❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ▼❆❚❖ ●❘❖❙❙❖ ❉❖ ❙❯▲ ■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲
▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲
❙■▲❱■❖ ❘❖●➱❘■❖ ❆▲❱❊❙ ❊❙◗❯■◆❈❆
❆▲●❖❘■❚▼❖❙✿ ❘❊❙❖▲❯➬➹❖ ❉❊ P❘❖❇▲❊▼❆❙
❇➪❙■❈❖❙
❈❆▼P❖ ●❘❆◆❉❊
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ▼❆❚❖ ●❘❖❙❙❖ ❉❖ ❙❯▲ ■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲
▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲
❙■▲❱■❖ ❘❖●➱❘■❖ ❆▲❱❊❙ ❊❙◗❯■◆❈❆
❆▲●❖❘■❚▼❖❙✿ ❘❊❙❖▲❯➬➹❖ ❉❊ P❘❖❇▲❊▼❆❙
❇➪❙■❈❖❙
❖r✐❡♥t❛❞♦r❛✿ Pr♦❢✳➟ ❉r✳➟ ❊❧✐s❛❜❡t❡ ❙♦✉s❛ ❋r❡✐t❛s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ■▼✴❯❋▼❙✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡✳
❆▲●❖❘■❚▼❖❙✿ ❘❊❙❖▲❯➬➹❖ ❉❊ P❘❖❇▲❊▼❆❙
❇➪❙■❈❖❙
❙■▲❱■❖ ❘❖●➱❘■❖ ❆▲❱❊❙ ❊❙◗❯■◆❈❆
❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❞❛ ❯♥✐✈❡r✲ s✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼❛t♦ ●r♦ss♦ ❞♦ ❙✉❧✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡✳
❆♣r♦✈❛❞♦ ♣❡❧❛ ❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛✿
Pr♦❢✳➟ ❉r✳➟ ❊❧✐s❛❜❡t❡ ❙♦✉③❛ ❋r❡✐t❛s ✲ ❯❋▼❙
Pr♦❢✳➟ ❉r✳➟ ❏❛♥❡t❡ ❞❡ P❛✉❧❛ ❋❡rr❛r❡③❡ ❙✐❧✈❛✲ ❯❋▼❙ Pr♦❢✳ ❉r✳ ❘✉✐ ❙❡✐♠❡t③ ✲ ❯◆❇
❈❆▼P❖ ●❘❆◆❉❊
❊♣í❣r❛❢❡
▼❛s ♦s q✉❡ ❡s♣❡r❛♠ ♥♦ s❡♥❤♦r r❡♥♦✈❛rã♦ ❛s s✉❛s ❢♦rç❛s✱ s✉❜✐rã♦ ❝♦♠ ❛s❛s ❝♦♠♦ á❣✉✐❛s✱ ❝♦rr❡rã♦ ❡ ♥ã♦ s❡ ❝❛♥s❛rã♦✱ ❝❛♠✐♥❤❛rã♦ ❡ ♥ã♦ s❡ ❢❛t✐❣❛rã♦✳
❆●❘❆❉❊❈■▼❊◆❚❖❙
Pr✐♠❡✐r❛♠❡♥t❡✱ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s ❡ ❛♦ ♥♦ss♦ ❙❡♥❤♦r ❏❡s✉s ❈r✐st♦✱ ♣♦r t❡r❡♠ ♠❡ ❝♦♥❝❡❞✐❞♦ ❛ ❣r❛ç❛ ❞❡ ✐♥❣r❡ss❛r ❡ ❞❡ ♣r♦ss❡❣✉✐r ❛té ♦ ✜♠ ♥❡st❡ ♠❡str❛❞♦✳
❆❣r❛❞❡ç♦ ❛✐♥❞❛ ❛ ♠✐♥❤❛ ❡s♣♦s❛ ❏♦s✐❛♥❡ ❈♦❧♦♠❜♦ P❡❞r✐♥✐ ❊sq✉✐♥❝❛✱ ♣♦r ❛❥✉❞❛r✲♠❡ ❞✉r❛♥t❡ ❡st❡ ❝✉rs♦✱ ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❝❤❡❣❛r ❛té ♦ ✜♠ ❞❡❧❡✳
❆♦s ♠❡✉s ♣❛✐s✱ ♣♦r ❛❝r❡❞✐t❛r❡♠ ❡♠ ♠✐♠ ❡ ♣♦r s✉❛s ♦r❛çõ❡s✱ ❛s q✉❛✐s✱ ❝♦♠ ❝❡r✲ t❡③❛✱ s✉rt✐r❛♠ ❡❢❡✐t♦✱ ♥♦ s❡♥t✐❞♦ ❞❡ ❡✉ s❡r ❛♣r♦✈❛❞♦ ♥❛s ❞✐s❝✐♣❧✐♥❛s ❞♦ ❝✉rs♦ ❡ ♥♦ ❡①❛♠❡ ❞❡ q✉❛❧✐✜❝❛çã♦✳
❆ ♠✐♥❤❛ ♣r♦❢❡ss♦r❛ ❡ ♦r✐❡♥t❛❞♦r❛✱ ❊❧✐s❛❜❡t❡ ❙♦✉③❛ ❋r❡✐t❛s✱ q✉❡✱ ❝♦♠ r❡s♣❡✐t♦✱ ❞❡t❡r♠✐♥❛çã♦✱ ❞❡❞✐❝❛çã♦ ❡ ❛♠♣❛r♦✱ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❛ ❝♦♥str✉çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❞❡❞✐❝♦✉✲s❡✱ s❡♠ ♠❡❞✐r ❡s❢♦rç♦s✱ ❡♠ ♦r✐❡♥t❛r✲♠❡✱ ♣❛r❛ q✉❡ ❡st❡ tr❛❜❛❧❤♦ ❢♦ss❡ ❝♦♥❝❧✉í❞♦ ❝♦♠ ê①✐t♦✳
❆♦ ♣r♦❢❡ss♦r ❈❧❛✉❞❡♠✐r ❆♥✐③✱ ♣♦r t❡r ❛❞♠✐♥✐str❛❞♦ ❡st❡ ❝✉rs♦✱ ❝♦♥❢♦r♠❡ ♦s ♣r✐♥✲ ❝í❝♣✐♦s q✉❡ r❡❣❡♠ ❛ ❛❞♠✐♥✐str❛çã♦ ♣ú❜❧✐❝❛✱ ❣❛r❛♥t✐❞♦ s❡r✐❡❞❛❞❡ ❡ r❡s♣❡✐t♦ ❡♠ t♦❞❛s ❛s ❡t❛♣❛s ❞❡st❡ ♠❡str❛❞♦✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❡st❡ ❝✉rs♦✱ ♣♦r t❡r❡♠ ♠❡ tr❛♥s♠✐t✐❞♦ ❝♦♥❤❡❝✐♠❡♥t♦s ♣r❡❝✐♦s♦s ❡ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❞❡s❡♠♣❡♥❤♦ ❞❛ ❢✉♥çã♦ ❞❡ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡✲ ♠át✐❝❛✳
❆♦ ♣r♦❣r❛♠❛ Pr♦❢♠❛t✱ ♣♦r ❞❛r✲♠❡ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❛❞q✉✐r ♦ tít✉❧♦ ❞❡ ♠❡str❡✱ ❛❧♠❡❥❛❞♦ ❤á ♠✉✐t♦ t❡♠♣♦✱ ♦ q✉❛❧✱ ❝♦♠ ❝❡rt❡③❛✱ ❛♠♣❧✐❛rá ❛s ♦♣♦rt✉♥✐❞❛❞❡s ❞❡ tr❛❜❛❧❤♦✱ ❛❧é♠ ❞❡ ♣r♦♣♦r❝✐♦♥❛r✲♠❡ ✉♠❛ ♠❛✐♦r ❜❛❣❛❣❡♠ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦✳
❆ ❈❛♣❡s✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ✜♥❛♥❝✐❛♠❡♥t♦ ❞♦ ❝✉rs♦✳
❆♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❞❡ ❣r❛❞✉❛çã♦✱ ♣♦r t❡r❡♠ ❞❛❞♦✲♠❡ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ ❣❛✲ r❛♥t✐✉ ♠✐♥❤❛ ♣r♦✜ssã♦ ❞❡ ♣r♦❢❡ss♦r ❡ q✉❡ ♠❡ ❛❥✉❞♦✉ ❝❤❡❣❛r ❛té ♦ Pr♦❢♠❛t✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛❣r❛❞❡ç♦ ❛ ❆♥tô♥✐♦ ❈❛r❧♦s ❚❛♠❛r♦③③✐✱ ♣♦r t❡r ❛❝r❡❞✐t❛❞♦ ❡♠ ♠✐♠✱ ♦r✐❡♥t❛♥❞♦✲♠❡ ❞♦ ✐♥í❝✐♦ ❛♦ ✜♠ ❞❛ ❣r❛❞✉❛çã♦✱ s❡♠ ♠❡❞✐r ❡s❢♦rç♦s✱ ✐♥❝❡♥t✐✈❛♥❞♦✲♠❡ ❛ ♣r♦ss❡❣✉✐r ♥♦s ❡st✉❞♦s✳
❆♦s ❝♦❧❡❣❛s ❞❡ t✉r♠❛ ❞❡ ♠❡str❛❞♦✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ❛♣♦✐♦ ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✱ q✉❡ ❧❡✈♦✉ à tr♦❝❛ ❞❡ ❡①♣❡r✐ê♥❝✐❛s✳
❆♦s ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦✱ ❧♦t❛❞♦s ♥❛ ❊s❝♦❧❛ ❊st❛❞✉❛❧ ❏♦sé ❋❡rr❡✐r❛ ❇❛r❜♦s❛✱ ▼❛✲ r✐♦♠❛r ❘❡③❡♥❞❡ ❉✐♥✐③ ❏✉♥✐♦r✱ ❙✉❡❧② ❖❧✐✈❡✐r❛ ❞❡ ❆ss✐s✱ ❊❞❡r ❘é❣✐s ❡ ❏♦❤♥♥② ▼❛tt♦s✱ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦ ♣r❡st❛❞♦ ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❝♦♥❝❧✉✐r ❡st❡ ❝✉rs♦✳
❘❡s✉♠♦
❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ✉♠❛ ✐♥tr♦❞✉çã♦ ❛♦ ❡st✉❞♦ ❞❡ ❛❧❣♦r✐t♠♦✳ ❆tr❛✈és ❞❡ ❡①❡♠♣❧♦s✱ ♦ ❝✉st♦ ❞❡ ❛❧❣♦r✐t♠♦s s❡rá ❛♥❛❧✐s❛❞♦ ❞❡ ❢♦r♠❛ s✐♠♣❧✐✜❝❛❞❛✳ ❙❡rã♦ ❞✐s❝✉t✐❞♦s ♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✱ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❙tr❛ss❡♥✱ ♣❛r❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s✱ ❡ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❝❧áss✐❝♦s ❞❛ ❝♦♠♣✉t❛çã♦✳
❆❜str❛❝t
❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ ♣r❡s❡♥t ❛♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ st✉❞② ♦❢ ❛❧❣♦r✐t❤♠✳ ❚❤✲ r♦✉❣❤ ❡①❛♠♣❧❡s✱ ❝♦st ❛❧❣♦r✐t❤♠s ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞ ✐♥ s✐♠♣❧✐✜❡❞ ❢♦r♠✳ ❉✐s❝✉ss❡❞ t❤❡ ❞✐✈✐s✐♦♥ ❛❧❣♦r✐t❤♠✱ ❙tr❛ss❡♥✬s ❛❧❣♦r✐t❤♠ ❢♦r ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❛♥❞ s♦♠❡ ❝❧❛ss✐❝❛❧ ♣r♦❜❧❡♠s ♦❢ ❝♦♠♣✉t✐♥❣✳
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶
✷ ❆❧❣♦r✐t♠♦s ✷
✷✳✶ ❆❧❣♦r✐t♠♦s ♣❛r❛ ❡①♣❛♥sã♦ ♥❛ ❜❛s❡ ❞❡❝✐♠❛❧ ❡ ❜✐♥ár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✷✳✷ ❆❧❣♦r✐t♠♦ ♣❛r❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✷✳✸ ❆❧❣♦r✐t♠♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ ✈❛❧♦r ❞❡ ✉♠❛ ♣♦tê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷✳✹ ◆♦t❛çã♦ ❆ss✐♥tót✐❝❛ O ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✹✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ ◆♦t❛çã♦ O ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✸ ❆❧❣♦r✐t♠♦s ♥❛ ▼❛t❡♠át✐❝❛ ❇ás✐❝❛ ✷✸
✸✳✶ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✸✳✷ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❞❡ ❉♦✐s ◆ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✸✳✸ ❉✐✈✐sã♦ ❞❡ P♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✸✳✸✳✶ ❉✐s♣♦s✐t✐✈♦ Prát✐❝♦ ❞❡ ❇r✐♦t✲❘✉✣♥✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✸✳✹ ◆ú♠❡r♦s Pr✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✸✳✺ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ▼❛tr✐③❡s ◗✉❛❞r❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✹ ❊①❡♠♣❧♦s ❈❧áss✐❝♦s ❞❡ ❆❧❣♦r✐t♠♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ✹✼
✹✳✶ ▲♦❝❛❧✐③❛çã♦ ❞❡ ■t❡♠ ❡♠ ✉♠❛ ▲✐st❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✹✳✷ ❖r❞❡♥❛çã♦ ❞♦s ❊❧❡♠❡♥t♦s ✭◆ú♠❡r♦s✮ ❞❡ ✉♠ ❱❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
✹✳✷✳✶ ❖r❞❡♥❛çã♦ ❇♦❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
✹✳✷✳✶✳✶ ❈✉st♦ ❞❛ ❖r❞❡♥❛çã♦ ❇♦❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✹✳✷✳✷ ❖r❞❡♥❛çã♦ P♦r ❙❡❧❡çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✹✳✷✳✷✳✶ ❈✉st♦ ❞❛ ❖r❞❡♥❛çã♦ P♦r ❙❡❧❡çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✹✳✷✳✸ ❖r❞❡♥❛çã♦ P♦r ■♥s❡rçã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✹✳✷✳✸✳✶ ❈✉st♦ ❞❛ ❖r❞❡♥❛çã♦ P♦r ■♥s❡rçã♦ ✭P✐♦r ❈❛s♦✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✹✳✷✳✹ ❖r❞❡♥❛çã♦ P♦r ■♥t❡r❝❛❧❛çã♦ ✭▼❡r❣❡✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✹✳✷✳✹✳✶ ❈✉st♦ ❞❛ ❖r❞❡♥❛çã♦ P♦r ■♥t❡r❝❛❧❛çã♦ ✭P✐♦r ❈❛s♦✮ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
✶
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡ss❡ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ✉♠❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ ▼❛t❡♠át✐❝❛✱ ❛tr❛✈és ❞❡ ✉♠❛ ✐♥✐❝✐❛çã♦ ❛♦ ❡st✉❞♦ ❞❡ ❛❧❣♦r✐t♠♦s✳
❏á ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦✱ ❛♣r❡♥❞❡✲s❡ ❛❧❣♦r✐t♠♦s ♣❛r❛ s♦❧✉❝✐♦♥❛r ❝❡rt♦s ♣r♦❜❧❡♠❛s✳ P♦r ❡①❡♠♣❧♦✱ ❝á❧❝✉❧♦ ❞♦ q✉♦❝✐❡♥t❡ ❡ ❞♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ✐♥t❡✐r♦ ♣♦r ♦✉tr♦ ✭❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✮ ♦✉ ❝á❧❝✉❧♦ ❞♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ ❞♦✐s ✐♥t❡✐r♦s ✭❛❧❣♦r✐t♠♦ ❡✉❝❧✐❞✐❛♥♦✮✳ ❊st❡s ❞♦✐s ❛❧❣♦r✐t♠♦s ❥á ❛♣❛r❡❝❡♠ ❞❡s❝r✐t♦s ♥♦s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✱ ❡s❝r✐t♦ ♣♦r ✈♦❧t❛ ❞❡ ✸✵✵ ❛✳❈✳
❯♠ ❛❧❣♦r✐t♠♦ r❡♣r❡s❡♥t❛ ♦s ♣❛ss♦s ♥❡❝❡ssár✐♦s ♣❛r❛ r❡❛❧✐③❛r ✉♠❛ t❛r❡❢❛✳ ❙✉❛ ❡①❡✲ ❝✉çã♦ ♣♦❞❡ s❡r ❢❡✐t❛ ♣♦r ❝♦♠♣✉t❛❞♦r❡s✱ ❝❛♣❛③❡s ❞❡ ❡❢❡t✉❛r ❡♥♦r♠❡s ❝á❧❝✉❧♦s✱ ♦✉ ♣♦r ♣❡ss♦❛s✳ ❆t✉❛❧♠❡♥t❡ ❡①✐st❡♠ ♥❛s ❡s❝♦❧❛s ❧❛❜♦r❛tór✐♦s ❞❡ ❝♦♠♣✉t❛çã♦ ❡ ❛❧✉♥♦s ❡ ♣r♦❢❡s✲ s♦r❡s sã♦ ❡st✐♠✉❧❛❞♦s ❛ ✉s❛r ♣r♦❣r❛♠❛s ♦✉ s♦❢t✇❛r❡s ♣❛r❛ ❛✉①✐❧✐❛r ♥♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦✲ ❛♣r❡♥❞✐③❛❣❡♠✳ ❯♠ ♣r♦❣r❛♠❛ ❞❡ ❝♦♠♣✉t❛❞♦r é ❡ss❡♥❝✐❛❧♠❡♥t❡ ✉♠ ❛❧❣♦r✐t♠♦ q✉❡ ❞✐③ ❛♦ ❝♦♠♣✉t❛❞♦r ♦s ♣❛ss♦s ❡s♣❡❝í✜❝♦s ❡ ❡♠ q✉❡ ♦r❞❡♠ ❡❧❡s ❞❡✈❡♠ s❡r ❡①❡❝✉t❛❞♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠✲ ♣❧♦✱ ♦s ♣❛ss♦s ❛ s❡r❡♠ t♦♠❛❞♦s ♣❛r❛ ❝❛❧❝✉❧❛r ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ ❞♦✐s ✐♥t❡✐r♦s✳ ❖✉tr♦ ❡①❡♠♣❧♦✱ ♦s ♣❛ss♦s ♣❛r❛ ❝❛❧❝✉❧❛r ❛s ♠é❞✐❛s ✜♥❛✐s ❞♦s ❛❧✉♥♦s ❞❡ ✉♠❛ ❡s❝♦❧❛✳
❆♥t❡s ❞❡ ❡①❡❝✉t❛r ✉♠ ♣r♦❣r❛♠❛ ❡♠ ✉♠ ❝♦♠♣✉t❛❞♦r✱ ♦ ♣r♦❣r❛♠❛ ❞❡✈❡ s❡r ❝♦❞✐✜✲ ❝❛❞♦ ❡♠ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❞❡ ♣r♦❣r❛♠❛çã♦ ❞❛ ❡s❝♦❧❤❛ ❞♦ ♣r♦❣r❛♠❛❞♦r✳ ❊✱ ❛♥t❡s ❞❡ ❝♦❞✐✜❝❛r ♦ ♣r♦❣r❛♠❛✱ ♦ ♣r♦❣r❛♠❛❞♦r ❞❡✈❡ t❡r ❡s❝♦❧❤✐❞♦ ✭❝r✐❛❞♦✱ ❞❡s❝♦❜❡rt♦✮ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❛ r❡s♦❧✉✲ çã♦ ❞♦ ♣r♦❜❧❡♠❛✳ P♦rt❛♥t♦✱ ❛ ❝r✐❛çã♦ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ ♣r❡❝❡❞❡ ❛ ♣ró♣r✐❛ ❡s❝r✐t❛ ❞♦ ♣r♦❣r❛♠❛ ♥❛ ❧✐♥❣✉❛❣❡♠ ❞❡ ♣r♦❣r❛♠❛çã♦✳ ❆♥t❡s ❞❡ ❝r✐❛r ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ r❡s♦❧✈❡r ✉♠ ♣r♦❜❧❡♠❛✱ ♦✉ ❡s❝♦❧❤❡r ✉♠ ❛❧❣♦r✐t♠♦ ❥á ♣r♦♥t♦✱ ♥ã♦ ❤á ❝♦♠♦ ❝♦❞✐✜❝á✲❧♦ ♥✉♠❛ ❧✐♥❣✉❛❣❡♠ ❞❡ ♣r♦❣r❛♠❛çã♦✱ ♠✉✐t♦ ♠❡♥♦s ❝♦♠♦ ❡①❡❝✉tá✲❧♦ ❡♠ ✉♠ ❝♦♠♣✉t❛❞♦r✳
❖s ❛❧❣♦r✐t♠♦s ♣♦❞❡♠ r❡❛❧✐③❛r ❛ ♠❡s♠❛ t❛r❡❢❛✱ ❝❛❞❛ ✉♠ ❣❛st❛♥❞♦ ♠❛✐s ♦✉ ♠❡♥♦s t❡♠♣♦✱ ❡s♣❛ç♦ ♦✉ ❡s❢♦rç♦ ❞♦ q✉❡ ♦ ♦✉tr♦✳ P❛r❛ q✉❛❧q✉❡r ♣r♦❝❡ss♦ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ♦ ❛❧❣♦r✐t♠♦ ♣r❡❝✐s❛ ❡st❛r ❜❡♠ ❞❡✜♥✐❞♦✱ ❛ ❝♦rr❡t✐✈✐❞❛❞❡ ❞♦ ❛❧❣♦r✐t♠♦ ♣r❡❝✐s❛ s❡r ❡st✉❞❛❞❛ ♠❛t❡♠❛t✐❝❛✲ ♠❡♥t❡✱ ❛ q✉❛♥t✐❞❛❞❡ ❛ss✐♥tót✐❝❛ ❞❡ t❡♠♣♦ ❡ ❡s♣❛ç♦ ✭❝✉st♦✮ ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ s✉❛ ❡①❡❝✉çã♦ ❞❡✈❡ s❡r ❛♥❛❧✐s❛❞❛✳
❯♠❛ ✐♥tr♦❞✉çã♦ s♦❜r❡ ❛❧❣♦r✐t♠♦s é ❛♣r❡s❡♥t❛❞❛ ♥♦ ❝❛♣ít✉❧♦ ✷✱ tr❛t❛♥❞♦ ❞♦ ❝♦♥✲ ❝❡✐t♦✱ r❡♣r❡s❡♥t❛çã♦✱ ❝✉st♦ ❞❡ ❡①❡❝✉çã♦ ❡ ❡✜❝✐ê♥❝✐❛ ❞❡ ❛❧❣♦r✐t♠♦s✳
◆♦ ❝❛♣ít✉❧♦ ✸✱ sã♦ ❞✐s❝✉t✐❞♦s✱ ❛❧é♠ ❞❡ ♦✉tr♦s ❡①❡♠♣❧♦s✱ ♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✱ ❞❡ ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❙tr❛ss❡♥✱ ♣❛r❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s✳
✷
❈❛♣ít✉❧♦ ✷
❆❧❣♦r✐t♠♦s
■♥❢♦r♠❛❧♠❡♥t❡✱ ✉♠ ❛❧❣♦r✐t♠♦ é ✉♠❛ s❡q✉ê♥❝✐❛ ♣r❡❝✐s❛ ✭s❡♠ ❞ú✈✐❞❛s s♦❜r❡ ♦ q✉❡ ❞❡✈❡ s❡r ❢❡✐t♦✮ ❞❡ ✐♥str✉çõ❡s q✉❡✱ s❡ ❧✐❞❛ ❡ ❡①❡❝✉t❛❞❛ ♣♦r q✉❛❧q✉❡r ♣❡ss♦❛ ♦✉ ♣♦r ✉♠ ❝♦♠✲ ♣✉t❛❞♦r✱ ♣r♦❞✉③ ♦ r❡s✉❧t❛❞♦ ❡s♣❡r❛❞♦✱ ✐st♦ é✱ ❛ s♦❧✉çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛✳ ❊st❛ s❡q✉ê♥❝✐❛ ❞❡ ✐♥str✉çõ❡s é ✉♠❛ ❞❡s❝r✐çã♦ ❞❛ s❡q✉ê♥❝✐❛ ❞♦s ♣❛ss♦s ♥❡❝❡ssár✐♦s✱ ♦s q✉❛✐s ❞❡✈❡♠ s❡r ❡①❡❝✉t❛✲ ❞♦s✱ ♣❛r❛ ♠❛♥✐♣✉❧❛r ✐♥❢♦r♠❛çõ❡s✱ ♦✉ ❞❛❞♦s✱ ♣❛r❛ s❡ ❝❤❡❣❛r ♥❛ r❡s♣♦st❛ ❞♦ ♣r♦❜❧❡♠❛✳
▼❛✐s ❢♦r♠❛❧♠❡♥t❡✱ ✉♠ ❛❧❣♦r✐t♠♦ é ✉♠❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛A= (i1, i2, i3, ..., in)❞❡
✐♥str✉çõ❡sik, k ∈ {1, 2, ..., n}✱ t❛✐s q✉❡✿
✶✳ ik ♥ã♦ ❛♣r❡s❡♥t❛ ❛♠❜✐❣✉✐❞❛❞❡✱ ∀k ∈ {1, 2, ..., n}❀
✷✳ ❞❡♣♦✐s ❞❛ ❡①❡❝✉çã♦ ❞❡ ik✱ ♥ã♦ ❤❛✈❡rá ❛♠❜✐❣✉✐❞❛❞❡ s♦❜r❡ ij, j ∈ {k+ 1, k+ 2, ..., n}❀
✸✳ ❛ ✐♥str✉çã♦ ❞❡ ♣❛r❛r é s❡♠♣r❡ ❛❧❝❛♥ç❛❞❛ ❞❡♣♦✐s ❞❛ ❡①❡❝✉çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ✐♥str✉çõ❡s❀ ❡
✹✳ ❤á ♣r♦❞✉çã♦ ❞❡ r❡s✉❧t❛❞♦ ❝♦rr❡t♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❛ s❡r s♦❧✉❝✐♦♥❛❞♦✳
❊①❡♠♣❧♦ ✶✳ ❆❜❛✐①♦ ❡stã♦ ❛s ✐♥str✉çõ❡s ✐♥s❝r✐t❛s ❡♠ ✉♠❛ ♣❧❛❝❛ ❞❡ ♠❡t❛❧ ❛♥❡①❛❞❛ à ❢r❡♥t❡ ❞❡ ✉♠ ✈✐❞❡♦❣❛♠❡✿
P❛ss♦ ✶ (i1)✿ ■♥s✐r❛ ✈✐♥t❡ ❡ ❝✐♥❝♦ ❝❡♥t❛✈♦s ♥❛ ❢❡♥❞❛ ♣❛r❛ ♠♦❡❞❛s ❛♦ ❧❛❞♦ ❞❛
♠áq✉✐♥❛✳
P❛ss♦ ✷(i2)✿ Pr❡ss✐♦♥❡ ♦ ❜♦tã♦ ✈❡r❞❡ ♥♦ t♦♣♦ ❞❛ ♠áq✉✐♥❛ q✉❛♥❞♦ ❡st✐✈❡r ♣r♦♥t♦
♣❛r❛ ✐♥✐❝✐❛r✳
❱❡❥❛ q✉❡ ❡①✐st❡♠ ❞✉❛s ✐♥str✉çõ❡s✱ i1 ❡ i2✱ ❜❡♠ ❞❡✜♥✐❞❛s✱ s❡♠ ❛♠❜✐❣✉✐❞❛❞❡s✳ ❍á
❝❧❛r❡③❛ s♦❜r❡ ♦ ♣ró①✐♠♦ ♣❛ss♦ ❛ s❡r ❡①❡❝✉t❛❞♦ ✭i2✮✳ ❆ ✐♥str✉çã♦ ❞❡ ♣❛r❛r ❢♦✐ ♦♠✐t✐❞❛✱ ♣♦✐s
✜❝❛ ❝❧❛r♦ q✉❡ ❝❛❞❛ ✐♥str✉çã♦ é ❡①❡❝✉t❛❞❛ ❛♣❡♥❛s ✉♠❛ ✈❡③ ♣❛r❛ ❝❛❞❛ ❥♦❣♦✳ ■st♦ ♣❡r♠✐t❡ ❞✐③❡r✱ ❝♦♥❢♦r♠❡ ❛ ❞❡✜♥✐çã♦✱ q✉❡ t❛✐s ♣❛ss♦s r❡♣r❡s❡♥t❛♠ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ✐♥✐❝✐❛r ✉♠ ❥♦❣♦✱ s❡♥❞♦ ❡st❡ ✐♥❞✐❝❛❞♦ ♣♦r A(i1, i2)✳
❊①❡♠♣❧♦ ✷✳ ❙✉♣♦♥❤❛ q✉❡ ✉♠❛ t❡r❝❡✐r❛ ✐♥str✉çã♦✱i3✱ s❡❥❛ ❛❞✐❝✐♦♥❛❞❛ ♥♦ ❡①❡♠♣❧♦✶✱ ❝♦♥❢♦r♠❡
♦ q✉❡ s❡❣✉❡ ❛❜❛✐①♦✿
P❛ss♦ ✸ (i3)✿ ◗✉❛♥❞♦ ❝❛❞❛ ❥♦❣♦ ❢♦r ✜♥❛❧✐③❛❞♦✱ ❡s❝r❡✈❛ s✉❛s ✐♥✐❝✐❛✐s ❡ ♣r❡ss✐♦♥❡ ♦
✸
❖ ♣❛ss♦ ✸ ❝✉♠♣r❡ ♦ ♣❛♣❡❧ ❞❡ ✉♠❛ ✐♥str✉çã♦ ❞❡ ♣❛r❛❞❛ ❞♦ ❥♦❣♦ ❢♦r♥❡❝✐❞♦ ♣❡❧❛ ♠áq✉✐♥❛✱ ♣❛r❛ ♥ã♦ ❞✉r❛r ♣❛r❛ s❡♠♣r❡✱ ❡st❛♥❞♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ t❡r❝❡✐r❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❞❡✜♥✐çã♦ ❢♦r♠❛❧ ❞❡ ❛❧❣♦r✐t♠♦✱ ❡ ♠♦str❛ q✉❡ ✈♦❝ê ♥ã♦ ♣♦❞❡rá ❡❢❡t✉❛r ✉♠ s❡❣✉♥❞♦ ❥♦❣♦ ❝♦♠ ♦ ❞✐♥❤❡✐r♦ ♣❛❣♦ ✐♥✐❝✐❛❧♠❡♥t❡ ♣❛r❛ ♦ ♣r✐♠❡✐r♦ ❥♦❣♦✳
❊①❡♠♣❧♦ ✸✳ ❙✉♣♦♥❤❛ ❛✐♥❞❛ q✉❡ s❡❥❛ ✐♥s❡r✐❞❛ ❛ s❡❣✉✐♥t❡ ✐♥str✉çã♦✱i4✱ ♥♦ ❡①❡♠♣❧♦ ✶✿
P❛ss♦ ✹ (i4)✿ P❛r❛ ❝❛❞❛ ✶✵✳✵✵✵ ♣♦♥t♦s q✉❡ ✈♦❝ê ❛❝✉♠✉❧❛r✱ ❣❛♥❤❛rá ✉♠ ❥♦❣♦
❣rát✐s✳ ◗✉❛♥❞♦ ♦ ❥♦❣♦ ❡♠ ❡①❡❝✉çã♦ ✜♥❛❧✐③❛r✱ s❡ ✈♦❝ê t❡♠ ❞✐r❡✐t♦ ❛ ✉♠ ❥♦❣♦ ❣rát✐s✱ ✈á ♣❛r❛ ♦ ♣❛ss♦ ✷ ✭i2✮✳
❖ ♣❛ss♦ ✹ ❝♦♥tr❛r✐❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ❛❧❣♦r✐t♠♦✱ ♣♦✐s s❡r✐❛ ♣♦ssí✈❡❧ ♦❜t❡r ❛ s❡q✉ê♥❝✐❛ ✐♥✜♥✐t❛ A = (i1, i2, i3, i4, i2, i3, i4, i2, i3, i4, ...)✳ P♦rt❛♥t♦✱ A = (i1, i2, i3, i4) ♥ã♦
r❡♣r❡s❡♥t❛ ✉♠ ❛❧❣♦r✐t♠♦✳ ❖ ♣❛ss♦ ✹ é ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ❧♦♦♣ ✭❧❛ç♦ ❝♦♥❞✐❝✐♦♥❛❧✮✱ ♣♦r ♣❡r♠✐t✐r ❛ ❡①❡❝✉çã♦ ❞❡ ✉♠ ♣❛ss♦ ♠❛✐s ❞♦ q✉❡ ✉♠❛ ✈❡③✱ t♦❞❛ ✈❡③ q✉❡ ❛ ❛✜r♠❛çã♦ ❞❡❧❡ ❢♦r ✈❡r❞❛❞❡✐r❛✳ ❖s ❡①❡♠♣❧♦s✶❡✷sã♦ ❝❤❛♠❛❞♦s ❞❡ ❛❧❣♦r✐t♠♦s s❡q✉❡♥❝✐❛✐s✱ ♣♦✐s ♥ã♦ ♣❡r♠✐t❡♠ ♦ r❡t♦r♥♦ ♣❛r❛ ❛ ❡①❡❝✉çã♦ ❞❡ ✉♠❛ ✐♥str✉çã♦ ❛♥t❡r✐♦r✱ ♠❛✐s ❞❡ ✉♠❛ ✈❡③✳
P❛r❛ tr❛♥s❢♦r♠❛r ❛ ✐♥❝❧✉sã♦ ❡ ♣❡r♠✐t✐r q✉❡A= (i1, i2, i3, i4)s❡❥❛ ✉♠ ❛❧❣♦r✐t♠♦✱
♣♦❞❡✲s❡ r❡❡s❝r❡✈ê✲❧♦ ❝♦♥❢♦r♠❡ ♦ q✉❡ s❡❣✉❡ ❛❜❛✐①♦✳
❊①❡♠♣❧♦ ✹✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ✐♥str✉çã♦i4 t❡♥❤❛ s✐❞♦ ♠♦❞✐✜❝❛❞❛✱ ❝♦♥❢♦r♠❡ ♦ q✉❡ s❡❣✉❡✿
P❛ss♦ ✹ (i4)✿ P❛r❛ ❝❛❞❛ ✶✵✳✵✵✵ ♣♦♥t♦s q✉❡ ❛❝✉♠✉❧❛r✱ ✈♦❝ê ❣❛♥❤❛rá ✉♠ ❥♦❣♦ ❞❡
❣r❛ç❛✱ s❡♥❞♦ ♦ ♥ú♠❡r♦ ♠á①✐♠♦ ❞❡ ❥♦❣♦s ❣r❛t✉✐t♦s ❣❛♥❤♦s ✐❣✉❛❧ ❛ ✶✵ ♣❛r❛ ❝❛❞❛ ❥♦❣♦ ♣❛❣♦✳ ◗✉❛♥❞♦ ♦ ❥♦❣♦ ❡♠ ❡①❡❝✉çã♦ ✜♥❛❧✐③❛r✱ s❡ ✈♦❝ê t❡♠ ❞✐r❡✐t♦ ❛ ✉♠ ❥♦❣♦ ❣rát✐s✱ ✈á ♣❛r❛ ♦ ♣❛ss♦ ✷✳
❱❡❥❛ ❛❣♦r❛ q✉❡ ♦ ♥♦✈♦ ♣❛ss♦ ✹ ✭❧♦♦♣✮ s❡rá ❡①❡❝✉t❛❞♦ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ✈❡③❡s✳ ❆❣♦r❛✱ ♥❡st❛s ❝♦♥❞✐çõ❡s✱ A= (i1, i2, i3, i4) é ✉♠ ❛❧❣♦r✐t♠♦✳
❆ s❡q✉ê♥❝✐❛ ❞❡ ✐♥str✉çõ❡s ♥ã♦ s❡rá ✉♠ ❛❧❣♦r✐t♠♦ ❝♦rr❡t♦ s❡ ❢❛❧t❛r❡♠ ✐♥str✉çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ❛ ♣r♦❞✉çã♦ ❞❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✳
❙✉♣♦♥❤❛ q✉❡x❡y s❡❥❛♠ ❞✉❛s ✈❛r✐á✈❡✐s ❛ss✉♠✐♥❞♦ ♦s r❡s♣❡❝t✐✈♦s ✈❛❧♦r❡s ✺ ❡ ✷✳ ➱
✉♠❛ t❛r❡❢❛ ❝♦♠✉♠ tr♦❝❛r ♦s ✈❛❧♦r❡s ❞❛s ✈❛r✐á✈❡✐s✱ ❝♦♠♦ s❡ ❡st✐✈❡ss❡ ❛rr❛♥❥❛♥❞♦ ♦s ♥ú♠❡r♦s ✷ ❡ ✺ ♣❛r❛ ❛s ♣♦s✐çõ❡sx ❡ y✱ ✜①❛❞❛s✳ ◆❡st❡ ❝❛s♦✱ ❛♣ós ❛ tr♦❝❛✱ t❡♠✲s❡ q✉❡ x= 2 ❡ y= 5✳ ❊①❡♠♣❧♦ ✺✳ ❖s ♣❛ss♦s ❛❜❛✐①♦ r❡♣r❡s❡♥t❛♠ ✉♠❛ ❢♦r♠❛ ♣❛r❛ s❡ tr♦❝❛r ♦s ✈❛❧♦r❡s ❛ss✉♠✐❞♦s ♣❡❧❛s ✈❛r✐á✈❡✐sx ❡ y✳
P❛ss♦ ✶✿ ❋❛ç❛ x ❛ss✉♠✐r ♦ ✈❛❧♦r ❞❡ y✳ P❛ss♦ ✷✿ ❋❛ç❛ y ❛ss✉♠✐r ♦ ✈❛❧♦r ❞❡ x✳ P❛ss♦ ✸✿ P❛r❡✳
❙❡rá q✉❡ ♦ ❛❧❣♦r✐t♠♦ ❞❡st❡ ❡①❡♠♣❧♦ ❡stá ❝♦rr❡t♦❄ ❱❡❥❛ ❝♦♠♦ ❡❧❡ ❢✉♥❝✐♦♥❛rá ♣❛r❛ ❛ s✉♣♦s✐çã♦ ❞❛❞❛ ❛❝✐♠❛✳
■♥str✉çã♦ ❱❛❧♦r ❛ss✉♠✐❞♦ ♣❛r❛ ① ❱❛❧♦r ❛ss✉♠✐❞♦ ♣❛r❛ ②
❆♥t❡s ❞❡ i1 ✺ ✷
❉❡♣♦✐s ❞❡ i1 ✷ ✷
✹
P♦r q✉❡ ❞❡♣♦✐s ❞♦ ♣❛ss♦ ✷ ❛ ✈❛r✐á✈❡❧y♥ã♦ ❛ss✉♠✐✉ ♦ ✈❛❧♦r ✺❄ ❆❝♦♥t❡❝❡ q✉❡ ❞❡♣♦✐s
❞♦ ♣❛ss♦ ✶ ♦ ✈❛❧♦r ✺ ❢♦✐ ❡sq✉❡❝✐❞♦✳ ▼❛s é ♣r❡❝✐s♦ ♦❜t❡r y = 5✱ ❞❡♣♦✐s ❞♦ ♣❛ss♦ ✷✳ P♦ré♠✱
❝♦♠♦ ❢❛③❡r ✐ss♦ s❡ ♦ ✈❡❧❤♦ ✈❛❧♦r ❞❡x✱ ✺✱ ❢♦✐ r❡❣r❛✈❛❞♦ ♣❛r❛ ♦ ✈❛❧♦r ✷❄ ❯♠❛ s♦❧✉çã♦ ♣❛r❛ t❛❧
♣r♦❜❧❡♠❛ é ❛♣r❡s❡♥t❛❞❛ ♥♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦✳
❊①❡♠♣❧♦ ✻✳ ❙❡❣✉❡ ❛❜❛✐①♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✐♥str✉çõ❡s q✉❡ r❡❛❧✐③❛rá ❛ tr♦❝❛ ❞♦s ✈❛❧♦r❡s ❝♦♥❢♦r♠❡ ❞❡s❡❥❛❞♦✳
❊♥tr❛❞❛✿ x ❡ y✳
❙❛í❞❛✿ x ❡y ❝♦♠ ✈❛❧♦r❡s tr♦❝❛❞♦s✳
P❛ss♦ ✶✿ ❆tr✐❜✉❛ ♦ ✈❛❧♦r ❞❡ y ❛ ✉♠❛ ✈❛r✐á✈❡❧ ❛✉①✐❧✐❛r ❝❤❛♠❛❞❛ ②❴✈❡❧❤♦✳ P❛ss♦ ✷✿ ❋❛ç❛ y ❛ss✉♠✐r ♦ ✈❛❧♦r ❞❡ x (y :=x)✳
P❛ss♦ ✸✿ ❋❛ç❛ x ❛ss✉♠✐r ♦ ✈❛❧♦r ❞❡ ②❴✈❡❧❤♦✳ P❛ss♦ ✹✿ P❛r❡✳
❖ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ é ♠♦str❛❞♦ ❧♦❣♦ ❛❜❛✐①♦✳
P❛ss♦s ❱❛❧♦r ❛ss✉♠✐❞♦ ♣❛r❛ ① ❱❛❧♦r ❛ss✉♠✐❞♦ ♣❛r❛ ②
P❛ss♦ ✶ ✺ ✷
P❛ss♦ ✷ ✺ ✺
P❛ss♦ ✸ ✷ ✺
P❛ss♦ ✹ P❛r❡✳
P❛r❛ s♦❧✉❝✐♦♥❛r ✉♠ ♣r♦❜❧❡♠❛✱ ✉♠ ❛❧❣♦r✐t♠♦ ♣r❡❝✐s❛ ❞❡ ✉♠❛ ❡♥tr❛❞❛✱ r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠❛ ♦✉ ♠❛✐s ✈❛r✐á✈❡✐s✱ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ♦ ♣r♦❜❧❡♠❛✳ ❆♣ós ❛ ❡①❡❝✉çã♦ ❞❡ t♦❞❛s ❛s ✐♥str✉çõ❡s✱ ♦ ❛❧❣♦r✐t♠♦ ♣r♦❞✉③ ✉♠❛ s❛í❞❛ q✉❡ é ❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✳
❊①❡♠♣❧♦ ✼✳ ✭❈♦♥❥❡❝t✉r❛ ❞❡ ▲♦t❤❛r ❈♦❧❧❛t③✮ ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ♣r♦❝❡❞✐♠❡♥t♦✿
❊♥tr❛❞❛✿ ❖ ♥ú♠❡r♦ ♣♦s✐t✐✈♦ z✳ ❙❛í❞❛✿ ❖ ♥ú♠❡r♦ ✶✳
P❛ss♦ ✶✿ ❊♥tr❡ ❝♦♠ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ③✳ P❛ss♦ ✷✿ ❙❡ ③ ❢♦r ♣❛r✱ tr♦q✉❡ ♦ ✈❛❧♦r ❞❡ ③ ♣♦r z
2✳
P❛ss♦ ✸✿ ❙❡ z = 1✱ ❡♥tã♦ ❛ s❛í❞❛ s❡rá ③ ❡ ♣❛r❡✳
P❛ss♦ ✹✿ ❙❡ ③ ❢♦r í♠♣❛r✱ tr♦q✉❡ ♦ ✈❛❧♦r ❞❡ ③ ♣♦r 3z+ 1✳
P❛ss♦ ✺✿ ❱á ♣❛r❛ i2✳
❊st❡ é ✉♠ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦ ♣❡❧♦ ❛❧❡♠ã♦ ▲♦t❤❛r ❈♦❧❧❛t③ ♣❛r❛ ♦ q✉❛❧ ♥ã♦ s❡ s❛❜❡ ❛✐♥❞❛ s❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♣r♦❞✉③✐❞❛ ✜♥❞❛rá ❡♠ ✶✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦✳ ❊❧❡ só s❡rá ✉♠ ❛❧❣♦r✐t♠♦✱ s❡ ❤♦✉✈❡r ♣❛r❛❞❛ ❝♦♠ ♣r♦❞✉çã♦ ❞❡ r❡s✉❧t❛❞♦ ✜♥❛❧z = 1✳
❊①❡♠♣❧♦ ✽✳ ❊①❡❝✉t❛♥❞♦ ❛ ❝♦♥❥❡❝t✉r❛ ❞❡ ❈♦❧❧❛t③ ♣❛r❛z = 1, z= 20 ❡ z = 7✱ t❡♠✲s❡✿ ❙❡q✉ê♥❝✐❛ ♦❜t✐❞❛ ♣❛r❛ z = 1✿ ✭✶✮✳
❙❡q✉ê♥❝✐❛ ♦❜t✐❞❛ ♣❛r❛ z = 20✿ ✭✷✵✱ ✶✵✱ ✺✱ ✶✻✱ ✽✱ ✹✱ ✷✱ ✶✮✳
✺
❊①❡♠♣❧♦ ✾✳ ❆ s❡q✉ê♥❝✐❛ (1,1,2,3,5,8,13,21, ...) é ❝♦♠♣♦st❛ ♣❡❧♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✱ ♦❜❡❞❡❝❡♥❞♦ à r❡❝♦rrê♥❝✐❛ an = an−1 +an−2✱ ♣❛r❛ a1 = a2 = 1 ❡ n > 3✱ ♦✉ s❡❥❛✱ ♣❛r❛ s❡
♦❜t❡r ♦ ♥✲és✐♠♦ t❡r♠♦ ❞❡st❛ s❡q✉ê♥❝✐❛✱ r❡❝♦rr❡✲s❡ ❛♦s ❞♦✐s t❡r♠♦s ❛♥t❡r✐♦r❡s ❛an✱ s♦♠❛♥❞♦✲
♦s ♣❛r❛ ❞❛r ♦ ✈❛❧♦r ❞❡ an✳ ❙❡❣✉❡ ❛❜❛✐①♦ ✉♠ ❛❧❣♦r✐t♠♦ q✉❡ r❡t♦r♥❛ ♦s n ♣r✐♠❡✐r♦s ♥ú♠❡r♦s
❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ ♣❛r❛n>3✳ ❊♥tr❛❞❛✿ ❖ ♥ú♠❡r♦ n✱ s❡♥❞♦ n>3✳
❙❛í❞❛✿ ❖s n ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✳ P❛ss♦ ✶✿ ❈♦♥s✐❞❡r❡ a1 =a2 = 1✳
P❛ss♦ ✷✿ P❛r❛ i := 3 ❛té n✱ ❢❛ç❛ ai :=ai−1+ai−2✳
P❛ss♦ ✸✿ P❛r❛ i := 1 ❛té n✱ r❡t♦r♥❡ ai✳
P❛ss♦ ✹✿ P❛r❡✳
❊①❡♠♣❧♦ ✶✵✳ ❯♠❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛ ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s (a1, a2, ..., an) s❡rá ❝r❡s❝❡♥t❡ s❡
a1 6a2 6a3 6...6an✳ ❖ ❛❧❣♦r✐t♠♦ ❛ s❡❣✉✐r ❧ê ♦s ♥ú♠❡r♦s ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛ ❡ ❞✐③ s❡
❡❧❛ é ❝r❡s❝❡♥t❡ ♦✉ ♥ã♦✳
❊♥tr❛❞❛✿ ❆ q✉❛♥t✐❞❛❞❡ n ❞❡ ❡❧❡♠❡♥t♦s ❞❛ s❡q✉ê♥❝✐❛ ❡ ♦s s❡✉s ❡❧❡♠❡♥t♦s✿ a1, a2, ..., an✳
❙❛í❞❛✿ ❆♣❡♥❛s ✉♠❛ ❞❛s ❞✉❛s ❢r❛s❡s✿ ✏❊st❛ s❡q✉ê♥❝✐❛ ♥ã♦ é ❝r❡s❝❡♥t❡✑ ♦✉ ✏❊st❛ s❡q✉ê♥❝✐❛
é ❝r❡s❝❡♥t❡✑✳
P❛ss♦ ✶✿ P❛r❛ i := 1 ❛té n −1✱ s❡ ai > ai+1 ❡s❝r❡✈❛✿ ✏❊st❛ s❡q✉ê♥❝✐❛ ♥ã♦ é ❝r❡s❝❡♥t❡✑✳
❈❛s♦ ❝♦♥trár✐♦✱ ❡s❝r❡✈❛✿ ✏❊st❛ s❡q✉ê♥❝✐❛ é ❝r❡s❝❡♥t❡✑✳
P❛ss♦ ✷✿ P❛r❡✳
P❛r❛ ❝♦♠♣r❡❡♥❞❡r ♠❡❧❤♦r ❝♦♠♦ sã♦ ❡①❡❝✉t❛❞❛s ❛s ✐♥str✉çõ❡s ❞♦ ❛❧❣♦r✐t♠♦✱ ♣♦❞❡✲ s❡ ❡❢❡t✉❛r ♦ ❝❤❛♠❛❞♦ t❡st❡ ❞❡ ♠❡s❛✱ q✉❡ ❝♦♥s✐st❡ ♥❛ ❡①❡❝✉çã♦ ♠❡♥t❛❧ ❞❡ss❡ ❛❧❣♦r✐t♠♦✱ ♦ q✉❛❧ t❛♠❜é♠ é ❝❤❛♠❛❞♦ ❞❡ ❝ó❞✐❣♦✱ s❡♥❞♦ ♦s ♣❛ss♦s ❛♥♦t❛❞♦s✱ ❡♠ s❡q✉ê♥❝✐❛ ❞❡ ❡①❡❝✉çã♦✱ ♥♦ ♣❛♣❡❧✳ ◆❡ss❡ t❡st❡✱ ❛ ♣❡ss♦❛ s❡rá ♦ ♣r♦❝❡ss❛❞♦r ❡ ✜❝❛rá r❡s♣♦♥sá✈❡❧ ♣♦r ❡①❡❝✉t❛r ❝❛❞❛ ❧✐♥❤❛ ❞♦ ❝ó❞✐❣♦✳ ❖ t❡st❡ ♣❡r♠✐t❡ ✈❡r✐✜❝❛r s❡ ♦ ❝ó❞✐❣♦ ❡stá r❡❛❧♠❡♥t❡ ❡①❡❝✉t❛♥❞♦ ❛ ❢✉♥çã♦ ♣❛r❛ ❛ q✉❛❧ ❢♦✐ ❝r✐❛❞♦✱ ❡ s❡ ❡①✐st❡ ❛❧❣✉♠ ❡rr♦ ❞❡ ❧ó❣✐❝❛ ♥♦s ♣❛ss♦s ❛♣r❡s❡♥t❛❞♦s✳
❖ t❡st❡ s✉r❣✐✉ ❡♠ ✉♠❛ é♣♦❝❛ ♥❛ q✉❛❧ ♣r♦❣r❛♠❛r ✉♠ ❝♦♠♣✉t❛❞♦r ❡r❛ ❡①tr❡♠❛✲ ♠❡♥t❡ ❝♦♠♣❧✐❝❛❞♦✳ ❆♣❡♥❛s ❛❝❡✐t❛r q✉❡ ✉♠ ❝ó❞✐❣♦ ❡st❛✈❛ ❝♦rr❡t♦ ❡r❛ ✉♠ ❞❡s♣❡r❞í❝✐♦ ❞❡ t❡♠♣♦ ❡✱ ♣♦rt❛♥t♦✱ ❡r❛ ♥❡❝❡ssár✐♦ t❡r ❝❡rt❡③❛ ❞❡ q✉❡ ❛q✉❡❧❡ ♣r♦❣r❛♠❛ ❢✉♥❝✐♦♥❛r✐❛✳ ▼✉♥✐❞♦s✱ ❡♥tã♦✱ ❞❡ ♣❛♣❡❧ ❡ ❧á♣✐s✱ ♦s ♣r♦❣r❛♠❛❞♦r❡s ❢❛③✐❛♠ s✐♠✉❧❛çõ❡s ❞❡ ❡①❡❝✉çã♦ ❡ ❝♦rr✐❣✐❛♠ ❡rr♦s ❡♥❝♦♥tr❛✲ ❞♦s ❛♥t❡s ♠❡s♠♦ ❞❡ ❡①❡❝✉t❛r ♦ ❝ó❞✐❣♦ ♥❛ ♠áq✉✐♥❛ r❡❛❧✳ ❊ss❛ ♣rát✐❝❛ s❡ ♣❡r❞❡✉ ❝♦♠ ♦ t❡♠♣♦✱ ❝♦♠ ♦ ❛❞✈❡♥t♦ ❞❛ ♣r♦❣r❛♠❛çã♦ ✐♥t❡r❛t✐✈❛✱ ♥❛ q✉❛❧ ♦ ♣r♦❣r❛♠❛❞♦r t❡♠ ❛❝❡ss♦ à ♠áq✉✐♥❛ ❡ ♣♦❞❡ t❡st❛r ❛❧t❡r❛çõ❡s✳
❆ ✈❛♥t❛❣❡♠ ❞❡ss❡ t✐♣♦ ❞❡ t❡st❡ é q✉❡ ❡❧❡ ❛❥✉❞❛ ❛ ❞❡s❡♥✈♦❧✈❡r ♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦ ❞♦ ♣r♦❣r❛♠❛❞♦r✱ q✉❡✱ ❞❡♣♦✐s ❞❡ ✉♠ ❝❡rt♦ t❡♠♣♦ ❞❡ ♣rát✐❝❛✱ ❝♦♥s❡❣✉❡ ✐❞❡♥t✐✜❝❛r ♣r♦❜❧❡♠❛s ❛♣❡♥❛s ♦❧❤❛♥❞♦ ♣❛r❛ ♦ ❝ó❞✐❣♦ ✐♠♣r❡ss♦✱ s❡♠ ♥❡❝❡ss✐t❛r ❡①❡❝✉tá✲❧♦✳
❊①❡♠♣❧♦ ✶✶✳ ❉❛❞❛ ✉♠❛ ❧✐st❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s✿ a1, a2, ..., an✱ ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞♦s
♥ú♠❡r♦s ❞❡st❛ ❧✐st❛ é ❞❛❞❛ ♣♦rM A= a1+a2+...+an
n ✳ ❱❡❥❛ ❛❜❛✐①♦✱ à ❡sq✉❡r❞❛✱ ♦ ❛❧❣♦r✐t♠♦
✻
✲ ❊♥tr❛❞❛✿ n ❡ ♦s ✈❛❧♦r❡s ❞❡ a1, a2, ..., an✳ ❱❡❥❛ ♦s ♣❛ss♦s ❡①❡❝✉t❛❞♦s
✲ ❙❛í❞❛✿ ❆ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞♦s n ♥ú♠❡r♦s✳ ❧♦❣♦ ❛❜❛✐①♦✿
✲ P❛ss♦ ✶✿ ❈♦♥s✐❞❡r❡ x= 0✳ ✲ P❛ss♦ ✶✿✲ P❛ss♦ ✷✿ x := 0✳
✲ P❛ss♦ ✷✿ P❛r❛ ❝❛❞❛ ✈❛❧♦r ❞❡i✱ ✈❛r✐❛♥❞♦ ❞❡ ✶ i := 1 ❡ x := 0 +a1 =a1✳
❛té n✱ ❢❛ç❛ x ❛ss✉♠✐r ♦ ✈❛❧♦r ❞♦ ❛♥t✐❣♦ x s♦✲ i := 2 ❡ x := a1
|{z}
x velho
+a2✳
♠❛❞♦ ❝♦♠ ai (x :=x+ai)✳ i := 3 ❡ x :=a1+a2
| {z }
x velho
+a3
✳✳✳
✲ P❛ss♦ ✸✿ ❉✐✈✐❞❛ x♣♦r n ❡ ❝♦♥s✐❞❡r❡ ♦ r❡s✉❧✲ i :=n ❡ x := t❛❞♦ ♦❜t✐❞♦ ❝♦♠♦M A M A= x
n
✳ =a1+a2+...+an−1
| {z }
x velho
+an
✲ P❛ss♦ ✹✿ ❘❡t♦r♥❡M A ❡ ♣❛r❡✳ ✲ P❛ss♦ ✸✿ M A := x n✳
✲ P❛ss♦ ✹✿ ❘❡t♦r♥❡ M A✳
❊①❡♠♣❧♦ ✶✷✳ P❛r❛ ❝❛❧❝✉❧❛r ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❞♦ ♣❧❛♥♦✿ A(x1, y1)❡ B(x2, y2)✱
✉t✐❧✐③❛✲s❡ ❛ ❢ór♠✉❧❛DA,B =
p
(x1−x2)2+ (y1−y2)2✱ ♥❛ q✉❛❧ DA,B ✐♥❞✐❝❛ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡
♦s ♣♦♥t♦sA❡B✳ ❆s ✐♥str✉çõ❡s ❛❜❛✐①♦ r❡♣r❡s❡♥t❛♠ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❝❛❧❝✉❧❛r ❡ss❛ ❞✐stâ♥❝✐❛✿ ❊♥tr❛❞❛✿ ❈♦♥s✐❞❡r❡ ♦s ✈❛❧♦r❡s✿ x1, x2, y1 ❡ y2 ✭❡♥tr❛❞❛✮✳
❙❛í❞❛✿ ❆ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❞♦✐s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ q✉❡ ♣♦ss✉❡♠ ❡ss❛s ❝♦♦r❞❡♥❛❞❛s✳ P❛ss♦ ✶✿ a :=x1−x2
P❛ss♦ ✷✿ b :=y1−y2
P❛ss♦ ✸✿ a :=a∗a ✭♥♦✈♦ ✈❛❧♦r ❞❡ a✮ P❛ss♦ ✹✿ b :=b∗b ✭♥♦✈♦ ✈❛❧♦r ❞❡ b✮ P❛ss♦ ✺✿ m :=a+b
P❛ss♦ ✻✿ DA,B :=sqrt(m)
P❛ss♦ ✼✿ ❊s❝r❡✈❛ ♦ ✈❛❧♦r ❞❡ DA,B ✭s❛í❞❛✮✳
P❛ss♦ ✽✿ P❛r❡✳
❖❜s❡r✈❛çã♦ ✶✳ sqrt✭♠✮ s✐❣♥✐✜❝❛√m✳
✷✳✶ ❆❧❣♦r✐t♠♦s ♣❛r❛ ❡①♣❛♥sã♦ ♥❛ ❜❛s❡ ❞❡❝✐♠❛❧ ❡ ❜✐♥ár✐❛
❖ s✐st❡♠❛ ❞❡❝✐♠❛❧ ♣♦s✐❝✐♦♥❛❧ é ✉t✐❧✐③❛❞♦ ✉♥✐✈❡rs❛❧♠❡♥t❡ ♣❡❧❛s ♣❡ss♦❛s ♣❛r❛ r❡✲ ♣r❡s❡♥t❛r ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❈❤❛♠❛✲s❡ ❞❡❝✐♠❛❧ ♣♦r s❡r r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦s ❞❡③ ❛❧❣❛r✐s♠♦s✿ ✵✱ ✶✱ ✷✱ ✸✱ ✹✱ ✺✱ ✻✱ ✼✱ ✽ ❡ ✾✱ ❝❤❛♠❛❞♦s ❞❡ ❛❧❣❛r✐s♠♦s ✐♥❞♦✲❛rá❜✐❝♦s✱ ♣♦✐s ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞♦s ♣❡❧♦ ♣♦✈♦ ✐♥❞✐❛♥♦ ❡ ❞✐✈✉❧❣❛❞♦s ♣❡❧♦ ♣♦✈♦ ár❛❜❡✳✼
❇✐♥ár✐♦ ●❧✐❢♦ ❇✐♥ár✐♦ ●❧✐❢♦
✵✵✶✵ ✶✵✶✶ ✰ ✵✵✶✶ ✵✶✵✵ ✹
✵✵✶✵ ✶✶✶✶ ✴ ✵✵✶✶ ✵✶✵✶ ✺
✵✵✶✶ ✵✵✵✵ ✵ ✵✵✶✶ ✵✶✶✵ ✻
✵✵✶✶ ✵✵✵✶ ✶ ✵✵✶✶ ✵✶✶✶ ✼
✵✵✶✶ ✵✵✶✵ ✷ ✵✵✶✶ ✶✵✵✵ ✽
✵✵✶✶ ✵✵✶✶ ✸ ✵✵✶✶ ✶✵✵✶ ✾
❖ s✐st❡♠❛ t❛♠❜é♠ é ❝❤❛♠❛❞♦ ♣♦s✐❝✐♦♥❛❧✱ ♣♦✐s ❝❛❞❛ ❛❧❣❛r✐s♠♦✱ ❛❧é♠ ❞♦ s❡✉ ✈❛❧♦r ✐♥trí♥s❡❝♦✱ ♣♦ss✉✐ ✉♠ ♣❡s♦ q✉❡ ❧❤❡ é ❛tr✐❜✉í❞♦ ❡♠ ❢✉♥çã♦ ❞❛ ♣♦s✐çã♦ q✉❡ ❡❧❡ ♦❝✉♣❛ ♥♦ ♥ú♠❡r♦✳ ◆♦ ❝❛s♦ ❞♦ s✐st❡♠❛ ❞❡❝✐♠❛❧✱ ❡ss❡ ♣❡s♦✱ é ✉♠❛ ♣♦tê♥❝✐❛ ❞❡ ❞❡③ ❡ ❡❧❡ ✈❛r✐❛ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿
❖ ❛❧❣❛r✐s♠♦ ❞❛ ❡①tr❡♠❛ ❞✐r❡✐t❛ t❡♠ ♣❡s♦ 100 ✭♣♦s✐çã♦ ✵✮❀ ♦ s❡❣✉✐♥t❡✱ s❡♠♣r❡ ❞❛
❞✐r❡✐t❛ ♣❛r❛ ❛ ❡sq✉❡r❞❛✱ t❡♠ ♣❡s♦ 101 ✭♣♦s✐çã♦ ✶✮❀ ♦ s❡❣✉✐♥t❡ t❡♠ ♣❡s♦ 102 ✭♣♦s✐çã♦ ✷✮❀ ♦
s❡❣✉✐♥t❡ t❡♠ ♣❡s♦103 ✭♣♦s✐çã♦ ✸✮✱ ❡t❝✳
❊①❡♠♣❧♦ ✶✸✳ ❈♦♥s✐❞❡r❡ ♦ ♥ú♠❡r♦ ✶✷✵✶✾ ♥❛ ❜❛s❡ ✶✵✳ ❆ss✐♠✱
12019 = 1·104+ 2·103 + 0·102 + 1·101 + 9·100 = 1·104 + 2·103 + 1·10 + 9
❖s s✐st❡♠❛s ❞❡ ♥✉♠❡r❛çã♦ ♣♦s✐❝✐♦♥❛✐s ❜❛s❡✐❛♠✲s❡ ♥♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r✱ q✉❡ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✳
❚❡♦r❡♠❛ ✶✳ ❉❛❞♦s a, b ∈ N ∪ {0}✱ ❝♦♠ b > 1✱ ❡①✐st❡♠ ♥ú♠❡r♦s ♥❛t✉r❛✐s c0, c1, ..., cn✱
♠❡♥♦r❡s ❞♦ q✉❡b✱ ✉♥✐✈♦❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦s✱ t❛✐s q✉❡ a=c0+c1b+c2b2 +...+cnbn✳
❉❡♠♦♥str❛çã♦✿
❖ t❡♦r❡♠❛ s❡rá ❞❡♠♦♥str❛❞♦ ♣♦r ♠❡✐♦ ❞❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✱ s♦❜r❡a✳ ❙❡ a= 0✱ ♦✉ s❡ a= 1✱ ❜❛st❛ t♦♠❛rn= 0 ❡c0 =a✱ ❡ ✈❛❧❡ ❛ ✉♥✐❝✐❞❛❞❡✳
❙✉♣♦♥❞♦ ♦ r❡s✉❧t❛❞♦ ✈á❧✐❞♦ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧ ♠❡♥♦r ❞♦ q✉❡a >1✱ s❡rá ♣r♦✈❛❞♦
s✉❛ ✈❛❧✐❞❛❞❡ t❛♠❜é♠ ♣❛r❛a✳ P❡❧❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✱ ❡①✐st❡♠q ❡r✱ ú♥✐❝♦s✱ t❛✐s q✉❡ a =bq+r✱ ❝♦♠ 06r < b✳
❚❡♠✲s❡✿
b >1⇒ab > a⇒a < ab6ab+r ⇒bq+r < ab+r⇒q < a✳
❈♦♠♦ q < a✱ ♣❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ s❡❣✉❡ q✉❡ ❡①✐st❡♠ ♥ú♠❡r♦s ♥❛t✉r❛✐s n′ ❡ d0, d1, ..., dn′ ✱ ❝♦♠dj < b✱ ♣❛r❛ t♦❞♦ j ✱ ✉♥✐✈♦❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦s✱ t❛✐s q✉❡
q =d0+d1b+· · ·+dn′bn ′
✳
▲❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ❛s ✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛ ❞❡st❛❝❛❞❛s✱ t❡♠✲s❡ q✉❡
a =bq+r=b(d0+d1b+· · ·+dn′bn ′
) +r✳
❖ r❡s✉❧t❛❞♦ s❡❣✉❡ t♦♠❛♥❞♦ c0 = r, n = n′ + 1 ❡ cj = dj❂1 ♣❛r❛ j = 1, ..., n✱ ♦s
✽
❈♦r♦❧ár✐♦ ✶✳ ❚♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ s❡ ❡s❝r❡✈❡ ❞❡ ♠♦❞♦ ú♥✐❝♦ ❝♦♠♦ s♦♠❛ ❞❡ ♣♦tê♥❝✐❛s ❞✐s✲ t✐♥t❛s ❞❡ ✷✳
❆ r❡♣r❡s❡♥t❛çã♦ ❞❛❞❛ ♥♦ t❡♦r❡♠❛ ❛❝✐♠❛ é ❝❤❛♠❛❞❛ ❞❡ ❡①♣❛♥sã♦ r❡❧❛t✐✈❛ à ❜❛s❡
b✳ ◗✉❛♥❞♦ b= 10✱ ❡ss❛ ❡①♣❛♥sã♦ é ❝❤❛♠❛❞❛ ❡①♣❛♥sã♦ ❞❡❝✐♠❛❧✱ ❡ q✉❛♥❞♦b = 2✱ ❡❧❛ r❡❝❡❜❡ ♦
♥♦♠❡ ❞❡ ❡①♣❛♥sã♦ ❜✐♥ár✐❛✳
❆ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ t❛♠❜é♠ ❢♦r♥❡❝❡ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ❡①♣❛♥sã♦ ❞❡ ✉♠ ♥ú♠❡r♦ q✉❛❧q✉❡r r❡❧❛t✐✈❛♠❡♥t❡ à ❜❛s❡b✳
❚r❛t❛✲s❡ ❞❡ ❛♣❧✐❝❛r✱ s✉❝❡ss✐✈❛♠❡♥t❡✱ ❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✱ ❝♦♠♦ s❡❣✉❡✿
a=bq0+r0, r0 < b,
q0 =bq1+r1, r1 < b,
q1 =bq2+r2, r2 < b,
❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❈♦♠♦ a > q0 > q1 > · · ·✱ ❞❡✈❡✲s❡✱ ❡♠ ✉♠ ❝❡rt♦ ♣♦♥t♦✱ t❡r qn❂1 < b ❡✱
♣♦rt❛♥t♦✱ ❞❡
qn❂1 =bqn+rn,
❞❡❝♦rr❡ q✉❡ qn = 0✱ ♦ q✉❡ ✐♠♣❧✐❝❛ 0 =qn=qn+1=qn+2 =· · ·✱ ❡ ♣♦rt❛♥t♦
0 =rn+1 =rn+2 =· · ·✳
❚❡♠✲s❡ ❡♥tã♦ q✉❡
a=r0+r1b+· · ·+rnbn✳
❆ ❡①♣❛♥sã♦ ♥✉♠❛ ❞❛❞❛ ❜❛s❡ b ❢♦r♥❡❝❡ ✉♠ ♠ét♦❞♦ ♣❛r❛ r❡♣r❡s❡♥t❛r ♦s ♥ú♠❡r♦s
♥❛t✉r❛✐s✳ P❛r❛ t❛♥t♦✱ ❡s❝♦❧❤❛ ✉♠ ❝♦♥❥✉♥t♦ S t❛❧ q✉❡ S = {1,2,3, ...,(b−1)}✳ ❯♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ a ♥❛ ❜❛s❡ b s❡ ❡s❝r❡✈❡ ❞❛ ❢♦r♠❛ xnxn❂1...x1x0, ❝♦♠ x0, ..., xn ∈ S✱ ❡ n ✈❛r✐❛♥❞♦✱
❞❡♣❡♥❞❡♥❞♦ ❞❡a✱ r❡♣r❡s❡♥t❛♥❞♦ ♦ ♥ú♠❡r♦ x0+x1b+· · ·+xnbn✳
◆♦ s✐st❡♠❛ ❞❡❝✐♠❛❧✱ ✐st♦ é✱ ❞❡ ❜❛s❡b = 10✱ ✉s❛✲s❡ S ❂ ④✵✱ ✶✱ ✷✱ ✸✱ ✹✱ ✺✱ ✻✱ ✼✱ ✽✱ ✾⑥✳
❊①❡♠♣❧♦ ✶✹✳ ❆ ❡①♣❛♥sã♦ ❞❡❝✐♠❛❧ ❞♦ ♥ú♠❡r♦ ✷✸✹✺✻ é ❞❛❞❛ ♣♦r✿ 2·104+ 3·103+ 4·102+ 5·101+ 6·100✳
❊①❡♠♣❧♦ ✶✺✳ ◆♦ s✐st❡♠❛ ❞❡ ❜❛s❡ b = 2✱ t❡♠✲s❡ q✉❡ S = {0, 1} ❡ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ é r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✵ ❡ ✶✳ P♦r ❡①❡♠♣❧♦✱ ♦ ♥ú♠❡r♦ ✶✵✱ ♥❛ ❜❛s❡ ✷✱ r❡♣r❡s❡♥t❛ ♦ ♥ú♠❡r♦ ✷ ✭♥❛ ❜❛s❡ ✶✵✮✳ ❚❡♠✲s❡ t❛♠❜é♠ q✉❡
✾
❆ s❡q✉ê♥❝✐❛ ❞❡ ✐♥str✉çõ❡s ❛❜❛✐①♦ é ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❝♦♥✈❡rt❡r ♥ú♠❡r♦ ❜✐♥ár✐♦ ♣❛r❛ ❛ ❢♦r♠❛ ❞❡❝✐♠❛❧ ❡q✉✐✈❛❧❡♥t❡✳
❇✐♥ár✐♦ ♣❛r❛ ❉❡❝✐♠❛❧
❊♥tr❛❞❛✿ ◆ú♠❡r♦ ❜✐♥ár✐♦ s=sjsj−1...s1s0✳
❙❛í❞❛✿ ◆ú♠❡r♦ ❞❡❝✐♠❛❧ m ❡q✉✐✈❛❧❡♥t❡ à s✳
P❛ss♦ ✶✿ ❈♦♠❡❝❡ ❝♦♥s✐❞❡r❛♥❞♦ j = 0✱ r❡♣r❡s❡♥t❛♥❞♦ ❛ ♣♦s✐çã♦ ❞♦ ♣r✐♠❡✐r♦ ❛❧❣❛r✐s♠♦ ❞❡ s✱
❞❛ ❞✐r❡✐t❛ ♣❛r❛ ❛ ❡sq✉❡r❞❛✳
P❛ss♦ ✷✿ ❈♦♥s✐❞❡r❡ t❛♠❜é♠ m = 0✳ ❊❧❡ r❡♣r❡s❡♥t❛rá ♦ ♥ú♠❡r♦ ❞❡❝✐♠❛❧ ❡q✉✐✈❛❧❡♥t❡ ❛♦s
❛❧❣❛r✐s♠♦s ♦❜t✐❞♦s✱ ❞❛ ❞✐r❡✐t❛ ♣❛r❛ ❛ ❡sq✉❡r❞❛✱ ❛té ❛ ♣♦s✐çã♦j✳
P❛ss♦ ✸✿ ❙❡ ❛ ♣♦s✐çã♦ j ♥ã♦ ♣♦ss✉✐r ❛❧❣❛r✐s♠♦ ❡♥tã♦ ♣❛r❡ ❡ r❡t♦r♥❡ ♦ ✈❛❧♦r ❞❡ m ❝♦♠♦
r❡s♣♦st❛✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❝♦♥t✐♥✉❡✳
P❛ss♦ ✹✿ ❙❡ ♦ ❛❧❣❛r✐s♠♦ ❞❛ ♣♦s✐çã♦ j ❢♦r ✶ ❡♥tã♦ ♦ ♥ú♠❡r♦ ❞❡❝✐♠❛❧ m s❡rá ❞❛❞♦ ♣♦r m
|{z}
◆♦✈♦
=|{z}m
❱❡❧❤♦ + 2j✳
P❛ss♦ ✺✿ ■♥❝r❡♠❡♥t❡ j✱ ♦✉ s❡❥❛✱ ❛✉♠❡♥t❡ ✉♠❛ ✉♥✐❞❛❞❡ ♥♦ j ❛♥t❡r✐♦r✱ ♣❛r❛ tr❛❜❛❧❤❛r ❝♦♠
♠❛✐s ❛❧❣❛r✐s♠♦s ❞❡s✳
P❛ss♦ ✻✿ ❱♦❧t❡ ❛ ❡①❡❝✉t❛r ♦ ♣❛ss♦ ✸✳
❊①❡♠♣❧♦ ✶✻✳ ❈♦♥✈❡rt❡♥❞♦ ♦ ♥ú♠❡r♦ ❜✐♥ár✐♦ s = 110 ♣❛r❛ ❛ ❢♦r♠❛ ❞❡❝✐♠❛❧ ❡q✉✐✈❛❧❡♥t❡✱ s❡❣✉✐♥❞♦ ♦s ♣❛ss♦s ❞❛❞♦s ❧♦❣♦ ❛❝✐♠❛✱ t❡♠✲s❡✿
P❛ss♦ ✶✿ j := 0✳
P❛ss♦ ✷✿ m := 0✳
P❛ss♦ ✸✿ ❊①✐st❡ ❛❧❣❛r✐s♠♦ ♥❛ ♣♦s✐çã♦ j = 0✱ ♦ ③❡r♦✳ ❊♥tã♦ ❝♦♥t✐♥✉❡✳ P❛ss♦ ✹✿ ❖ ❛❧❣❛r✐s♠♦ ❞❛ ♣♦s✐çã♦ j = 0 ♥ã♦ é ✶ (s0 6= 1)✳
P❛ss♦ ✺✿ j := 1✳
P❛ss♦ ✻✿ ❱♦❧t❡ ❛ ❡①❡❝✉t❛r ♦ ♣❛ss♦ ✸✳
P❛ss♦ ✸✿ ❊①✐st❡ ❛❧❣❛r✐s♠♦ ♥❛ ♣♦s✐çã♦ j = 1✱ s1 = 1✳ ❊♥tã♦ ❝♦♥t✐♥✉❡✳
P❛ss♦ ✹✿ ❖ ❛❧❣❛r✐s♠♦ ❞❛ ♣♦s✐çã♦ j = 1 é ✶✳ ❊♥tã♦ m ♣❛ss❛ ❛ ✈❛❧❡r m+ 2j = 0 + 21 = 2✱
♦✉ s❡❥❛✱ m := 2✳ P❛ss♦ ✺✿ j := 2✳
P❛ss♦ ✻✿ ❱♦❧t❡ ❛ ❡①❡❝✉t❛r ♦ ♣❛ss♦ ✸✳
P❛ss♦ ✸✿ ❊①✐st❡ ❛❧❣❛r✐s♠♦ ♥❛ ♣♦s✐çã♦ j = 2✱ s2 = 1✳ ❊♥tã♦ ❝♦♥t✐♥✉❡✳ P❛ss♦ ✹✿ ❖ ❛❧❣❛r✐s♠♦ ❞❛ ♣♦s✐çã♦ j = 2 é ✶✳ ❊♥tã♦ m := 6✳
P❛ss♦ ✺✿ j := 3✳
P❛ss♦ ✻✿ ❱♦❧t❡ ❛ ❡①❡❝✉t❛r ♦ ♣❛ss♦ ✸✳
P❛ss♦ ✸✿ ◆ã♦ ❡①✐st❡ s3✳ ❊♥tã♦ ♣❛r❡ ❡ ❛ r❡s♣♦st❛ ♣r♦❝✉r❛❞❛ ém = 6✱ ♥❛ ❢♦r♠❛ ❞❡❝✐♠❛❧✳
✶✵
❆❧❣♦r✐t♠♦ ❇✐♥ár✐♦ ♣❛r❛ ❉❡❝✐♠❛❧
✕❃ ❊♥tr❛❞❛✿ ❖ ♥ú♠❡r♦ ❜✐♥ár✐♦ s=sjsj−1...s1s0✱ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳
i1✿ j := 0✳
i2✿ m := 0
i3✿ ❙❡ ♥ã♦ ❡①✐st✐rsj ❡♥tã♦ ♣❛r❡ ❡ r❡t♦r♥❡m ❝♦♠♦ r❡s♣♦st❛✳
i4✿ ❙❡ sj = 1 ❡♥tã♦ m:=m+ 2j✳
i5✿ j :=j+ 1✳
i6✿ ❱á ♣❛r❛i3✳
P❛r❛ ❝♦♥✈❡rt❡r ✉♠ ♥ú♠❡r♦ ❞❡❝✐♠❛❧ ♣❛r❛ ❛ ❢♦r♠❛ ❜✐♥ár✐❛ ❡q✉✐✈❛❧❡♥t❡✱ ✉t✐❧✐③❛♥❞♦ ♦ q✉❡ ❢♦✐ ❝♦♠❡♥t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ♦❜té♠✲s❡ ♦ s❡❣✉✐♥t❡ ❡sq✉❡♠❛ ♥❛ t❛❜❡❧❛ ❛❜❛✐①♦✿
❈♦♥✈❡rsã♦ ❞❡ ◆ú♠❡r♦ ❉❡❝✐♠❛❧ ♣❛r❛ ❇✐♥ár✐♦ ✲ ❆❧❣♦r✐t♠♦ ◆ú♠❡r♦ ❛ s❡r ❞✐✈✐❞✐❞♦ ♣♦r ✷ ◗✉♦❝✐❡♥t❡ ♦❜t✐❞♦ ❘❡st♦ ♦❜t✐❞♦
m q0 r0
q0 q1 r1
q1 q2 r2
q2 q3 r3
✳✳✳ ✳✳✳ ✳✳✳
qn−2 qn−1 rn−1
qn−1 0 rn
❆ss✐♠✱ ♦s ❛❧❣❛r✐s♠♦s q✉❡ ❢♦r♠❛rã♦ ♦ ♥ú♠❡r♦ ❜✐♥ár✐♦ s❡rã♦ t♦❞♦s ♦s r❡st♦s ♦❜t✐❞♦s ♥❛ t❡r❝❡✐r❛ ❝♦❧✉♥❛ ❞❛ t❛❜❡❧❛ ❛❝✐♠❛✱ ♦✉ s❡❥❛✱ t❛❧ ♥ú♠❡r♦ ❜✐♥ár✐♦ s❡rárnrn−1...r3r2r1r0✱ ♥❡ss❛
♦r❞❡♠ ❞❡ ♣♦s✐çã♦✳
❊①❡♠♣❧♦ ✶✼✳ ❙❡❣✉❡ ❛❜❛✐①♦ ❛ ❝♦♥✈❡rsã♦ ❞♦ ♥ú♠❡r♦ ❞❡❝✐♠❛❧ ✷✶✵ ♣❛r❛ ❛ ❢♦r♠❛ ❜✐♥ár✐❛ ❡q✉✐✲ ✈❛❧❡♥t❡✱ ❝♦♥❢♦r♠❡ ❛ t❛❜❡❧❛ ❛❝✐♠❛✳
◆ú♠❡r♦ ❛ s❡r ❞✐✈✐❞✐❞♦ ♣♦r ✷ ◗✉♦❝✐❡♥t❡ ♦❜t✐❞♦ ❘❡st♦ ♦❜t✐❞♦
✷✶✵ ✶✵✺ ✵
✶✵✺ ✺✷ ✶↑
✺✷ ✷✻ ✵↑
✷✻ ✶✸ ✵↑
✶✸ ✻ ✶↑
✻ ✸ ✵↑
✸ ✶ ✶↑
✶ ✵ ✶↑
❈♦♥❢♦r♠❡ ❛ t❛❜❡❧❛ ❞❡st❡ ❡①❡♠♣❧♦✱ ♦ ♥ú♠❡r♦ ❜✐♥ár✐♦ ♦❜t✐❞♦ é✿ 11010010✳
❆ t❛❜❡❧❛ ❞❛❞❛ ❛♥t❡s ❞❡st❡ ❡①❡♠♣❧♦ ♣❡r♠✐t❡ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛ ❞❡ ✐♥str✉çõ❡s ♦✉ ♣❛ss♦s✱ ❜❡♠ ❞❡✜♥✐❞♦s✱ q✉❡ r❡♣r❡s❡♥t❛♠ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❝♦♥✈❡rt❡r ♥ú♠❡r♦ ❞❡❝✐♠❛❧ ♣❛r❛ ❛ ❢♦r♠❛ ❜✐♥ár✐❛ ❡q✉✐✈❛❧❡♥t❡✱ ❝♦♥❢♦r♠❡ ♦ q✉❡ s❡❣✉❡ ❛❜❛✐①♦✳
✶✶
❊♥tr❛❞❛✿ ❖ ♥ú♠❡r♦ ❞❡❝✐♠❛❧ m✳
❙❛í❞❛✿ ❖ ♥ú♠❡r♦ ❜✐♥ár✐♦ s=sjsj−1...s1s0✳
P❛ss♦ ✶✿ ❈♦♥s✐❞❡r❡ j = 0✱ ♦ q✉❛❧ r❡♣r❡s❡♥t❛ ❛ ♣♦s✐çã♦ ❞♦ ♣r✐♠❡✐r♦ ❛❧❣❛r✐s♠♦ ❞♦ r❡s✉❧t❛❞♦✱ ❞❛ ❞✐r❡✐t❛ ♣❛r❛ ❛ ❡sq✉❡r❞❛✳
P❛ss♦ ✷✿ ❉✐✈✐❞❛ m ♣♦r ✷✱ ♦❜t❡♥❞♦ q✉♦❝✐❡♥t❡ qj ❡ r❡st♦rj ✭✵ ♦✉ ✶✮✳ P❛ss♦ ✸✿ ■♥s✐r❛ rj ♥❛ ♣♦s✐çã♦ j ❞❡s✱ ♦✉ s❡❥❛✱ ❢❛ç❛ sj :=rj✳
P❛ss♦ ✹✿ ❙❡ qj = 0 ❡♥tã♦ ♣❛r❡ ❡ r❡t♦r♥❡s ❝♦♠♦ r❡s♣♦st❛✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❝♦♥t✐♥✉❡✳
P❛ss♦ ✺✿ ❋❛ç❛ m ❛ss✉♠✐r ✈❛❧♦r qj✱ ♦✉ s❡❥❛✱ m :=qj✳
P❛ss♦ ✻✿ ■♥❝r❡♠❡♥t❡ ♦ ✈❛❧♦r ❞❡ j✱ s♦♠❛♥❞♦ ❛ ❡❧❡ ✉♠❛ ✉♥✐❞❛❞❡✳ P❛ss♦ ✼✿ ❱♦❧t❡ ❛ ❡①❡❝✉t❛r ♦ ♣❛ss♦ ✷✳
❊①❡♠♣❧♦ ✶✽✳ ❙❡❣✉❡ ❛❜❛✐①♦ ❛ ❝♦♥✈❡rsã♦ ❞♦ ♥ú♠❡r♦ ❞❡❝✐♠❛❧ m = 21 ♣❛r❛ ❛ ❢♦r♠❛ ❜✐♥ár✐❛ ❡q✉✐✈❛❧❡♥t❡✱ ❛♣❧✐❝❛♥❞♦ ♦s ♣❛ss♦s ❞♦ ❛❧❣♦r✐t♠♦ ❧♦❣♦ ❛❝✐♠❛✳
P❛ss♦ ✶✿ ❈♦♥s✐❞❡r❡ j = 0✳
P❛ss♦ ✷✿ m= 21 ❞✐✈✐❞✐❞♦ ♣♦r ✷ ❢♦r♥❡❝❡ q✉♦❝✐❡♥t❡ q0 = 10 ❡ r❡st♦ r0 = 1✳ P❛ss♦ ✸✿ s0 ❛ss✉♠❡ ✈❛❧♦r r0 = 1✱ ♦✉ s❡❥❛✱ s0 := 1✳
P❛ss♦ ✹✿ q0 6= 0✳ ❊♥tã♦ ❝♦♥t✐♥✉❡✳
P❛ss♦ ✺✿ m ❛ss✉♠❡ ♦ ✈❛❧♦r ❞❡ q0✱ ✐st♦ é✱m := 10✳ P❛ss♦ ✻✿ j é ✐♥❝r❡♠❡♥t❛❞♦ ❞❡ ♠♦❞♦ q✉❡j := 1✳
P❛ss♦ ✼✿ ❱♦❧t❡ ❛ ❡①❡❝✉t❛r ♦ ♣❛ss♦ ✷✳
P❛ss♦ ✷✿ m= 10 ❞✐✈✐❞✐❞♦ ♣♦r ✷ ❢♦r♥❡❝❡ q✉♦❝✐❡♥t❡ q1 = 5 ❡ r❡st♦r1 = 0✳ P❛ss♦ ✸✿ s1 ❛ss✉♠❡ ✈❛❧♦r r1 = 0✱ ♦✉ s❡❥❛✱ s1 := 0✳
P❛ss♦ ✹✿ q1 6= 0✳ ❊♥tã♦ ❝♦♥t✐♥✉❡✳
P❛ss♦ ✺✿ m ❛ss✉♠❡ ♦ ✈❛❧♦r ❞❡ q1✱ ✐st♦ é✱m := 5✳ P❛ss♦ ✻✿ j := 2✳
P❛ss♦ ✼✿ ❱♦❧t❡ ❛ ❡①❡❝✉t❛r ♦ ♣❛ss♦ ✷✳
P❛ss♦ ✷✿ m= 5 ❞✐✈✐❞✐❞♦ ♣♦r ✷ ❢♦r♥❡❝❡ q✉♦❝✐❡♥t❡ q2 = 2 ❡ r❡st♦ r2 = 1✳ P❛ss♦ ✸✿ s2 :=r2 = 1✳
P❛ss♦ ✹✿ q2 6= 0✳ ❊♥tã♦ ❝♦♥t✐♥✉❡✳
P❛ss♦ ✺✿ m :=q2 = 2✳ P❛ss♦ ✻✿ j := 3✳
P❛ss♦ ✼✿ ❱♦❧t❡ ❛ ❡①❡❝✉t❛r ♦ ♣❛ss♦ ✷✳
P❛ss♦ ✷✿ m= 2 ❞✐✈✐❞✐❞♦ ♣♦r ✷ ❢♦r♥❡❝❡ q✉♦❝✐❡♥t❡ q3 = 1 ❡ r❡st♦ r3 = 0✳ P❛ss♦ ✸✿ s3 :=r3 = 0✳
P❛ss♦ ✹✿ q3 6= 0✳ ❊♥tã♦ ❝♦♥t✐♥✉❡✳
P❛ss♦ ✺✿ m :=q3 = 1✳ P❛ss♦ ✻✿ j := 4✳
P❛ss♦ ✼✿ ❱♦❧t❡ ❛ ❡①❡❝✉t❛r ♦ ♣❛ss♦ ✷✳
P❛ss♦ ✷✿ m= 1 ❞✐✈✐❞✐❞♦ ♣♦r ✷ ❢♦r♥❡❝❡ q✉♦❝✐❡♥t❡ q4 = 0 ❡ r❡st♦ r4 = 1✳ P❛ss♦ ✸✿ s4 :=r4 = 1✳
P❛ss♦ ✹✿ q4 = 0✳ ❊♥tã♦ ❛ s❛í❞❛ és=s4s3s2s1s0 = 10101✳
✶✷
❚❡♠✲s❡ ❛❜❛✐①♦ ♦ ♣s❡✉❞♦❝ó❞✐❣♦ ♣❛r❛ ❡ss❡ ♣r♦❝❡❞✐♠❡♥t♦✳ Pr♦❝❡❞✐♠❡♥t♦ ❇✐♥ár✐♦ ♣❛r❛ ❉❡❝✐♠❛❧
i1✿ j := 0
i2✿ ❉✐✈✐❞❛m ♣♦r ✷✱ ♣❛r❛ ♦❜t❡r ♦ q✉♦❝✐❡♥t❡ qj ❡ ♦ r❡st♦ rj ✭✵ ♦✉ ✶✮✳
i3 :❈♦❧♦q✉❡ rj ♥❛ ♣♦s✐çã♦j ❞❡ s=sjsj−1...s1s0✳
i4✿ ❙❡ qj = 0✱ ❡♥tã♦ ♣❛r❡ ❡ r❡t♦r♥❡ s ❝♦♠♦ r❡s♣♦st❛✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❝♦♥t✐♥✉❡✳
i5✿ m :=qj✳
i6✿ j :=j+ 1✳
i7✿ ❱á ♣❛r❛i2✳
✷✳✷ ❆❧❣♦r✐t♠♦ ♣❛r❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦s s✉❜❝♦♥❥✉♥t♦s ❞❡
✉♠ ❝♦♥❥✉♥t♦
Pr♦♣♦s✐çã♦ ✶✳ ❉❛❞♦ ✉♠ ❝♦♥❥✉♥t♦A✱ ❝♦♠n ❡❧❡♠❡♥t♦s✱ ❡♥tã♦ ♦ ♥ú♠❡r♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ A é2n✳
❉❡♠♦♥str❛çã♦✿
P♦r ■♥❞✉çã♦ ❋✐♥✐t❛ s♦❜r❡ n✱ t❡♠✲s❡✿
P(n) :❙❡A♣♦ss✉✐n❡❧❡♠❡♥t♦s✱ ❡♥tã♦ ♦ ♥ú♠❡r♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡Aé2n✳ ❉❡ss❛
❢♦r♠❛✱ ♣❛r❛ n= 1✱ t❡♠✲s❡✿
A = {a1} ❞❡ ♠♦❞♦ q✉❡ ∅ ❡ {a1} sã♦ ♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ A✳ ❈♦♠♦ 2n = 21 = 2✱
P(1) é ✈❡r❞❛❞❡✐r❛✳
❙✉♣♦♥❞♦ P(n) ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❛❧❣✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ n > 1✱ s❡rá ♣r♦✈❛❞♦ q✉❡
P(n+ 1) t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛✱ q✉❡ s❡A t❡♠n+ 1❡❧❡♠❡♥t♦s ❡♥tã♦A ♣♦ss✉✐rá2n+1
s✉❜❝♦♥❥✉♥t♦s✳ ❚❡♠✲s❡ q✉❡✿
A={a1, a2, ..., an, an+1}={a1, a2, ..., an}
| {z }
= B
∪ {an+1}✳
❈♦♠♦ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ B é ✐❣✉❛❧ ❛ n✱ s❡❣✉❡✱ ♣♦r ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱
q✉❡ B ♣♦ss✉✐ 2n s✉❜❝♦♥❥✉♥t♦s✿ B
1 = Ø✱ B2✱✳✳✳✱ B2n✱ ♦s q✉❛✐s t❛♠❜é♠ sã♦ s✉❜❝♦♥❥✉♥t♦s ❞❡
A✱ ❥á q✉❡ Bi ⊂ B ⊂ A, ∀i ∈ {1,2, ...,2
n}✳ ❆❧é♠ ❞✐ss♦✱ a
n+1 ∈/ Bi, ∀i ∈ {1,2, ...,2
n}✳ P❛r❛
❝♦♥str✉✐r ♦s s✉❜❝♦♥❥✉♥t♦s r❡st❛♥t❡s ❞❡A✱ ❜❛st❛ ✐♥s❡r✐r ♦ ❡❧❡♠❡♥t♦ an+1 ❡♠ ❝❛❞❛ s✉❜❝♦♥❥✉♥t♦
Bi✱ ♦❜t❡♥❞♦✲s❡ 2n ♥♦✈♦s s✉❜❝♦♥❥✉♥t♦s✳ ❆ss✐♠✱ ♦ t♦t❛❧ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ A é 2n + 2n =
2n.(1 + 1) = 2n. 2 = 2n+1✳ P♦rt❛♥t♦ P(n+ 1) é ✈❡r❞❛❞❡✐r❛✳
❆ ❞❡♠♦♥str❛çã♦ ❞❛ ♣r♦♣♦s✐çã♦ ✶ ❢♦r♥❡❝❡ ✉♠❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛ ❞❡ ♣❛ss♦s✱ ❜❡♠ ❞❡✜♥✐❞♦s✱ ♣❛r❛ ♣r♦❞✉③✐r ♦s s✉❜❝♦♥❥✉♥t♦s ❞❡A✱ ❡s❝r✐t❛ ♥❛ ❢♦r♠❛ ❞❡ ✉♠ ♣s❡✉❞♦❝ó❞✐❣♦✳
❆❧❣♦r✐t♠♦ ❙✉❜❝♦♥❥✉♥t♦s
i1✿ ❊♥tr❛❞❛✿ n ❡ a1, a2, ..., an✳ i6 : B(k+ 1) :=B(i)∪ {a(j)}
i2✿ B(1) =Ø i7 : k :=k+ 1
i3✿ k := 1 i7 : ❋✐♠P❛r❛
i4✿ P❛r❛ j := 1 ❛té n✱ ❢❛ç❛ i7 :❋✐♠P❛r❛
✶✸
❊①❡♠♣❧♦ ✶✾✳ ❋❛③❡♥❞♦ ♦ t❡st❡ ❞❡ ♠❡s❛ ❛♣❧✐❝❛♥❞♦ ♦ ❆❧❣♦r✐t♠♦ ❙✉❜❝♦♥❥✉♥t♦s ❡♠A={a1, a2}✱
♣❛r❛ ❧✐st❛r ♦s s✉❜❝♦♥❥✉♥t♦s ❞❡A✱ t❡♠✲s❡✿
■♥str✉çã♦ ❊①❡❝✉t❛❞❛ ❦ ❥ i ❙✉❜❝♦♥❥✉♥t♦ ❖❜t✐❞♦
✳✳✳
i2 B(1) = Ø
i3 ✶
i4 ✶ ✶
i5 ✶ ✶ ✶
i6 ✶ ✶ ✶ B(2) =B(1)∪ {a(1)}={a(1)}
i7 ✷ ✶ ✶
i4 ✷ ✷ ✶
i5 ✷ ✷ ✶
i6 ✷ ✷ ✶ B(3) =B(1)∪ {a(2)}={a(2)}
i7 ✸ ✷ ✶
i5 ✸ ✷ ✷
i6 ✸ ✷ ✷ B(4) =B(2)∪ {a(2)}={a(1), a(2)}
i8 P❛r❡
✷✳✸ ❆❧❣♦r✐t♠♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ ✈❛❧♦r ❞❡ ✉♠❛ ♣♦tê♥❝✐❛
❈♦♥s✐❞❡r❛♥❞♦ q✉❡ ❛s ♦♣❡r❛çõ❡s ❛r✐t♠ét✐❝❛s ❜ás✐❝❛s ❞❡ ✉♠ ❝♦♠♣✉t❛❞♦r sã♦✿ ❛❞✐çã♦✱ s✉❜tr❛çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❞✐✈✐sã♦✱ s❡❣✉❡ ❛❜❛✐①♦ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ s❡ ♦❜t❡r ♦ ✈❛❧♦r ❞❡ ✉♠❛ ♣♦tê♥❝✐❛ ❞❡ ❜❛s❡ x✳❆❧❣♦r✐t♠♦ ❊①♣♦♥❡♥❝✐❛çã♦
i1✿ ❊♥tr❡ ❝♦♠ ♦s ✈❛❧♦r❡s ❞❡x6= 0 ❡ ❞❡ n i5✿ x(i) :=x(i−1) ∗ x
✭n ✉♠❛ ✈❛r✐á✈❡❧ ✐♥t❡✐r❛ ♥ã♦ ♥❡❣❛t✐✈❛✮✳ i6✿ i :=i+ 1
i2✿ i := 1 i7✿ ❙❡ n = 0 ❡♥tã♦ r❡t♦r♥❡ xn = 1✳
i3✿ x(i−1) := 1 i8✿ ❙❡♥ã♦ r❡t♦r♥❡✿ xn=x(i−1)✳
i4✿ ❊♥q✉❛♥t♦ i6n✱ ❢❛ç❛ i9✿ P❛r❡✳
❊ss❡ ❛❧❣♦r✐t♠♦ ❝♦♥s✐❞❡r❛ x 6= 0 ❡ n ∈ N∪ {0}✱ ✐♥✐❝✐❛♥❞♦ ❝♦♠ i = 1 ❡ x(0) = 1✳
❉❡♣♦✐s✱ ♣❛r❛ ❝❛❞❛ ✈❛❧♦r ❛ss✉♠✐❞♦ ♣♦r i✱ ❞❡ 1❛té n✱ t❡♠✲s❡✿ x(1) = 1|{z}
x(0)
·x−→ x(2) =|{z}x
x(1)
·x−→ x(3) =|{z}x2
x(2)
·x
x(4) =|{z}x3
x(3)
·x−→ · · · −→ x(n) = xn−1
|{z}
x(n−1)
·x✱
s❡♥❞♦x(n)r❡t♦r♥❛❞♦ ❝♦♠♦ s♦❧✉çã♦✳
✶✹
i1✿ x= 5 ❡ n= 3 i6✿ i= 3
i2✿ i= 1 i4✿ ❚❡st❡ ✈❡r❞❛❞❡✐r♦
i3✿ x(0) = 1 i5✿ x(3) =x(2)·5 = 25·5 = 125
i4✿ ❚❡st❡ ✈❡r❞❛❞❡✐r♦ i6✿ i= 4
i5✿ x(1) =x(0)·5 = 1·5 = 5 i4✿ ❚❡st❡ ❢❛❧s♦
i6✿ i= 2 i7✿ ❚❡st❡ ❢❛❧s♦
i4✿ ❚❡st❡ ✈❡r❞❛❞❡✐r♦ i8✿ 53 = 125
i5✿ x(2) =x(1)·5 = 5·5 = 25 i9✿ P❛r❡✳
❉❡✜♥✐çã♦ ✶✳ ❈❤❛♠❛✲s❡ ♣✐s♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ x✱ ❡ ❞❡♥♦t❛✲s❡ ♦ ♣✐s♦ ❞❡ x ♣♦r⌊x⌋ ♦ ♠❛✐♦r
✐♥t❡✐r♦ ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛x✳
❊①❡♠♣❧♦ ✷✶✳ ❈♦♥❢♦r♠❡ ❛ ❞❡✜♥✐çã♦ ✶✱ t❡♠✲s❡✿
⌊7⌋= 7❀ ⌊4,23⌋= 4 ❡ ⌊−5,23⌋=−6✳
❉❡✜♥✐çã♦ ✷✳ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ r❡❛❧ x✱ ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♠❛✐♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ x s❡rá
r❡♣r❡s❡♥t❛❞♦ ♣♦r⌈x⌉✳
❊①❡♠♣❧♦ ✷✷✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ t❡♠✲s❡✿ ⌈7⌉= 7❀⌈4,23⌉= 5 ❡⌈−5,23⌉=−5✳
❙❡rá ✈✐st♦ ❛ s❡❣✉✐r ✉♠ ♦✉tr♦ ❛❧❣♦r✐t♠♦✱ ❞❡♥♦♠✐♥❛❞♦ ❡①♣♦♥❡♥❝✐❛çã♦ rá♣✐❞❛✱ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ♣♦tê♥❝✐❛s✳ ❊❧❡ é ❜❛s❡❛❞♦ ❡♠ s✉❝❡ss✐✈❛s ❞✐✈✐sõ❡s ❞❡ ❡①♣♦❡♥t❡ ♣❡❧♦ ♥ú♠❡r♦ ✷✱ ❝♦♥❢♦r♠❡ ❛s ✐❣✉❛❧❞❛❞❡s ❛❜❛✐①♦✱ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡xn✳
xn = x2q0+r0
= (x·x)q0
·xr0
= (x2)2q1+r1
·xr0
= (x2·x2)q1
·(x2)r1
·xr0
= (x4)2q2+r2
·(x2)r1
·xr0
= (x4·x4)q2
·(x4)r2
·(x2)r1
·xr0
✳✳✳
= x2k−22·1+rk−2
·x2k−3rk−3
·x2k−4rk−4
·...·(x2)r1
·xr0
= x2k−2
·x2k−21
·x2k−2rk−2
· xk−3rk−3
·...·(x2)r1
·xr0
= x2k−12·0+1
·x2k−2rk−2
·x2k−3rk−3
·...·(x2)r1
·xr0
▼❛s✱ ❡ss❡ ♣r♦❝❡❞✐♠❡♥t♦ ❢❛③ ❝♦♠ q✉❡ n s❡❥❛ ❡s❝r✐t♦ ♥❛ ❢♦r♠❛ ❜✐♥ár✐❛✱ ♦✉ s❡❥❛✱ n =rk−1rk−2...r1r0✱ s❡♥❞♦ rk−1 = 1✱ rj ∈ {0,1}, ∀j ∈ {0,1,2, ...,(k−2)} ❡ k−1 ♦ ♥ú♠❡r♦
✶✺
2k−1 62k−1+r
k−22k−2+...+r12 +r0
| {z }
=n
6
62k−1 + 2k−2 + ... + 2 + 1
| {z }
Soma dos k termos da P G de raz˜ao 2
= 1· 22k−−11 = 2k
−1<2k✳
P♦rt❛♥t♦✱ 2k−1 6n < 2k ❡
2k−1 6n <2k ⇔log
22
k−1 6log
2n < log22
k ⇔
k−16log2n < k ⇔k−1 = ⌊log2n⌋ ⇔ k = 1 +⌊log2n⌋✱
s❡♥❞♦⌊log2n⌋ ♦ ♠❛✐♦r ✐♥t❡✐r♦ ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ log2n✳
P❛r❛ ❡ss❡ ❛❧❣♦r✐t♠♦✱ t❡♠✲s❡✿ ❼ ❚♦t❛❧ ❞❡ ❞✐✈✐sõ❡s ❂k✳
❼ ❚♦t❛❧ ❞❡ ♠✉❧t✐♣❧✐❝❛çõ❡s ❂ (k−1) + (k−1) = 2k−2✳ ❼ ❚♦t❛❧ ❞❡ ♦♣❡r❛çõ❡s ❂ 3k−2✳
❍á ♣♦rt❛♥t♦3k−2 = 3 (1 +⌊log2n⌋)−263 (1 +log2n)−2 = 1 + 3log2n=g(n)
♠✉❧t✐♣❧✐❝❛çõ❡s ❡ ❞✐✈✐sõ❡s✱ ♥♦ t♦t❛❧✳ ❆❧❣♦r✐t♠♦ ❊①♣♦♥❡♥❝✐❛çã♦ ❘á♣✐❞❛
i1✿ ❊♥tr❡ ❝♦♠ ♦s ✈❛❧♦r❡s ❞❡x ❡n✱ ❝♦♥s✐❞❡r❛♥❞♦ resp := 1✳
i2✿ ❉✐✈✐❞❛n ♣♦r ✷ ♣❛r❛ ♦❜t❡r ♦ q✉♦❝✐❡♥t❡ q ❡ ♦ r❡st♦ r✳
i3✿ ❙❡ r= 1✱ ❝♦♥s✐❞❡r❡ resp :=resp∗x✳
i4✿ ❙❡ q= 0✱ ❡♥tã♦ ♣❛r❡✳
i5✿ n :=q✳
i6✿ x :=x∗x✳
i7✿ ❱á ♣❛r❛i2✳
❊①❡♠♣❧♦ ✷✸✳ ❆ s❡❣✉✐r é ❡①❡❝✉t❛❞♦ ♦ t❡st❡ ❞❡ ♠❡s❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ x25✳
■♥str✉çã♦ ❊①❡❝✉t❛❞❛ ❱❛r✐á✈❡✐s
x n resp q r
i1 x ✷✺ ✶ ✲ ✲
i2 x ✷✺ ✶ ✶✷ ✶
i3 x ✷✺ x ✶✷ ✶
i5 x ✶✷ x ✶✷ ✶
