡t❛♠❡♥t❡✱ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♥ã♦ ♥✉❧♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ✉♠ ❥♦❣❛❞♦r ❡s❝♦❧❤❡✉ ♦ ♥ú♠❡r♦31✱ ❡ ♦ ♦✉tr♦✱
♦ ♥ú♠❡r♦ 7✳ ❯♠ ❞♦s ❥♦❣❛❞♦r❡s é s♦rt❡❛❞♦ ♣❛r❛ ✐♥✐❝✐❛r ♦ ❥♦❣♦✳ ❊❧❡ r❡❝❡❜❡rá ♦ ♥ú♠❡r♦
❡s❝♦❧❤✐❞♦ ♣❡❧♦ ❝♦❧❡❣❛ ❡ ❞❡✈❡rá s✉❜tr❛✐r ❞♦ ♠❛✐♦r ♥ú♠❡r♦✱31✱ ✉♠ ♠ú❧t✐♣❧♦ ♥ã♦ ♥✉❧♦ ❞♦
♠❡♥♦r✱(n7 = 7,14,21 ou 28) ❞❡ ♠♦❞♦ q✉❡ ♦ r❡s✉❧t❛❞♦ ❛✐♥❞❛ s❡❥❛ ♣♦s✐t✐✈♦✳ ❖ s❡❣✉♥❞♦
❥♦❣❛❞♦r r❡❝❡❜❡rá ♦ ♥♦✈♦ ♣❛r ❞❡ ♥ú♠❡r♦s31−n7✱7❡ r❡♣❡t✐rá ♦ ♣r♦❝❡ss♦✱ s✉❜tr❛✐♥❞♦ ❞♦
♠❛✐♦r ♥ú♠❡r♦ ✉♠ ♠ú❧t♣❧♦ ❞♦ ♠❡♥♦r✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ●❛♥❤❛rá ♦ ❥♦❣♦ q✉❡♠ ♦❜t✐✈❡r ♣r✐♠❡✐r♦ ♦ ♥ú♠❡r♦0❡♠ ✉♠ ♣❛r✳
❊①❡♠♣❧♦ ✸✳✷ ❱❛♠♦s s✐♠✉❧❛r ✉♠❛ ♣❛rt✐❞❛ ❝♦♠ ♦s ♥ú♠❡r♦s ♠❡♥❝✐♦♥❛❞♦s ♥❛ ❞❡s❝r✐çã♦ ❞♦ ❥♦❣♦✱ ♦✉ s❡❥❛✱31 ❡ 7✳
❖ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r t❡rá ✈ár✐❛s ♦♣çõ❡s ❞❡ ❥♦❣♦✿ [24,7]✱[17,7]✱[10,7]✱[3,7]✳ ❙✉♣♦♥❤❛✲
♠♦s q✉❡ ❡s❝♦❧❤❛[10,7]✳ ◆❡st❡ ❝❛s♦✱ ♦ s❡❣✉♥❞♦ ❥♦❣❛❞♦r só t❡rá ✉♠❛ ❛❧t❡r♥❛t✐✈❛✿ [3,7]✳
❙❡rá ❛ ✈❡③✱ ♥♦✈❛♠❡♥t❡✱ ❞♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r q✉❡ ♣♦❞❡rá ❡s❝♦❧❤❡r[3,4]♦✉[3,1]✳ ❙❡ ❥♦❣❛r [3,4]✱ ♦ s❡❣✉♥❞♦ ❥♦❣❛❞♦r s❡rá ♦❜r✐❣❛❞♦ ❛ ❥♦❣❛r [3,1] ❡✱ ♥❛ ❥♦❣❛❞❛ s❡❣✉✐♥t❡✱ ♦ ♣r✐♠❡✐r♦
❥♦❣❛❞♦r ❥♦❣❛rá[1,0]❡ s❡rá ♦ ✈❡♥❝❡❞♦r✳
❊①❡♠♣❧♦ ✸✳✸ ❙✉♣♦♥❤❛♠♦s q✉❡ ♦s ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s t❡♥❤❛♠ s✐❞♦ 49 ❡ 5✳
❖ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r t❡rá ✈ár✐❛s ♦♣çõ❡s ❞❡ ❥♦❣♦✿ [4,5]✱ [9,5]✱ [14,5]✱ [19,5]✱ [24,5]✱ [29,5]✱ [34,5]✱ [39,5]✱ [44,5]✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡s❝♦❧❤❛ [4,5]✳ ◆❡st❡ ❝❛s♦✱ ♦ s❡❣✉♥❞♦
❥♦❣❛❞♦r ♣♦❞❡rá t❡r só ✉♠❛ ❛❧t❡r♥❛t✐✈❛[4,1]✳ ❙❡rá ❛ ✈❡③✱ ♥♦✈❛♠❡♥t❡✱ ❞♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r
q✉❡ ❥♦❣❛rá[0,1]❡ s❡rá ♦ ✈❡♥❝❡❞♦r✳
❊①❡♠♣❧♦ ✸✳✹ ❙❡❥❛♠ 50 ❡ 8 ♦s ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s✳
❖ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r t❡rá ✈ár✐❛s ♦♣çõ❡s ❞❡ ❥♦❣♦✿
[2,8],[10,8],[18,8],[26,8],[34,8],[42,8]
❙✉♣♦♥❤❛♠♦s q✉❡ ❡s❝♦❧❤❛[10,8]✳ ◆❡st❡ ❝❛s♦✱ ♦ s❡❣✉♥❞♦ ❥♦❣❛❞♦r só t❡rá ✉♠❛ ❛❧t❡r♥❛t✐✈❛ [2,8]✳ ❙❡rá ❛ ✈❡③✱ ♥♦✈❛♠❡♥t❡✱ ❞♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r✱ q✉❡ ❡s❝♦❧❤❡rá[2,0]❡ s❡rá ♦ ✈❡♥❝❡❞♦r✳
◗✉❡stã♦ ✸✳✹✳✶ ◆♦s ❡①❡♠♣❧♦s ✉♠✱ ❞♦✐s ❡ três ♦s ♥ú♠❡r♦s ✐♥✐❝✐❛✐s sã♦31 ❡7✱49❡ 5✱50
❡ 8✱ ❛s ❥♦❣❛❞❛s ✈❡♥❝❡❞♦r❛s sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ [1,0],[0,1] ❡ [2,0]✳ ◗✉❛❧ r❡❧❛çã♦ ❡①✐st❡
❡♥tr❡ ♦s ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s ❡ ♦ ♥ú♠❡r♦ ♥ã♦ ♥✉❧♦ ♦❜t✐❞♦ ♥❛ ❥♦❣❛❞❛ ✈❡♥❝❡❞♦r❛❄
❘❡s♣♦st❛ ❈♦♠❡♥t❛❞❛✿ ❖ ♥ú♠❡r♦ ♥ã♦ ♥✉❧♦ é ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞♦s ♥ú✲ ♠❡r♦s ✐♥✐❝✐❛✐s✳ ❖ ♣r♦❢❡ss♦r ♣♦❞❡ ❝❤❛♠❛r ❛ ❛t❡♥çã♦ ❞♦s ❛❧✉♥♦s ♣❛r❛ ❡st❡ ❢❛t♦✱ q✉❡ ♦ ❥♦❣♦ t❡r♠✐♥❛ ❝♦♠ ♦ ♣❛r {n,0}✱ ♦♥❞❡ n é ♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ❞♦s ❞♦✐s ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s ✐♥✐❝✐❛❧♠❡♥t❡✳ ❉❡ ❢❛t♦✱ s❡ ❞❡♥♦t❛r♠♦s ♣♦r a ❡ b ♦s ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s ❡ ✉♠ ♥ú♠❡r♦ ❞✐✈✐❞✐r a ❡ b✱ ❡st❡ ♥ú♠❡r♦ t❛♠❜é♠ ❞✐✈✐❞✐rá a−mb ❡ b✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ ✉♠ ♥ú♠❡r♦ ❞✐✈✐❞✐r a −mb ❡ b✱ ❡st❡ ♥ú♠❡r♦ t❛♠❜é♠ ❞✐✈✐❞✐rá a ❡ b✳ P♦rt❛♥t♦✱ ♦s ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡ a ❡ b ❡ ♦s a−mb ❡ b sã♦ ♦s ♠❡s♠♦s ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱
M DC(a, b) =M DC(a−mb, b) =· · ·=M DC(n,0) =n✳ ❖ ♥♦♠❡ ❞♦ ❥♦❣♦✱ ❥♦❣♦ ❞❡ ❊✉✲ ❝❧✐❞❡s✱ t❛♠❜é♠ s✉❣❡r❡ ♦❜s❡r✈❛r ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ ❞♦✐s ♥ú♠❡r♦s✿
✹ ✷ ✸ ✸✶ ✼ ✸ ✶
✸ ✶ ✵
❖♥❞❡✱ ❡♠ ❝❛❞❛ ♣❛ss❛❣❡♠✱ ❞♦ ♠❛✐♦r ♥ú♠❡r♦ s✉❜tr❛✐✲s❡ ✉♠ ♠ú❧t✐♣❧♦ ❞♦ ♠❡♥♦r ✭♥♦ ❥♦❣♦✱ ❡ss❡ ♠ú❧t✐♣❧♦ ♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♦ ♠❛✐♦r ♣♦ssí✈❡❧✮✳ ❯♠ r❡s✉❧t❛❞♦ q✉❡ ❡♥✉♥✲ ❝✐❛ ❡①❛t❛♠❡♥t❡ ♣❛rt❡ ❞❛ r❡❣r❛ ❞♦ ❥♦❣♦ é ♦ ▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s ♦ q✉❛❧ ❡♥✉♥❝✐❛r❡♠♦s ❡ ♣r♦✈❛r❡♠♦s ❛ s❡❣✉✐r✿
▲❡♠❛ ✸✳✵✳✶ ❙❡❥❛♠ a, b, n ∈ N ❝♦♠ a < na < b✳ ❙❡ ❡①✐st❡ (a, b−na)✱ ❡♥tã♦ (a, b)
❡①✐st❡✱ ❡(a, b) = (a, b−na)✳
❖❜s❡r✈❛çã♦✿ (a, b)r❡♣r❡s❡♥t❛ ♦ M DC ❡♥tr❡a❡ b✳
❉❡♠♦♥str❛çã♦ ✸✳✶ ❙❡❥❛ d= (a, b−na)✳ ❈♦♠♦ d|a ❡ d|(b−na)✱ s❡❣✉❡ q✉❡ d ❞✐✈✐❞❡ b=b−na+na✳ ▲♦❣♦✱ dé ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a❡ b✳ ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ c s❡❥❛ ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a❡ b❀ ❧♦❣♦✱c é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡a ❡ b−na ❡ ♣♦rt❛♥t♦✱ c|d✳ ■ss♦ ♣r♦✈❛ q✉❡d= (a, b)✳
◗✉❡stã♦ ✸✳✹✳✷ ❈✉r✐♦s❛♠❡♥t❡ ♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r ❣❛♥❤♦✉ ❛s três ♣❛rt✐❞❛s✳ ❙✉♣♦♥❤❛ q✉❡ ❛♦ ✐♥✈és ❞❛s ❥♦❣❛❞❛s ❢❡✐t❛s ♥❛ ♣r✐♠❡✐r❛ r♦❞❛❞❛ ❡❧❡ t✐✈❡ss❡ ❥♦❣❛❞♦ ♦s ♣❛r❡s[17,7] ♥♦
♣r✐♠❡✐r♦ ❡①❡♠♣❧♦✱[29,5]♥♦ s❡❣✉♥❞♦ ❡①❡♠♣❧♦ ❡ [34,8] ♥♦ t❡r❝❡✐r♦ ❡①❡♠♣❧♦✳ ▼♦♥t❡ ✉♠❛
❡str❛té❣✐❛ ✈❡♥❝❡❞♦r❛ ♣❛r❛ ♦ s❡❣✉♥❞♦ ❥♦❣❛❞♦r✳ ❘❡s♣♦st❛ ❝♦♠❡♥t❛❞❛✿
1◦ ❊①❡♠♣❧♦✿ ❖ ♣r✐♠❡r♦ ❥♦❣❛❞♦r ❥♦❣❛ [17,7]✳ ❖ s❡❣✉♥❞♦ ❥♦❣❛❞♦r t❡♠ ❛s s❡❣✉✐♥t❡s
♣♦ss✐❜✐❧✐❞❛❞❡s✿ [3,7] ♦✉ [10,7]✳ ❙✉♣♦♥❤❛ q✉❡ ❡❧❡ ❡s❝♦❧❤❛ [10,7]✳ ❙❡rá ❛ ✈❡③✱
♥♦✈❛♠❡♥t❡ ❞♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r q✉❡ só t❡rá ✉♠❛ ❛❧t❡r♥❛t✐✈❛✿ [3,7]✳ ❆❣♦r❛✱ ♦
s❡❣✉♥❞♦ ❥♦❣❛❞♦r ♣♦❞❡ ❥♦❣❛r✿ [3,4] ♦✉ [3,1]✳ ❙✉♣♦♥❤❛ q✉❡ ❡❧❡ ❡s❝♦❧❤❛ [3,4]✳ ❖
♣r✐♠❡✐r♦ ❥♦❣❛❞♦r s❡rá ♦❜r✐❣❛❞♦ ❛ ❥♦❣❛r [3,1]✳ ❖ s❡❣✉♥❞♦ ❥♦❣❛❞♦r ❥♦❣❛rá [0,1] ❡
✈❡♥❝❡rá ❛ ♣❛rt✐❞❛✳
2◦ ❊①❡♠♣❧♦✿ ❖ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r ❥♦❣❛ [29,5]✳ ❖ s❡❣✉♥❞♦ t❡♠ ❛s s❡❣✉✐♥t❡ ♣♦ss✐❜✐❧✐❞❛✲
❞❡s✿ [4,5]✱[9,5]✱[13,5]✱[18,5]✱ ❡ [24,5]✳ ❙✉♣♦♥❤❛ q✉❡ ❡❧❡ ❡s❝♦❧❤❛[4,5]✳ ❙❡rá ❛ ✈❡③
♥♦✈❛♠❡♥t❡ ❞♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r q✉❡ só t❡rá ✉♠❛ ❛❧t❡r♥❛t✐✈❛✿ [4,1]✳ ❖ s❡❣✉♥❞♦
❥♦❣❛rá [0,1]❡ ✈❡♥❝❡rá ❛ ♣❛rt✐❞❛✳
3◦ ❊①❡♠♣❧♦✿ ❖ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r ❥♦❣❛ [34,8]✳ ❖ s❡❣✉♥❞♦ t❡♠ ❛s s❡❣✉✐♥t❡s ♣♦ss✐❜✐❧✐✲
❞❛❞❡s✿ [2,8]✱ [10,8]✱ [18,8]✱ [26,8]✳ ❙✉♣♦♥❤❛ q✉❡ ❡❧❡ ❡s❝♦❧❤❛ [10,8]✳ ❙❡rá ❛ ✈❡③
♥♦✈❛♠❡♥t❡ ❞♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r q✉❡ só t❡rá ✉♠❛ ❛❧t❡r♥❛t✐✈❛✿ [2,8]✳ ❖ s❡❣✉♥❞♦
❥♦❣❛❞♦r ❥♦❣❛rá [2,0]❡ ✈❡♥❝❡rá ❛ ♣❛rt✐❞❛✳
◗✉❡stã♦ ✸✳✹✳✸ ◆❛s ♣❛rt✐❞❛s ❡♠ q✉❡ ♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r ✈❡♥❝❡✉✱ ❡❧❡ ❡♥tr❡❣♦✉ ♣❛r❛ ♦ ❛❞✈❡rsár✐♦ ♦s s❡❣✉✐♥t❡s ♣❛r❡s✿
❊①❡♠♣❧♦1✿ [10,7],[3,4]✳
❊①❡♠♣❧♦2✿ [4,5]✳
❊①❡♠♣❧♦3✿ [10,8]✳
P♦r ♦✉tr♦ ❧❛❞♦ ❛s ❥♦❣❛❞❛s ❞♦ s❡❣✉♥❞♦ ❥♦❣❛❞♦r ✭♣❡r❞❡❞♦r✮ ❢♦r❛♠✿ ❊①❡♠♣❧♦1✿ [3,7],[3,1]✳
❊①❡♠♣❧♦2✿ [4,1]✳
❊①❡♠♣❧♦3✿ [2,8]✳
❛✳ ❆ ♣r✐♠❡✐r❛ ❝♦❧✉♥❛ ❞❛ t❛❜❡❧❛ ✐♥❞✐❝❛ ❛s ❥♦❣❛❞❛s ❞♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r ✭✈❡♥❝❡❞♦r✮❀ ❞✐✈✐❞❛ ♦ ♠❛✐♦r ♥ú♠❡r♦ ❞♦ ♣❛r ♣❡❧♦ ♠❡♥♦r ❡ ✐♥s✐r❛ ♦ r❡s✉❧t❛❞♦ ❛♣r♦①✐♠❛❞♦ ♥❛ t❡r❝❡✐r❛ ❝♦❧✉♥❛✿
❊①✳✶ ❬✶✵✱✼❪ ✶✱✹✷✽ ❊①✳✶ ❬✸✱✹❪ ✶✱✸✸✸ ❊①✳✷ ❬✹✱✺❪ ✶✱✷✺ ❊①✳✸ ❬✶✵✱✽❪ ✶✱✷✺
❜✳ ❆ ♣r✐♠❡✐r❛ ❝♦❧✉♥❛ ❞❛ t❛❜❡❧❛ ✐♥❞✐❝❛ ❛s ❥♦❣❛❞❛s ❞♦ s❡❣✉♥❞♦ ❥♦❣❛❞♦r ✭♣❡r❞❡❞♦r✮❀ ❞✐✈✐❞❛ ♦ ♠❛✐♦r ♥ú♠❡r♦ ❞♦ ♣❛r ♣❡❧♦ ♠❡♥♦r ❡ ✐♥s✐r❛ ♦ r❡s✉❧t❛❞♦ ❛♣r♦①✐♠❛❞♦ ♥❛ t❡r❝❡✐r❛ ❝♦❧✉♥❛✳
❊①✳✶ ❬✸✱✼❪ ✷✱✸✸✸ ❊①✳✶ ❬✸✱✶❪ ✸ ❊①✳✷ ❬✹✱✶❪ ✹ ❊①✳✸ ❬✷✱✽❪ ✹
❝✳ ◆❛ q✉❡stã♦ ❞♦✐s ❝r✐♦✉✲s❡ ✉♠❛ ❡str❛té❣✐❛ ♣❛r❛ q✉❡ ♦ s❡❣✉♥❞♦ ❥♦❣❛❞♦r ✈❡♥❝❡ss❡ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❥♦❣❛❞❛ ❞✐❢❡r❡♥t❡ ❞❛ ♣r♦♣♦st❛✱ ♥♦s ❡①❡♠♣❧♦s✱ ♣❡❧♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r✳ ❆ss✐♠✱ ♠♦♥t❡ ❞✉❛s t❛❜❡❧❛s✱ ✉♠❛ ✐♥❞✐❝❛♥❞♦ ♥❛ ♣r✐♠❡✐r❛ ❝♦❧✉♥❛ ❛s ❥♦❣❛❞❛s ♣r♦♣♦st❛s ♣♦r ✈♦❝ê ♣❛r❛ ♦ s❡❣✉♥❞♦ ❥♦❣❛❞♦r ✭✈❡♥❝❡❞♦r✮ ❡ ❛ ♦✉tr❛ ❝♦♠ ❛s ❥♦❣❛❞❛s ❞♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r ✭♣❡r❞❡❞♦r✮✳ ❆ss✐♠ ❝♦♠♦ ♥♦ ❡①❡r❝í❝✐♦ ❜✳ ✐♥s✐r❛ ♥❛ t❡r❝❡✐r❛ ❝♦❧✉♥❛ ♦ r❡s✉❧t❛❞♦ ❛♣r♦①✐♠❛❞♦ ❞❛ ❞✐✈✐sã♦ ❞♦ ♠❛✐♦r ♥ú♠❡r♦ ❞♦ ♣❛r ❥♦❣❛❞♦ ♣❡❧♦ ♠❡♥♦r✳ ❙❡❣✉♥❞♦ ❥♦❣❛❞♦r ✭✈❡♥❝❡❞♦r✮✿
❊①✳✶ ❬✶✵✱✼❪ ✶✱✹✷✽ ❊①✳✶ ❬✸✱✹❪ ✶✱✸✸✸ ❊①✳✷ ❬✹✱✺❪ ✶✱✷✺ ❊①✳✸ ❬✶✵✱✽❪ ✶✱✷✺ Pr✐♠❡✐r♦ ❥♦❣❛❞♦r ✭♣❡r❞❡❞♦r✮✿
❊①✳✶ ❬✶✼✱✼❪ ✷✱✹✷✽ ❊①✳✶ ❬✸✱✼❪ ✷✱✸✸✸ ❊①✳✶ ❬✸✱✶❪ ✸ ❊①✳✷ ❬✷✾✱✺❪ ✺✱✽ ❊①✳✷ ❬✹✱✶❪ ✹ ❊①✳✸ ❬✸✹✱✽❪ ✹✱✷✺ ❊①✳✸ ❬✷✱ ✽❪ ✹
❞✳ ◗✉❛❧ ❢♦✐ ♦ ♠❛✐♦r r❡s✉❧t❛❞♦ ❡♥❝♦♥tr❛❞♦ ♥❛ ❞✐✈✐sã♦ ❞♦s ♥ú♠❡r♦s ❞♦s ♣❛r❡s ❞❛s ❥♦❣❛❞❛s ❞♦ ✈❡♥❝❡❞♦r❄
▼❛✐♦r r❡s✉❧t❛❞♦✿ 1,428✳
❡✳ ◗✉❛❧ ❢♦✐ ♦ ♠❡♥♦r r❡s✉❧t❛❞♦ ❡♥❝♦♥tr❛❞♦ ♥❛ ❞✐✈✐sã♦ ❞♦s ♥ú♠❡r♦s ❞♦s ♣❛r❡s ❞❛s ❥♦❣❛❞❛s ❞♦ ♣❡r❞❡❞♦r❄
▼❡♥♦r r❡s✉❧t❛❞♦✿ 2,333✳
◗✉❡stõ❡s 3.4.2 ❡ 3.4.3 ❝♦♠❡♥t❛❞❛s✿ ❖ ♣r♦❢❡ss♦r ❞❡✈❡ ✐♥st✐❣❛r ♦s ❛❧✉♥♦s ❛ ♣❡r✲
❝❡❜❡r❡♠ q✉❡ ♦ ❢❛t♦ ❞❡ ♦ r❡s✉❧t❛❞♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ♣❛r ✈❡♥❝❡❞♦r✱ ♥♦s ❡①❡♠♣❧♦s✱ s❡r s❡♠♣r❡ ♠❡♥♦r q✉❡ 1,428 ♥ã♦ é ✉♠❛ ❝♦✐♥❝✐❞ê♥❝✐❛✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠❛ ❡str❛té❣✐❛ ♣❛r❛
✈❡♥❝❡r ♦ ❥♦❣♦✳ ❉❡ ❢❛t♦✱ ❛ ❡s❝♦❧❤❛ ❞♦ ♣❛r ❛ s❡r ❥♦❣❛❞♦ é ✉♠❛ ❡str❛té❣✐❛ ♣❛r❛ ✈❡♥❝❡r ♦ ❥♦❣♦✳ ❙❡ ✉♠ ❥♦❣❛❞♦r r❡❝❡❜❡ ✉♠ ♣❛r[a, b]❝♦♠
a b <
1 +√5
2 ≈1,618
❡❧❡ t❡rá ❛s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ❛❞♦t❛r ❛ ❡str❛té❣✐❛ q✉❡ ❧❤❡ ❣❛r❛♥t❡ ❛ ✈✐tór✐❛✳ ❆ ✐❞❡✐❛ é q✉❡ ♦s ❛❧✉♥♦s ❡st❡❥❛♠ ♠♦t✐✈❛❞♦s ❛ ❛❝♦♠♣❛♥❤❛r ❡ ❛ ❡♥t❡♥❞❡r ❛ ❞❡♠♦♥str❛çã♦ ▼❛t❡♠át✐❝❛ ❞❛ ✈❛❧✐❞❛❞❡ ❞❛ ❡str❛té❣✐❛ ♣❛r❛ q✉❛❧q✉❡r ♣❛r ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡s❝♦❧❤✐❞♦s✳ Pr♦✈❛r❡♠♦s ❛ ❛✜r♠❛çã♦ ❛ s❡❣✉✐r✳ ❈♦♥s✐❞❡r❛çõ❡s ✐♥✐❝✐❛✐s✿
❞❛❞♦ ✉♠ ♣❛r [a, b]✱ ❝♦♠a > b✱ ♦s ♣❛r❡s
[a−b, b],[a−2b, b],· · · ,[a−qb, b]
❝♦♠a−qb≥0✱ ❝❤❛♠❛♠✲s❡ ♣❛r❡s ❞❡r✐✈❛❞♦s ❞❡[a, b]✳ ❆ss✐♠✱ [24,7],[17,7],[10,7],[3,7]
sã♦ ♦s ♣❛r❡s ❞❡r✐✈❛❞♦s ❞❡ [31,7]✳ ❙❡ a−qb≥0 ❡ a−(q+ 1)b <0 [a−qb, b]❝❤❛♠❛✲s❡
♣❛r ❞❡r✐✈❛❞♦ ♠í♥✐♠♦ ❞❡ [31,7]✳
❖❜s❡r✈❡ q✉❡✱ ❞❡♥tr❡ t♦❞♦s ♦s ♣❛r❡s ❞❡r✐✈❛❞♦s ❞❡ ✉♠ ♣❛r[a, b]✱ ❝♦♠a > b✱ ♦s ♥ú♠❡✲ r♦s ❞♦ ♣❛r ❞❡r✐✈❛❞♦ ♠í♥✐♠♦ sã♦b❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡a ♣♦rb✳
❙❡ [a−qb, b]❢♦r ♦ ♣❛r ❞❡r✐✈❛❞♦ ♠í♥✐♠♦✱ ❞✐r❡♠♦s q✉❡ ♦ ♣❛r [a−(q−1)b, b]é ♦ ♣❛r
❛♥t❡r✐♦r ❛♦ ♣❛r ❞❡r✐✈❛❞♦ ♠í♥✐♠♦✳
❖❜s❡r✈❡✱ ♠❛✐s ✉♠❛ ✈❡③✱ ♦ ❡①❡♠♣❧♦✿ ❉❛❞♦ ♦ ♣❛r [31,7]✱ ♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r t❡♠
❛♣❡♥❛s ❞✉❛s ♦♣çõ❡s s✐❣♥✐✜❝❛t✐✈❛s✿
• ❡❧❡ ❡s❝♦❧❤❡ ♦ ♣❛r ❛♥t❡r✐♦r ❛♦ ♣❛r ❞❡r✐✈❛❞♦ ♠í♥✐♠♦✱ ✐st♦ é✱ [10,7]✱ ♦❜r✐❣❛♥❞♦ ♦
❛❞✈❡rsár✐♦ ❛ ❥♦❣❛r [3,7]✳ ◗✉❛❧q✉❡r ♦✉tr❛ ❡s❝♦❧❤❛ ❞❛r✐❛ ❡st❛s ♠❡s♠❛s ❞✉❛s ♦♣çõ❡s
❛♦ ❛❞✈❡rsár✐♦✳
❆♥❛❧✐s❛♥❞♦ ♦ ❝❛s♦ ❣❡r❛❧✿ s✉♣♦♥❤❛♠♦s q✉❡ ♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r r❡❝❡❜❛ ♦ ♣❛r [n, m]
❝♦♠m < n✳ ❙❡ mn ❢♦r ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ k✱ ♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r ❣❛♥❤❛rá ♦ ❥♦❣♦ ❝♦♠ ❛ ❥♦❣❛❞❛[n−km, m] = [0, m]✳ ❙✉♣♦♥❤❛♠♦s ❡♥tã♦✱ n=qm+r✱ 0< r < m✳ ❖ ❥♦❣❛❞♦r ❞❡✈❡rá ♦♣t❛r ♣❡❧♦ ♣❛r ❞❡r✐✈❛❞♦ ♠í♥✐♠♦ ♦✉ ♣❡❧♦ ♣❛r ❛♥t❡r✐♦r ❛ ❡st❡✱ ✐st♦ é✱ ❞❡✈❡rá ♦♣t❛r ❡♥tr❡✿
[nm−qm, m] = [qm+r−qm, m] = [r, m], ❝♦♠ 0< r < m.
[n−(q−1)m, m] = [qm+r−qm+m, m] = [m+r, m] ❝♦♠ m < m+r ❈♦♠♦ ♦ ❛❞✈❡rsár✐♦ ✈❛✐ ♣r♦ss❡❣✉✐r✱ t✐r❛♥❞♦ ❞❡m ✉♠ ♠ú❧t✐♣❧♦ ❞❡r ♦✉ t✐r❛♥❞♦ ❞❡m+r ✉♠ ♠ú❧t✐♣❧♦ ❞❡m✱ ♦❧❤❡♠♦s ♣❛r❛ ❛s r❛③õ❡s mr ❡
m+r
m ✳ ❋❛③❡♥❞♦ m
r =x✱ t❡r❡♠♦s m+r
m = 1 +
r
m = 1 +
1
x.
◗✉❛❧ ❞❛s r❛③õ❡s✱x= mr ♦✉1 +
1
x = m+r
m é ✈❛♥t❛❥♦s❛ ♣❛r❛ ♦ ❥♦❣❛❞♦r❄ ❖❜s❡r✈❛♠♦s✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ q✉❡ ❛s r❛③õ❡s s❡r✐❛♠ ✐❣✉❛✐s s❡
x= 1 + 1
x
♦✉ s❡❥❛✱ s❡x2
−x−1 = 0✱ ♦✉ ❛✐♥❞❛✱ ❞❛❞♦ q✉❡x >0✱ s❡
x= 1 +
√
5
2 .
❊st❡ ♥ú♠❡r♦ 1+√5
2 ≈1,618✱ t❡rá ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥❛ ❞✐s❝✉ssã♦✳ ❱❛♠♦s ❝❤❛♠á✲❧♦
❞❡r✳
❉❡♠♦♥str❛çã♦ ✸✳✷ • x < r⇒1 + 1
x > r
x < r⇒ 1 x >
1
r
⇒1 +1
x > 1 +
1
r = 1 +
1
1+√5 2
= 1 + 2
1 +√5 =
1 +√5 + 2 1 +√5
= 3 +
√
5 1 +√5 ·
1−√5 1−√5
= 3−3
√
5 +√5−5
1−5 =
−2−2√5 4
= −2(1 +
√
5)
−4 =
1 +√5 2
= r
• x > r⇒1 +1
x < r
❉❡♠♦♥str❛çã♦ ✸✳✸ x > r⇒ 1
x <
1
r
⇒1 + 1
x < 1 +
1
r = 1 +
1
1+√5 2
= 1 + 2
1 +√5 =
1 +√5 + 2 1 +√5
= 3 +
√
5 1 +√5·
1−√5 1−√5
= 3−3
√
5 +√5−5
1−5 =
−2−2√5 4
= −2(1 +
√
5)
−4 =
1 +√5 2 = r.
❘❡❢❛③❡♠♦s✱ ❡♥tã♦✱ ❛ ♣❡r❣✉♥t❛✿ ♦ ♣r✐♠❡✐r♦ ❥♦❣❛❞♦r ♣♦❞❡ ♦♣t❛r ❡♥tr❡ ✉♠ ♣❛r ❝✉❥❛ r❛③ã♦ é ♠❛✐♦r ❞♦ q✉❡ r✱ ♦✉ ✉♠ ♣❛r ❝✉❥❛ r❛③ã♦ é ♠❡♥♦r ❞♦ q✉❡ r✳ ◗✉❛❧ é ❛ ♠❡❧❤♦r ♦♣çã♦❄ ❆ r❡s♣♦st❛ s❡ ❡♥❝♦♥tr❛ ♥♦ s❡❣✉✐♥t❡ ❢❛t♦✿
❙❡ ✉♠ ❥♦❣❛❞♦r r❡❝❡❜❡r ✉♠ ♣❛r[a, b]❝♦♠1< ab < r✱ ♥❛q✉❡❧❛ ❥♦❣❛❞❛ ❡❧❡ ♥ã♦ ♣♦❞❡rá ❣❛♥❤❛r ♦ ❥♦❣♦ ❡ t❡rá ❝♦♠♦ ú♥✐❝❛ ♦♣çã♦ ♦ ♣❛r[a−b, b]❝♦♠ r❛③ã♦ b
a−b > r✳ ❉❡ ❢❛t♦✱ s❡
1< a b < r a
b ♥ã♦ é ✐♥t❡✐r♦ ❡a−2b é ♥❡❣❛t✐✈♦✳ P♦rt❛♥t♦✱ ❛ ú♥✐❝❛ ♦♣çã♦ s❡rá[a−b, b]❡
b a−b =
1
a b −1
> 1 r−1 =
1
1+√5 2 −1
= 1
1+√5−2 2
= 2
−1 +√5·
−1−√5
−1 +√5
= −2−2
√
5 1−5 =
−2(1 +√5)
−4
= 1 +
√
5
2 =r
✳
