Mestrado Profissional em Matemática em Rede Nacional
PROPMAT
DISSERTAÇÃO DE MESTRADO
Uma Matemática Voltada para o Professor do
Ensino Médio
Geraldo Perreira Barbosa Pilho
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❆▲❆●❖❆❙ ■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆
▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲
●❊❘❆▲❉❖ ❋❊❘❘❊■❘❆ ❇❆❘❇❖❙❆ ❋■▲❍❖
❯▼❆ ▼❆❚❊▼➪❚■❈❆ ❱❖▲❚❆❉❆ P❆❘❆ ❖ P❘❖❋❊❙❙❖❘ ❉❖
❊◆❙■◆❖ ▼➱❉■❖
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❆▲❆●❖❆❙ ■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆
▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲
●❊❘❆▲❉❖ ❋❊❘❘❊■❘❆ ❇❆❘❇❖❙❆ ❋■▲❍❖
❯▼❆ ▼❆❚❊▼➪❚■❈❆ ❱❖▲❚❆❉❆ P❆❘❆ ❖ P❘❖❋❊❙❙❖❘ ❉❖
❊◆❙■◆❖ ▼➱❉■❖
❚r❛❜❛❧❤♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏♦sé ❈❛r❧♦s ❞❡ ▲✐♠❛
✐✐✐
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦ Pr♦❢✳ ❉r✳ ❏♦sé ❈❛r❧♦s ❞❡ ▲✐♠❛✱ ♣❡❧❛ s❡r❡♥✐❞❛❞❡ ❡♠ s✉❛ ✈✐❞❛ ♣r♦✜ss✐♦♥❛❧ ❡ ❝♦✲ ♠✉♥✐❝❛çã♦ ❝♦♠ s❡✉s ❛❧✉♥♦s ❡ s✉❛ ✜❧❤❛ ❉♦✉t♦r❛♥❞❛ ❑❛r❧❛ ❦❛t❡r✐♥❡ ❇❛r❜♦③❛ ❞❡ ▲✐♠❛ ♣❡❧♦ ❣r❛♥❞✐♦s♦ ❛♣♦✐♦ ♥❛ ♦r✐❡♥t❛çã♦ ❞♦s ❝♦♠❛♥❞♦s ♥♦ ▲❛t❡①✳
❆♦s ♠❡✉s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ ❝❛♠✐♥❤❛❞❛✱ ❆❧❡①❛♥❞r❡ ❋❛r✐❛✱ ❈❧❛r✐❝② ❆❧✈❡s✱ ❈❧❛✉❞✐♦ ❆❣r❛✱ ❉✐♦❣♦ ▼❡✉r❡r✱ ❉❥❛❧♠❛ ●♦♠❡s✱ ❊❞✈❛♥ ❍♦rá❝✐♦✱ ❊✇❡rt♦♥ ❘♦♦s❡✇❡❧t✱ ❋❛❜rí❝✐❛ ❖♠❡♥❛✱ ❋❡r✲ ♥❛♥❞♦ ❱❛❧ér✐♦✱ ●❡r❛❧❞♦ ❇❛rr♦s✱ ●✐❧♠❛r ❚❡♦❞♦③✐♦✱ ●✐✈❛♥✐❧❞♦ ❙❛♥t♦s✱ ●r❛❝✐♥♦ ❋r❛♥❝✐s❝♦✱ ❏♦sé ❆♣❛r❡❝✐❞♦✱ ❏♦sé ❘❡✐♥❛❧❞♦✱ ▲❡❞✐✈❛❧❞♦ ●♦♠❡s✱ ▼❛r❝✐❡❧ ❞❛ ❙✐❧✈❛✱ ❖r❧❛♥❞♦ ❆♣rí❣✐♦✱ ❘♦s❡ ❙✐❧✈❛✱ ❚❤✐❛❣♦ ▲❡ss❛ ❡ ❱❛♥✐♥❛ ❈❛r✈❛❧❤♦ ♣❡❧❛s ❤♦r❛s ❞❡ ❡st✉❞♦s✱ ♣❡❧❛s ✐♥❢♦r♠❛çõ❡s ❡ ❝♦♠✲ ♣r❡❡♥sã♦✱ ❛ss✐♠ ❝♦♠♦✱ àq✉❡❧❡s q✉❡ ✐♥✐❝✐❛r❛♠ ❛ ❥♦r♥❛❞❛ ❝♦♥♦s❝♦✱ ♠❛s ♣♦r ❛❧❣✉♠ ♠♦t✐✈♦✱ ♥ã♦ ❝♦♥❝❧✉ír❛♠✱ ❊❞❝❛r❧♦s ❞❛ ❙✐❧✈❛✱ ❊♠❛♥♦❡❧ ▼❛r✐❛♥♦✱ ●❡♦✈❛♥ ❱✐❡✐r❛✱ ■s❛❛❝ ●♦♠❡s✱ ❏♦sé ❘♦♠✐❧❞♦✱ ❏✉❝é❧✐♦ ❋❡rr❡✐r❛✱ ▲ú❝✐♦ ❆✉❣✉st♦✱ ◆✐❧③❡r ❏❛❝q✉❡❧✐♥❡ ❡ ❚♦♥② ❋á❜✐♦✳
➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧✱ ♠✐♥❤❛ ❡s♣♦s❛✱ ✜❧❤♦s✱ s♦❣r❛✱ ❝✉♥❤❛❞❛ ❡ ❝♦✲❝✉♥❤❛❞♦ ♣❡❧♦s ❞✐❛s✱ ♥♦✐t❡s ❡ ✜♥s ❞❡ s❡♠❛♥❛ ❞❡ ❛✉①í❧✐♦✱ q✉❡ ❢♦✐ ❡ss❡♥❝✐❛❧ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡ss❡ ❝✉rs♦✳
❆♦s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ tr❛❜❛❧❤♦ q✉❡ ♠❡ ❞❡r❛♠ ❛♣♦✐♦ ❝♦♠ ❛ ♣❛rt❡ ♣rát✐❝❛ ❞❡ss❡ tr❛✲ ❜❛❧❤♦✱ ❆❧♠✐r✱ ❇r✉♥✐♦ ❆❧❜✐♥♦✱ ❈❛✐♦ ❈és❛r✱ ▲✉❝✐✈❛❧❞❛ ❇❛r❜♦③❛✱ ❘✐❧❞s♦♥✱ ❙ér❣✐♦ ❈❛❜r❛❧ ❡ ❲✐t❡♠❜❡r❣✱ ❛ss✐♠ ❝♦♠♦ ✉♠ ❛❧✉♥♦ ❡♠ ❡s♣❡❝✐❛❧ ❞❛ ❊s❝♦❧❛ ❊st❛❞✉❛❧ ❉r ❋❡r♥❛♥❞❡s ▲✐♠❛✱ ❆❧✐ss♦♥ ▲♦♣❡s✱ q✉❡ ♣♦ss✉✐ ✉♠ ❢✉t✉r♦ ♣r♦♠✐ss♦r ♥❛ ár❡❛ ❞❛s ❡①❛t❛s✳
❆♦s ♠❡✉s ❛❧✉♥♦s ❞❛s ❊s❝♦❧❛s ❊st❛❞✉❛❧ ❉r ❋❡r♥❛♥❞❡s ▲✐♠❛✱ ▼✉♥✐❝✐♣❛❧ ◆✐s❡ ❞❛ ❙✐❧✈❡✐r❛ ❡ ❞♦ ❈♦❧é❣✐♦ ❆♥❝❤✐❡t❛✱ ♣♦r ❝♦♥tr✐❜✉✐r❡♠✱ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛✱ ❝♦♠ ❡ss❛ ♠✐♥❤❛ ❝♦♥q✉✐st❛✳
❘❡s✉♠♦
▼✉✐t♦s sã♦ ♦s ❞❡s❛✜♦s q✉❡ ❛ ▼❛t❡♠át✐❝❛ ♣r♦♣õ❡ ❞✉r❛♥t❡ t♦❞❛ ❛ ♥♦ss❛ ✈✐❞❛ ♣r♦✜ss✐♦♥❛❧✱ s❡❥❛ ❝♦♠♦ ❡st✉❞❛♥t❡ ♦✉ ❝♦♠♦ ♣r♦✜ss✐♦♥❛❧ ❞❛ ár❡❛✳ ❆ ❜✉s❝❛ ♣❡❧♦ ❝♦♥❤❡❝✐♠❡♥t♦ é ♥❡❝❡ssár✐❛ ♣❛r❛ ♣♦❞❡r ❞❛r♠♦s ❝♦♥t❛ ❞❛s ♥♦✈✐❞❛❞❡s q✉❡ s✉r❣❡♠ ❛ ❝❛❞❛ ❞✐❛✳ ❙❡♠♣r❡ ♣r❡❝✐s❛♠♦s ❞❡ ♠♦t✐✈❛çã♦ ♣❛r❛ ♣♦❞❡r ✜❝❛r♠♦s ❛t✉❛❧✐③❛❞♦s ❝♦♠ ♦s ❛ss✉♥t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ss❛s ♥♦✈✐❞❛❞❡s✳ ❆ s❡❣✉✐r✱ ♠♦str❛r❡♠♦s ✉♠ ♠❛t❡r✐❛❧ q✉❡ ♠♦t✐✈❛rá ♦ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛ q✉❡ ❜✉s❝❛ ♦ ❛♣r❡♥❞✐③❛❞♦ ♦ t❡♠♣♦ t♦❞♦✱ r❡s❣❛t❛♥❞♦ ♦ ❡stí♠✉❧♦ q✉❡ ♠✉✐t❛s ✈❡③❡s s❡ ♣❡r❞❡ ♣❡❧♦ ❢❛t♦ ❞♦ ♣r♦✜ss✐♦♥❛❧ ❞❛ ❡❞✉❝❛çã♦ ♥ã♦ ❝♦♥s❡❣✉✐r r❡s♦❧✈❡r ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s q✉❡ s✉r❣❡♠ ♥❛ s✉❛ ❝❛♠✐♥❤❛❞❛✳ ❖s ❛ss✉♥t♦s q✉❡ s❡rã♦ ❛❜♦r❞❛❞♦s ❛q✉✐ sã♦ ❞❡ ❣r❛♥❞❡ ✉t✐❧✐❞❛❞❡ ♥❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛ ❡ ♣♦r ❡❧❡s t❡r❡♠ ✉♠❛ ❜❡❧❡③❛ ❡ ❣r❛❝✐♦s✐❞❛❞❡ ♥❛ r❡s♦❧✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s ❞❡✈❡♠ s❡r ❛♣r❡❝✐❛❞♦s ❝♦♠ ♠❛✐♦r ✐♥t❡r❡ss❡✳ ❆ ❧✐♥❣✉❛❣❡♠ ✉t✐❧✐③❛❞❛ ❛q✉✐ é ♠✉✐t♦ ❛❣r❛❞á✈❡❧ ♣❛r❛ ♦ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ❡st✐♠✉❧❛♥❞♦✲♦ ❛✐♥❞❛ ♠❛✐s ❡♠ q✉❡r❡r ❛❞q✉✐r✐r ♦ ❝♦♥❤❡❝✐♠❡♥t♦✳ ❊st❡ ♠❛t❡r✐❛❧ ❞❡✈❡ s❡r ✉s❛❞♦ ❞❡ ♠❛♥❡✐r❛ ✐♥t❡♥s❛ ❡ ♦st❡♥s✐✈❛ ♣❛r❛ q✉❡ ✈♦❝ê ❝♦♠♦ ♣r♦❢❡ss♦r t❡♥❤❛ t♦t❛❧ ❝♦♥✜❛♥ç❛ ❡ ❞♦♠í♥✐♦ ❞♦ ❝♦♥t❡ú❞♦ ♣❛r❛ ♣r♦♣♦r té❝♥✐❝❛s ❡ ❡str❛té❣✐❛s ♥♦s ♣❧❛♥❡❥❛♠❡♥t♦s ❞❡ s✉❛s ❛✉❧❛s✳
❆❜str❛❝t
❚❤❡r❡ ❛r❡ ♠❛♥② ❝❤❛❧❧❡♥❣❡s t❤❛t ♠❛t❤❡♠❛t✐❝s ♦✛❡rs t❤r♦✉❣❤♦✉t ♦✉r ✇♦r❦✐♥❣ ❧✐✈❡s✱ ❡✐t❤❡r ❛s ❛ st✉❞❡♥t ♦r ❛s ❛♥ ❛r❡❛ ♦❢ ✇♦r❦✳ ❚❤❡ q✉❡st ❢♦r ❦♥♦✇❧❡❞❣❡ ✐s ♥❡❝❡ss❛r② t♦ ❜❡ ❛❜❧❡ t♦ ❣✐✈❡ ❛❝❝♦✉♥t ♦❢ t❤❡ ♥❡✇ ❞❡✈❡❧♦♣♠❡♥ts t❤❛t ❛r✐s❡ ❡✈❡r② ❞❛②✳ ❆❧✇❛②s ♥❡❡❞ ♦❢ ♠♦t✐✈❛t✐♦♥ ✐♥ ♦r❞❡r t♦ st❛② ✉♣❞❛t❡❞ ✇✐t❤ t❤❡ ♠❛tt❡rs ♥❡❝❡ss❛r② ❢♦r t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡s❡ ♥❡✇s✳ ◆❡①t✱ ✇❡ ✇✐❧❧ s❤♦✇ ❛ ♠❛t❡r✐❛❧ t❤❛t ✇✐❧❧ ♠♦t✐✈❛t❡ t❤❡ ♠❛t❤ t❡❛❝❤❡r ✇❤♦ s❡❡❦s ❧❡❛r♥✐♥❣ ❛❧❧ t❤❡ t✐♠❡✱ r❡s❝✉✐♥❣ t❤❡ st✐♠✉❧✉s t❤❛t ♦❢t❡♥ ❣❡ts ❧♦st ❜❡❝❛✉s❡ ♦❢ ♣r♦❢❡ss✐♦♥❛❧ ❡❞✉❝❛t✐♦♥ ❝❛♥ ♥♦t s♦❧✈❡ s♦♠❡ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❜❧❡♠s t❤❛t ❛r✐s❡ ✐♥ ②♦✉r ✇❛❧❦✳ ❚❤❡ s✉❜❥❡❝ts t♦ ❜❡ ❞✐s❝✉ss❡❞ ❤❡r❡ ❛r❡ ♦❢ ❣r❡❛t ✉s❡ ✐♥ ❜❛s✐❝ ❡❞✉❝❛t✐♦♥ ❛♥❞ t❤❡② ❤❛✈❡ ❛ ❜❡❛✉t② ❛♥❞ ❣r❛❝❡ ✐♥ s♦❧✈✐♥❣ ♣r♦❜❧❡♠s ♠✉st ❜❡ ❡①❛♠✐♥❡❞ ✇✐t❤ ❣r❡❛t ✐♥t❡r❡st✳ ❚❤❡ ❧❛♥❣✉❛❣❡ ✉s❡❞ ❤❡r❡ ✐s ✈❡r② ♣❧❡❛s❛♥t ❢♦r t❤❡ t❡❛❝❤❡r ♦❢ ▼❛t❤❡♠❛t✐❝s✱ st✐♠✉❧❛t✐♥❣ ✐t ❡✈❡♥ ♠♦r❡ ✐♥ ✇❛♥t✐♥❣ t♦ ❛❝q✉✐r❡ ❦♥♦✇❧❡❞❣❡✳ ❚❤✐s ♠❛t❡r✐❛❧ s❤♦✉❧❞ ❜❡ ✉s❡❞ ✐♥ ❛♥ ✐♥t❡♥s❡ ❛♥❞ ♦✈❡rt ✇❛② ❢♦r ②♦✉ ❛s ❛ t❡❛❝❤❡r ❤❛✈❡ ❢✉❧❧ ❝♦♥✜❞❡♥❝❡ ❛♥❞ ❝♦♥t❡♥ts ♦❢ t❤❡ ❞♦♠❛✐♥ t♦ ♣r♦♣♦s❡ t❡❝❤♥✐q✉❡s ❛♥❞ str❛t❡❣✐❡s ✐♥ ♣❧❛♥♥✐♥❣ t❤❡✐r ❧❡ss♦♥s✳
❙✉♠ár✐♦
✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✹
✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ❞♦s ◆ú♠❡r♦s ■♥t❡✐r♦s ✶✽ ✷✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✸ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❡ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸✳✶ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ♠❞❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸✳✸ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✹ ◆ú♠❡r♦s Pr✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✺ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✸✾
✸✳✶ ❈♦♥❣r✉ê♥❝✐❛ ❞❡ ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❡ ❈♦♥❣r✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✸ ❆ ❉❡s✐❣✉❛❧❞❛❞❡ ❚r✐❛♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✹ ◗✉❛❞r✐❧át❡r♦s ◆♦tá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✹✳✶ P❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✹✳✷ ❚r❛♣é③✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✹✳✸ ❘❡tâ♥❣✉❧♦✱ ❧♦s❛♥❣♦ ❡ q✉❛❞r❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✺ ▲✉❣❛r❡s ●❡♦♠étr✐❝♦s ❇ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✻ ❚❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✸✳✼ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✸✳✾ ➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷
❙❯▼➪❘■❖ ✈✐✐✐
✸✳✾✳✶ ➪r❡❛ ❞♦ P❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✸✳✾✳✷ ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✸✳✾✳✸ ➪r❡❛ ❞♦ ❚r❛♣é③✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✸✳✾✳✹ Pr♦♣r✐❡❞❛❞❡s ●❡♦♠étr✐❝❛s ■♠♣♦rt❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✸✳✶✵ ➪r❡❛ ❞♦ ❈ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾ ✸✳✶✵✳✶ ➪r❡❛s ❞❡ s❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵ ✸✳✶✶ Pr♦❜❧❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷
✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠ ✾✾
✺ ❙♦❧✉çõ❡s ❞♦s Pr♦❜❧❡♠❛s ✶✶✵
✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵ ✺✳✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵ ✺✳✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✵ ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✹ ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✽ ✺✳✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✺
✻ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✶✻✹
■♥tr♦❞✉çã♦
❊ss❡ tr❛❜❛❧❤♦ ✈✐s❛ ❝♦♥t❡♠♣❧❛r ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ❡ ár❡❛s ❛✜♥s✱ ❛tr❛✈és ❞❡ ❝✉rs♦s ❞❡ ❢♦r♠❛çã♦ ♣❛r❛ ♣r♦❢❡ss♦r❡s✱ ♠♦t✐✈❛♥❞♦✲♦s ♣❛r❛ ✉♠ ♠❡❧❤♦r ❞❡s❡♠♣❡♥❤♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ ❊st❡ tr❛❜❛❧❤♦ ❢❛③ ♣❛rt❡ ❞❡ ✉♠ s❡♥t✐♠❡♥t♦ ❞❡ ❛♥❣úst✐❛ q✉❡ t✐✈❡ ❤á ❛❧❣✉♥s ❛♥♦s✱ q✉❛♥❞♦ ♣❡r❝❡❜✐ q✉❡ ❡st❛✈❛ ❡♥s✐♥❛♥❞♦ ▼❛t❡♠át✐❝❛ ❞❡ ♠♦❞♦ ❡q✉✐✈♦❝❛❞♦✱ ❝♦♠ r❡s♦❧✉çõ❡s ❞❡ ❡①❡r❝í❝✐♦s✱ s❡♠ ♠✉✐t❛ ♣r❡♦❝✉♣❛çã♦ ❝♦♠ ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s✱ t❡♦r❡♠❛s ❡ ❛♣❧✐❝❛çõ❡s ❞❡✈✐❞❛s ❞❡ss❡s ❝♦♥❤❡❝✐♠❡♥t♦s✳
❊ss❡ s❡♥t✐♠❡♥t♦ ♠❡ ❡st✐♠✉❧♦✉ ❛ ❞❡s❡♥✈♦❧✈❡r ❡ss❡ tr❛❜❛❧❤♦✱ ♣♦✐s ♣❡♥s♦ q✉❡ ❡ss❛ ❛♥❣úst✐❛ ♥ã♦ é ♣r✐✈✐❧é❣✐♦ ♠❡✉✳ P♦rt❛♥t♦✱ ♠❡✉s ❝♦❧❡❣❛s t❛♠❜é♠ ❞❡✈❡♠ ♣❛ss❛r ♣♦r ✐ss♦✳ ❊ ❛♥t❡s ❞✐ss♦ s❡ ♣r♦❧♦♥❣❛r ♣❛r❛ ❡❧❡s✱ ❣♦st❛r✐❛ ❞❡ ❝♦♠♣❛rt✐❧❤❛r ❛s ✐❞❡✐❛s ❞❡ss❡ tr❛❜❛❧❤♦✱ q✉❡ ❡s♣❡r♦ s❡r ❞❡ ❣r❛♥❞❡ ✈❛❧✐❛ ♣❛r❛ ♠❡✉s ❝♦♠♣❛♥❤❡✐r♦s✳
❖ ♠❛t❡r✐❛❧ ❛ s❡r ❛♣r❡s❡♥t❛❞♦ ❞❡✈❡ s❡r ✉s❛❞♦ ♣❡❧♦ ♣r♦❢❡ss♦r ❝♦♥t✐♥✉❛♠❡♥t❡✱ ♥♦ ✐♥t✉✐t♦ ❞❡ ❡♥❝♦r❛❥á✲❧♦ ❛♦s ❞❡s❛✜♦s q✉❡ ❛ ▼❛t❡♠át✐❝❛ ♣r♦♣õ❡ ♣❛r❛ ❛ ❣❡♥t❡✳ P♦r ♠✉✐t❛s ✈❡③❡s ♣❛ss❛♠♦s ♣♦r s✐t✉❛çõ❡s q✉❡ ♥♦s ❞❡✐①❛♠ ❝♦♥str❛♥❣✐❞♦✱ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ♥❛q✉❡❧❡ ♠♦♠❡♥t♦ ❞❡ ❞✐✜❝✉❧❞❛❞❡ ♥ã♦ ❝♦♥s✐❣❛♠♦s ♦❜t❡r ✉♠❛ ✐❞❡✐❛ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦✳ P❛r❛ t♦❞♦ ❝♦♠❡♥tár✐♦ q✉❡ s❡ ♦✉✈❡✱ ❞❡✈❡♠♦s t❡r ❛t❡♥çã♦ ❡♠ ❝❛❞❛ ♣❛❧❛✈r❛ q✉❡ s❡ é ❢❛❧❛❞❛✳ ▼✉✐t❛s ❞❡ss❛s ♣❛❧❛✈r❛s ♣♦❞❡♠ ♥♦s ❧❡✈❛r ❛ ❞✐✈❡rs♦s ❝❛♠✐♥❤♦s✱ ❜❛st❛ ✈♦❝ê ❛ss✐♠✐❧❛r ❛s ✐♥❢♦r♠❛çõ❡s q✉❡ sã♦ út❡✐s ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ❞❡t❡r♠✐♥❛❞♦s ❛ss✉♥t♦s✳ ❱♦✉ ❝♦♥t❛r ✉♠ ❡①❡♠♣❧♦ q✉❡ ❛❝♦♥t❡❝❡✉ ❝♦♠✐❣♦✳ ❊✉ ❡st❛✈❛ ❛ss✐st✐♥❞♦ ✉♠❛ ❛✉❧❛ ❞❡ ❣❡♦♠❡tr✐❛✱ q✉❛♥❞♦ ♦ ♣r♦❢❡ss♦r ❝♦❧♦❝♦✉ ♥♦ q✉❛❞r♦ ✉♠ tr✐â♥❣✉❧♦ ❝♦♠ ♦s â♥❣✉❧♦s ❞❡ ♠❡❞✐❞❛s x✱ 2x ❡ 3x ♣❛r❛
❙❯▼➪❘■❖ ✷
❡♥tã♦ x 2x 3x✏180✆✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡x✏30✆✱ ❡ ❞❡ss❛ ❢♦r♠❛ 3x✏90✆ ❡ ♣♦rt❛♥t♦✱
♦ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦✳ P❡♥s❡✐ ✐ss♦ ❛ ♣r✐♥❝í♣✐♦✱ ♠❛s ❛❝❤♦ q✉❡ ♣❡❧❛ ❛❣✐❧✐❞❛❞❡ q✉❡ ❡❧❡ ❞❡✉ ❛ r❡s♣♦st❛✱ ❛❝r❡❞✐t♦ q✉❡ ♥ã♦ ❢♦✐ ❡ss❛ ❝♦♥t❛ q✉❡ ❡❧❡ ❢❡③ ♥❛ ❤♦r❛✳ ▲♦❣♦ ❡♠ s❡❣✉✐❞❛✱ ❡ss❡ ♠❡s♠♦ ❝♦♠♣❛♥❤❡✐r♦ ❢❡③ ❛ s❡❣✉✐♥t❡ ❛✜r♠❛çã♦✿ ❙❡ ❛ s♦♠❛ ❞❡ ❞♦✐s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r ❢♦r ✐❣✉❛❧ ❛♦ t❡r❝❡✐r♦✱ ❡♥tã♦ ♦ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦✳ ❊ ❡ss❛ ❝♦♥t❛ é ❜❡♠ ♠❛✐s ❢á❝✐❧ ❞❡ ❢❛③❡r ❞♦ q✉❡ ❛ ♣r✐♠❡✐r❛✱ ♣♦✐s x 2x✏3x✱ ♣♦rt❛♥t♦ ♦ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦✳
◆❛ é♣♦❝❛ ❞♦s ●r❡❣♦s✱ ♣♦❞✐❛✲s❡ ❢❛❧❛r ❞♦ ❝á❧❝✉❧♦ ❡ ❞❛ ❣❡♦♠❡tr✐❛ ❝♦♠♦ ♣❛rt❡s ú♥✐❝❛s ❞❡ ✉♠ ❝♦r♣♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s ❜❡♠ ❞❡❧✐♠✐t❛❞♦ ❡ ♥ã♦ ♠✉✐t♦ ❡①t❡♥s♦✳ ❍♦❥❡ ❡♠ ❞✐❛✱ ♣♦ré♠✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♥t❡ú❞♦s ❞❡ ♠❛t❡♠át✐❝❛ q✉❡ s❡ ❝♦♥❤❡❝❡ é ✐♠❡♥s❛ ❡ ❝r❡s❝❡ ❡①♣♦♥❡♥❝✐✲ ❛❧♠❡♥t❡✱ t♦r♥❛♥❞♦✲s❡ ❞✐❢í❝✐❧ ❞❡❝✐❞✐r q✉❛❧ ❞❡✈❡ s❡r ❛ ♠❛t❡♠át✐❝❛ q✉❡ s❡ ❛❝♦♥s❡❧❤❡ ❡♥s✐♥❛r ❡ ❝♦♠♦ ❞❡✈❡ s❡r ❛♣r❡s❡♥t❛❞❛ ♣❛r❛ s✉❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❡ s✉❛ ♠❡❧❤♦r ✉t✐❧✐❞❛❞❡ ♣❛r❛ ♦ ❢✉t✉r♦ ❞♦s ❛❧✉♥♦s ❬✾❪✳ ❊✉ ❛❝r❡s❝❡♥t♦✱ ❛✐♥❞❛✱ ❛ ❡ss❛ ❡✈♦❧✉çã♦ q✉❡ ❞✐❛ ❛♣ós ❞✐❛ ❛♣❛r❡✲ ❝❡♠ ♥♦✈❛s t❡❝♥♦❧♦❣✐❛s s❡♥❞♦ q✉❡❜r❛❞❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❛♥t❡s ❡r❛♠ ✐♥q✉❡❜rá✈❡✐s ❡ q✉❡ ♥ós✱ ❞♦❝❡♥t❡s✱ ❛❝❤❛♠♦s q✉❡ s❛❜❡♠♦s ✉♠ ♣♦✉❝♦ ❞❡ ♠❛t❡♠át✐❝❛✱ ♠❛s q✉❡ ♥♦ ❢✉♥❞♦ s♦♠♦s ✐♥s❡❣✉r♦s ❞❛q✉✐❧♦ q✉❡ tr❛♥s♠✐t✐♠♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❡ ♣r♦♣♦♠♦s ❝♦♥❤❡❝✐♠❡♥t♦s q✉❡ ♥ã♦ sã♦ ❞❡ ❣r❛♥❞❡ ✈❛❧✐❛ ♣❛r❛ ♦ ❛❧✉♥♦✱ t♦r♥❛♥❞♦✲♦ ✉♠ ❢r✉str❛❞♦ ❝♦♠ ❛ ❞✐s❝✐♣❧✐♥❛✳ ❖ ♣r♦❢❡ss♦r ❲❛❧t❡r ❙♣✐♥❡❧❧✐✱ ❡♠ ❡♥tr❡✈✐st❛ ❛ ❘❡✈✐st❛ ❈á❧❝✉❧♦ ❞❡❝❧❛r❛ q✉❡ s❡♥t❡ ♠❛✐s ❞✐✜❝✉❧❞❛❞❡ ♥♦s t❡♠♣♦s ❞❡ ❤♦❥❡ ❞♦ q✉❡ ❤á ❞♦✐s ❛♥♦s ❡ ❛✐♥❞❛ ♠❛✐s ❞✐✜❝✉❧❞❛❞❡ ❞♦ q✉❡ ❤á q✉❛tr♦ ❛♥♦s✳
▲❡♠❜r♦✲♠❡ q✉❡ ❛♦ ❡st✉❞❛r ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♦ ❡st✉❞♦ ❞❛ ❡q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ♣❡❞✐ ♣❛r❛ ♦ ♣r♦❢❡ss♦r q✉❡ ❞❡♠♦♥str❛ss❡ ❛ ❡q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡①♣♦st❛ ♣♦r ❡❧❡ ♥♦ q✉❛❞r♦ ♥❛q✉❡❧❛ é♣♦❝❛✳ ❆ r❡s♣♦st❛ q✉❡ ❡✉ ♦❜t✐✈❡ ❞❡ss❡ ♣r♦❢❡ss♦r✱ q✉❡ ❡r❛ ❝♦♥s✐❞❡r❛❞♦ ♦ ♠❡❧❤♦r ❞♦ ❝♦❧é❣✐♦ ❡ ✉♠ ❞♦s ♠❡❧❤♦r❡s ❞♦ ❡st❛❞♦✱ ❢♦✐ q✉❡ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ss❛ ❡q✉❛çã♦ ❞❡♠♦r❛r✐❛ ♥♦ ♠í♥✐♠♦ ❞✉❛s ❛✉❧❛s ❞❡ ✺✵ ♠✐♥✉t♦s ❡ q✉❡ ♥ã♦ ✈❛❧❡r✐❛ ❛ ♣❡♥❛ ❞❡♠♦♥strá✲❧❛✳ ❋✐q✉❡✐ ❝♦♠ ✐ss♦ ❣✉❛r❞❛❞♦ ❡♠ ♠✐♥❤❛ ❝❛❜❡ç❛✱ ❛❝❤❛♥❞♦ q✉❡ ✉♠ ❞✐❛ ♣♦❞❡r✐❛ ❞❡❞✉③í✲❧❛✱ ♣♦✐s ❥á ❤❛✈✐❛ ❢❡✐t♦ ✐ss♦ ❝♦♠ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s✳ ❍♦❥❡✱ ❡✉ ♣❡r❝❡❜♦ q✉❡ ♥ã♦ s❡r✐❛ ♥❡❝❡ssár✐❛ ❞✉❛s ❛✉❧❛s ♣❛r❛ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ss❛✱ ♣♦✐s ✉s❛ ❛♣❡♥❛s ✉♠ ♣♦✉❝♦ ❞❡ ❣❡♦♠❡tr✐❛ ❡ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✳ ❖✉ s❡❥❛✱ ✉♠❛ ❝♦✐s❛ s✐♠♣❧❡s ❡ ❛tr❛t✐✈❛ ♣❛r❛ ♠✐♠✳ ❚❡♥❤♦ ❝❡rt❡③❛ ❞❡ q✉❡ s❡ ❡ss❡ ♣r♦❢❡ss♦r ❝♦❧♦❝❛ss❡ ✐ss♦ ❝♦♠♦ ✉♠ ❞❡s❛✜♦ ♣❛r❛ ♠✐♠✱ ❡✉ ♣♦❞❡r✐❛ ♠❡ ❛♣r♦①✐♠❛r ♠❡❧❤♦r ❞❛ ▼❛t❡♠át✐❝❛ ❡ ♠❡ ❛♣❛✐①♦♥❛r ❛✐♥❞❛ ♠❛✐s ♣♦r ❡❧❛✳ P♦ré♠ ❝❛❞❛ ❝♦✐s❛ ❡♠ s❡✉ t❡♠♣♦✱ ❡ ❡✉ ❛❝❤♦ q✉❡ ♦ ♠❡✉ t❡♠♣♦✱ ❛♦s ♣♦✉❝♦s ❡stá ❝❤❡❣❛♥❞♦✳
❙❯▼➪❘■❖ ✸
❊♥t❡♥❞❡♠♦s q✉❡ ♥ã♦ é s✉✜❝✐❡♥t❡ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞♦ ❢✉t✉r♦ ❝✐❞❛❞ã♦ ✉♠ ❛♣r❡♥❞✐③❛❞♦ ❜✉r♦❝rát✐❝♦ ❞❛ ▼❛t❡♠át✐❝❛ ❡ ♣❡r❝❡❜❡♠♦s ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ❡st✐♠✉❧❛r ♦s ❛❧✉♥♦s ❞❡s❞❡ ❝❡❞♦ ❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♥♦✈♦s ❡ ❞❡s❛✜❛♥t❡s✱ ♣r♦♣✐❝✐❛♥❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ✐♠❛❣✐♥❛çã♦ ❡ ❞❛ ❝r✐❛t✐✈✐❞❛❞❡ ❬✶✵❪✳ ❖ ❝♦♥❤❡❝✐♠❡♥t♦ é ❛❞q✉✐r✐❞♦ ❛tr❛✈és ❞❛ ✈♦♥t❛❞❡ ❞❡ ❛♣r❡♥❞❡r ❞♦ ❛❧✉♥♦✳ ◆ã♦ ❡①✐st❡ ✉♠ ♣r♦❢❡ss♦r q✉❡ ❝♦♥s✐❣❛ tr❛♥s♠✐t✐r ❝♦♥❤❡❝✐♠❡♥t♦ ♣❛r❛ ♦ ❛❧✉♥♦ s❡ ❡ss❡ ♥ã♦ ❛❞♠✐t❡ ♦ ❛♣r❡♥❞✐③❛❞♦✳ ❉❡ss❛ ❢♦r♠❛ é ♠✉✐t♦ ❝ô♠♦❞♦ ♣❛r❛ q✉❡♠ ❡st❛r ❛♣r❡♥❞❡♥❞♦✱ ♣♦✐s ♥ã♦ t❡r✐❛ ❡s❢♦rç♦ ♥❡♥❤✉♠ ♣❛r❛ ❛❞q✉✐r✐r ♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ❡ ❛ss✐♠✱ ♥ã♦ ❞❛r✐❛ ✈❛❧♦r ♣❛r❛ ❛q✉✐❧♦ q✉❡ ❡st❛✈❛ s❡♥❞♦ ❛ss✐♠✐❧❛❞♦✳
❆❧é♠ ❞✐ss♦✱ ♣❡♥s♦ q✉❡ ♦ ❛❧✉♥♦ ❞❡✈❡ ❢❛③❡r ♦ s❡✉ ♣❛♣❡❧ ❞❡♥tr♦ ❞❡ s✉❛ ♣r♦✜ssã♦ ❞❡ ❡st✉❞❛♥t❡✳ ❖ q✉❡ q✉❡r♦ ❞✐③❡r é q✉❡ s❡ ♦ ❛❧✉♥♦ ♥ã♦ ❢♦r ❛trás ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♣♦r ✈♦♥t❛❞❡ ♣ró♣r✐❛✱ ♥ã♦ ❡①✐st✐rá ✉♠❛ ♣❡ss♦❛ q✉❡ ❝♦♥s✐❣❛ ❝♦❧♦❝❛r ✐ss♦ ♥❡❧❡✳ ❖ ♣❛♣❡❧ ❞❡ ✉♠ ♣r♦❢❡ss♦r é ❛❥✉❞❛r ♦ ❛❧✉♥♦ ❛ ✈❡r ❛s r❡❧❛çõ❡s ❡♥tr❡ ❛ ▼❛t❡♠át✐❝❛ q✉❡ ❡❧❡ ❡st✉❞❛ ❡ ♦✉tr❛s ❝♦✐s❛s✱ ❛✜r♠❛ ❙♣✐♥❡❧❧✐✳ P♦r ❡①❡♠♣❧♦✱ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s sã♦ ✉s❛❞♦s ♥❛ ❛♥á❧✐s❡ ❞❡ ❝✐r❝✉✐t♦s ❡❧étr✐❝♦s✱ ❧♦❣❛r✐t♠♦s sã♦ ✉s❛❞♦s ♥❛ ❊s❝❛❧❛ ❘✐❝❤t❡r✱ s❡♥♦ ❡ ❝♦ss❡♥♦ ♥♦s ❝á❧❝✉❧♦s ❞❡ r❛♠♣❛s✱ ❡♥tr❡ ♦✉tr❛s✳ ❖ ❛❧✉♥♦ ♥ã♦ ♣♦❞❡ s❡ ❧✐♠✐t❛r ❛ ✈❡r q✉❡ ✉♠ ❛ss✉♥t♦ ❡♠ ▼❛t❡♠át✐❝❛ só s❡r✈❡ ♣❛r❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❡st✉❞♦✳ ❊ ♦✉tr❛✱ q✉❡♠ ❞❡✈❡ ❡st❛❜❡❧❡❝❡r ❛s r❡❧❛çõ❡s ❡♥tr❡ ❛ ▼❛t❡♠át✐❝❛ ❡ ❛s s✐t✉❛çõ❡s ♥❛s q✉❛✐s ❡❧❛ s❡r✈❡ ♥ã♦ é ♦ ♣r♦❢❡ss♦r✱ ❡ s✐♠ ♦ ❛❧✉♥♦✱ ❛✜r♠❛ ♦ ♣r♦❢❡ss♦r ❲❛❧t❡r ❙♣✐♥❡❧❧✐ ❬✶✶❪✱ ♣r♦❢❡ss♦r ❝♦♠ ♠❛✐s ❞❡ ✹✵ ❛♥♦s ❞❡ ❡①♣❡r✐ê♥❝✐❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳
❈❛♣ít✉❧♦
1
❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r
◆❡st❡ ❝❛♣ít✉❧♦✱ ❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ ❜ás✐❝❛ ♥❡❝❡ssár✐❛ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ s❡rá ❞❡s❡♥✈♦❧✈✐❞❛ ❞❡ ❢♦r♠❛ q✉❡ ♦ ❧❡✐t♦r ♥ã♦ t❡rá ❞✐✜❝✉❧❞❛❞❡ ❡♠ ❝♦♠♣r❡❡♥❞❡r ♦ ♠❡s♠♦✳ ❯♠ ❞♦s ♠❛✐♦r❡s ❞❡s❛✜♦s q✉❡ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ q✉❛❧q✉❡r t❡♠❛ ♠❛t❡♠át✐❝♦ é q✉❛♥❞♦ ♥♦s ❞❡♣❛r❛♠♦s ❝♦♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❛r❣✉♠❡♥t❛r s♦❜r❡ ❢❛t♦s q✉❡ sã♦ ó❜✈✐♦s✱ ✐st♦ é✱ ✐♠❡❞✐❛t♦s✳
◆❡st❡ s❡♥t✐❞♦✱ q✉❛♥❞♦ ♣❡♥s❛♠♦s ❡♠ ❝♦♠♣r❡❡♥❞❡r ✉♠❛ t❡♦r✐❛ ♠❛t❡♠át✐❝❛✱ s❡❥❛ ❞❡ ❝✉♥❤♦ ❡❧❡♠❡♥t❛r ♦✉ ♥ã♦✱ ♣❛ss❛♠♦s ♣❡❧❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝♦♥s♦❧✐❞❞❛r ♦s ❝♦♥❝❡✐t♦s ❞❡ss❛ t❡♦r✐❛ ❞❡ ❢♦r♠❛ q✉❡ ♣♦ss❛♠♦s ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ❝♦♥s✐st❡♥t❡ ❡ q✉❡ s❡❥❛ ❝❛♣❛③ ❞❡ ❞❛r ✉♠ s❡♥t✐❞♦ ❧ó❣✐❝♦ ❛♦s ♥♦ss♦s q✉❡st✐♦♥❛♠❡♥t♦s✳
❉❡s❝♦❜r✐ ♥❛ ♣rát✐❝❛ q✉❡ ❛ s♦❧✉çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ♦✉ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ✉♠ ❡♥✉♥❝✐❛❞♦ ♣❡r♣❛ss❛✱ ♠✉✐t❛s ✈❡③❡s✱ ♣❡❧♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❤✐stór✐❝♦ ❞❡ ❝♦♠♦ s❡ ❞❡s❡♥✈♦❧✈❡ ❞❡t❡r♠✐♥❛❞♦ ❝♦♥t❡ú❞♦❬✶❪✳
❆ s❡❣✉✐r tr❛t❛r❡♠♦s s♦❜r❡ ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ ❛ss✉♥t♦ q✉❡ ♠❡ ❡♥❝❛♥t♦✉ ♣❡❧❛ ♣r❡❝✐sã♦ ❝♦♠♦ ♠ét♦❞♦ ❡ ♣❡❧❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ s❡✉s ❛r❣✉♠❡♥t♦s✳ ❙❡♠ ❞ú✈✐❞❛ ♥❡♥❤✉♠❛✱ ❡ss❡ ♣r✐♥❝í♣✐♦ ♠❡ ♠♦str♦✉ ♦ q✉❛♥t♦ ❛ ▼❛t❡♠át✐❝❛ ♣♦❞❡ s❡r ❛♣r❡❝✐❛❞❛ ❝♦♠ s❡✉s ❛r❣✉♠❡♥t♦s ❧ó❣✐❝♦s r❡✢❡t✐♥❞♦ ❛ ❝✐ê♥❝✐❛ ♠❛❣♥í✜❝❛ q✉❡ ❡❧❛ é✳
✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✺
✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛
◆♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳✱ ♦ ♠❛t❡♠át✐❝♦ ●✐✉s❡♣♣❡ P❡❛♥♦ ✭✶✽✺✽✲✶✾✸✷✮✱ ❝✐t❛❞♦ ♣♦r ❬✽❪✱ ❡st❛❜❡❧❡❝❡✉ ♦s ❛①✐♦♠❛s ♥❡❝❡ssár✐♦s ♣❛r❛ ❞❡s❝r❡✈❡r ❝♦♠ ♣r❡❝✐sã♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❖ ú❧t✐♠♦ ❆①✐♦♠❛ ❞❡ P❡❛♥♦ ❞✐③✿
❙❡❥❛ ❆ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ N ✭A ⑨N✮✳ ❙❡ 1P A ❡ s❡✱ ❛❧é♠ ❞✐ss♦✱ ❆ ❝♦♥té♠ t♦❞♦s
♦s s✉❝❡ss♦r❡s ❞♦s s❡✉s ❡❧❡♠❡♥t♦s ❡♥tã♦ A✏N✳
❊ss❡ ❛①✐♦♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❛①✐♦♠❛ ❞❡ ✐♥❞✉çã♦ ❡ s❡r✈❡ ♣❛r❛ ❡st❛❜❡❧❡❝❡r ♣r♦✈❛s ❞❡ ❛✜r♠❛çõ❡s ♠❛t❡♠át✐❝❛s ❝♦♠ ♦ r✐❣♦r ♠❛t❡♠át✐❝♦ ♥❡❝❡ssár✐♦✳ ❆❣♦r❛✱ ✐r❡♠♦s ❡♥✉♥❝✐❛r ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✳
❙❡❥❛P♣nq ✉♠❛ s❡♥t❡♥ç❛ ❛❜❡rt❛ s♦❜r❡ N✳ ❙✉♣♦♥❤❛ q✉❡✿
✭✐✮ ✭❇❛s❡ ❞❛ ■♥❞✉çã♦✮ ❙❡P♣n0q é ✈❡r❞❛❞❡✐r❛ ❡
✭✐✐✮ ♣❛r❛ t♦❞♦ n➙n0✱ s❡♠♣r❡ q✉❡ P♣nq é ✈❡r❞❛❞❡✐r❛✱ s❡❣✉❡ q✉❡P♣n 1q é ✈❡r❞❛❞❡✐r❛✳
❊♥tã♦✱ P♣nq é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ nP N✳
❊ss❡ ♣r✐♥❝í♣✐♦ ❢♦r♥❡❝❡ ✉♠❛ ❞❛s ♠❛✐s ♣♦❞❡r♦s❛s té❝♥✐❝❛s ❞❡ ❞❡♠♦♥str❛çã♦ ❞❛ ▼❛t❡♠á✲ t✐❝❛✿ ❛ ❞❡♠♦♥str❛çã♦ ♣♦r ✐♥❞✉çã♦✳ ❆ s❡❣✉✐r ✐❧✉str❛r❡♠♦s ✐ss♦ ❝♦♠ ❛❧❣✉♥s ❡①❡♠♣❧♦s✳
❊①❡♠♣❧♦ ✶✳✶✳✶✳ ▼♦str❡ q✉❡ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦ n t❡♠♦s
1 2 ☎ ☎ ☎ n✏ n☎ ♣n 1q
2 .
Pr♦✈❛r❡♠♦s ❛ ❛✜r♠❛çã♦ ❛❝✐♠❛ ✉s❛♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✳ ◆♦t❡ q✉❡
1✏ 1☎ ♣1 1q
2 .
P♦rt❛♥t♦✱ P♣1q é ✈❡r❞❛❞❡✐r❛✳
❆ss✉♠✐r❡♠♦s✱ ❛❣♦r❛✱ q✉❡P♣nq é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❛❧❣✉♠ nP N✱ ♦✉ s❡❥❛✱
1 2 ☎ ☎ ☎ n✏ n☎ ♣n 1q
2 .
❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ■♥❞✉çã♦✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ P♣n 1q
é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛✱ q✉❡
1 2 ☎ ☎ ☎ n ♣n 1q ✏ ♣n 1q ☎ r♣n 1q 1s
✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✻
❖❜s❡r✈❡ q✉❡
♣1 2 ☎ ☎ ☎ nq ♣n 1q ✏ n☎ ♣n 1q
2 ♣n 1q
✏ ♣n 1q ☎
✁n
2 1
✠
✏ ♣n 1q ☎
✂
n 2 2
✡
✏ ♣n 1q ☎ ♣n 2q
2 .
P♦rt❛♥t♦ P♣n 1q é ✈❡r❞❛❞❡✐r❛✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ P♣nq é
✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ nPN✳
❖ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ♠♦str❛r é q✉❡ ❞❛❞♦ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧n✱ ❛ s♦♠❛ ❞♦sn ♣r✐♠❡✐r♦s
♥❛t✉r❛✐s ❝♦♥s❡❝✉t✐✈♦s s❡rá ✐❣✉❛❧ ❛ n☎ ♣n 1q
2 ✳
❊①❡♠♣❧♦ ✶✳✶✳✷✳ Pr♦✈❡ q✉❡ n3
♣n 1q3
♣n 2q3
é ♠ú❧t✐♣❧♦ ❞❡3✳
▼♦str❛r❡♠♦s✱ ♣♦r ✐♥❞✉çã♦✱ q✉❡ ❛ ❛✜r♠❛çã♦ ❛❝✐♠❛ é ✈❡r❞❛❞❡✐r❛✳ P❡r❝❡❜❛ q✉❡
13 ♣1 1q3 ♣1 2q3 ✏36✏3☎12.
P♦rt❛♥t♦✱ P♣1q é ✈❡r❞❛❞❡✐r❛✳
❆ss✉♠✐r❡♠♦s✱ ❛❣♦r❛✱ q✉❡P♣nq é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❛❧❣✉♠ nP N✱ ♦✉ s❡❥❛
n3 ♣n 1q3 ♣n 2q3 ✏3k, kPN.
P❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡P♣n 1q é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛✱
q✉❡
♣n 1q3 ♣n 1 1q3 ♣n 2 1q3 ✏3t, tPN.
➱ ✈❡r❞❛❞❡✱
♣n 1q3
♣n 2q3
♣n 3q3
✁ rn3
♣n 1q3
♣n 2q3
s ✏ ♣n 3q3
✁n3
✏n3
3☎n2
☎3 3☎n☎32
33
✁n3
✏3☎ ♣3☎n2
32☎
n 32q
✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✼
▲♦❣♦✱
♣n 1q3 ♣n 2q3 ♣n 3q3 ✏3t, t✏3☎n2 32☎n 32, nP N.
P♦rt❛♥t♦✱ P♣n 1q é ✈❡r❞❛❞❡✐r❛✳ ❉❡ss❡ ♠♦❞♦ P♣nq é ✈❡r❞❛❞❡ ♣❛r❛ t♦❞♦ n P N✱ ♣❡❧♦
Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ■♥✜♥✐t❛✳ ❖✉ s❡❥❛✱ n3
♣n 1q3
♣n 2q3
é ❞❛ ❢♦r♠❛ 3k, k P N ❡
❝♦♥❝❧✉✐✲s❡ q✉❡ n3
♣n 1q3
♣n 2q3
é ♠ú❧t✐♣❧♦ ❞❡ 3 ♣❛r❛ t♦❞♦n PN✳
❊①❡♠♣❧♦ ✶✳✶✳✸✳ ▼♦str❡ q✉❡✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ n✱ Mn :✏n☎ ♣n2✁1q ☎ ♣3n 2q é
♠ú❧t✐♣❧♦ ❞❡ ✷✹✳
❚❛♠❜é♠ ✉s❛r❡♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ♣❛r❛ ♣r♦✈❛r ❡ss❛ ✐❣✉❛❧❞❛❞❡✳ ◆♦t❡ q✉❡
M1 ✏1☎ ♣1 2
✁1q ☎ ♣3.1 2q ✏ 0.
❉❡ss❛ ❢♦r♠❛✱ M1 é ♠ú❧t✐♣❧♦ ❞❡ ✷✹ ❡✱ ♣♦rt❛♥t♦✱P♣1q é ✈❡r❞❛❞❡✐r❛✳
❆ss✉♠✐r❡♠♦s ❛❣♦r❛ q✉❡P♣nq é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛✱ Mn ✏24☎t, tPN.
❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ■♥❞✉çã♦✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ P♣n 1q
é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛✱
Mn 1 ✏24☎k, k PN.
❉❡ ❢❛t♦✱
Mn 1✁Mn ✏ ♣n 1q ☎ r♣n 1q2✁1s ☎ r3☎ ♣n 1q 2s ✁n☎ ♣n2✁1q ☎ ♣3n 2q
✏ ♣n 1q ☎ rn2
2ns ☎ r3n 5s ✁n☎ ♣n2✁
1q ☎ ♣3n 2q
✏ ♣n 1q ☎ rn☎ ♣n 2qs ☎ r3n 5s ✁n☎ ♣n2✁
1q ☎ ♣3n 2q
✏ n☎ ♣n 1q ☎ ♣n 2q ☎ ♣3n 5q ✁n☎ ♣n✁1q ☎ ♣n 1q ☎ ♣3n 2q
✏ n☎ ♣n 1q ☎ r3n2
11n 10✁3n2
n 2s
✏ n☎ ♣n 1q ☎ ♣12n 12q
✏ 12n☎ ♣n 1q2
.
▲♦❣♦✱
Mn 1✁Mn✏12n☎ ♣n 1q 2
.
▲❡✈❛♥❞♦✲s❡ ❡♠ ❝♦♥s✐❞❡r❛çã♦ q✉❡ n ❡ n 1 sã♦ ♥ú♠❡r♦s ♥❛t✉r❛✐s ❝♦♥s❡❝✉t✐✈♦s✱ t❡♠♦s
q✉❡ ✉♠ ❞❡❧❡s é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♣❛r✳ ❉❡ss❛ ❢♦r♠❛✱
✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✽
❉❛í✱
Mn 1✁Mn ✏24s, sPN.
❉♦♥❞❡✱ s❡❣✉❡ q✉❡
Mn 1 ✏ 24s Mn
✏ 24s 24t
✏ 24☎ ♣s tq.
❉❡♥♦t❛♥❞♦ ♣♦r k ✏s t PN✱ t❡♠✲s❡ q✉❡
Mn 1 ✏24k, kPN.
P♦rt❛♥t♦✱P♣n 1q é ✈❡r❞❛❞❡✐r❛✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ t❡♠✲s❡
q✉❡ P♣nq é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ n✳
❊①❡♠♣❧♦ ✶✳✶✳✹✳ ✭P✐③③❛ ❞❡ ❙t❡✐♥❡r✮ ▼♦str❡ q✉❡ ♦ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ ♣❛rt❡s ♣♦ssí✈❡✐s q✉❡ ♣♦❞❡♠♦s ❞✐✈✐❞✐r ♦ ♣❧❛♥♦ ❝♦♠ n r❡t❛s ❞❡st❡ ♣❧❛♥♦ é ❞❛❞♦ ♣♦r P♣nq ✏ n☎ ♣n 1q
2 1✳
■r❡♠♦s ♠♦str❛r ♣❡❧♦ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ■♥❞✉çã♦ q✉❡ ❛ ❛✜r♠❛çã♦ ❛❝✐♠❛ é ✈❡r✲ ❞❛❞❡✐r❛✳ ❖❜s❡r✈❡ q✉❡
P♣1q ✏ 1☎ ♣1 1q
2 1✏2.
▲♦❣♦✱ P♣1q é ✈❡r❞❛❞❡✐r❛✳
❱❡❥❛ ♥❛ ♣rát✐❝❛ ❝♦♠♦ ✐ss♦ ❛❝♦♥t❡❝❡✳ ❙✉♣♦♥❤❛ ✉♠ ♣❧❛♥♦ ♥❛ ❢♦r♠❛ ❞❡ ✉♠❛ ♣✐③③❛ ❡ q✉❡r❡♠♦s ❞❛r ✉♠ ❝♦rt❡ ♥❡❧❛✳ ➱ ♦❜✈✐♦ q✉❡ t❡r❡♠♦s ❡ss❡ ♣❧❛♥♦ ✭♣✐③③❛✮ ❞✐✈✐❞✐❞♦ ❡♠ ❞✉❛s ♣❛rt❡s✳ ❱❡❥❛ ❛s ✜❣✉r❛s✳ ◆❛ ✜❣✉r❛ ♣aq ❛♣❛r❡❝❡ ♦ ♣❧❛♥♦ ❡♠ ❢♦r♠❛ ❞❡ ♣✐③③❛ s❡♠ ♦ ❝♦rt❡ ❡ ♥❛
✜❣✉r❛ ♣bq ❛♣❛r❡❝❡ ♦ ♣❧❛♥♦ ❞✐✈✐❞✐❞♦ ❡♠ ❞✉❛s ♣❛rt❡s ❝♦♠ ✉♠ ❝♦rt❡✳
F igura♣aq F igura♣bq
❆❣♦r❛✱ ❛ss✉♠✐r❡♠♦s q✉❡P♣nq é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛ P♣nq ✏ n☎ ♣n 1q
✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✾
❡ ✐r❡♠♦s ♠♦str❛r✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ q✉❡ P♣n 1q é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛✱
P♣n 1q ✏ ♣n 1q ☎ ♣n 2q
2 1.
❚r❛ç❛♥❞♦✲s❡ n r❡t❛s✱ ♦❜t❡♠♦s✱ ♣♦r ❤✐♣ót❡s❡✱ n☎ ♣n 1q
2 1 ♣❛rt❡s✳ ❖ q✉❡ s❡ ❞❡✈❡
❡♥tr❛r ❡♠ ❞✐s❝✉sã♦ é q✉❛❧ s❡r✐❛ ❛ ❢♦r♠❛ ❞❡ ❝♦rt❛r ♦ ♣❧❛♥♦ ❝♦♠ n 1 r❡t❛s ♣❛r❛ ♦❜t❡r♠♦s
♦ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ ♣❛rt❡s ♣♦ssí✈❡✐s✳
■r❡♠♦s ♦❜t❡r ♦ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ ♣❛rt❡s s❡ ❛ r❡t❛ ❞❡ ♦r❞❡♠ n 1❝♦rt❛r ❝❛❞❛ ✉♠❛ ❞❛s
♦✉tr❛s r❡t❛s ❡♠ ♣♦♥t♦s q✉❡ ♥ã♦ s❡❥❛♠ ❞❡ ✐♥t❡rs❡❝çã♦✳ ❖❜s❡r✈❡ ❛s ✜❣✉r❛s ❛❜❛✐①♦ ♣❛r❛ ❡ss❡ ❡♥t❡♥❞✐♠❡♥t♦✳
F igura♣cq F igura♣dq
❖❜s❡r✈❡ q✉❡ ♥❛ ✜❣✉r❛ ♣cq ❛ r❡t❛ ❞❡ ♦r❞❡♠ n 1 ❝♦rt❛ ❛s ♦✉tr❛s r❡t❛s ♥❛ ✐♥t❡rs❡❝çã♦✱
♦ q✉❡ ❞✐✈✐❞❡ ♦ ♣❧❛♥♦ ❡♠ s❡✐s ♣❛rt❡s✱ ❡♥q✉❛♥t♦ q✉❡ ♥❛ ✜❣✉r❛ ♣dq ❛ r❡t❛ ♥ã♦ ❝♦rt❛ ♥❛
✐♥t❡rs❡❝çã♦ ❡ ❞✐✈✐❞❡ ♦ ♣❧❛♥♦ ❡♠ s❡t❡ ♣❛rt❡s✳
❆❧é♠ ❞✐ss♦✱ ❛ r❡t❛ ❞❡ ♦r❞❡♠ n 1 ❛♦ ❝♦rt❛r ❛ ♣r✐♠❡✐r❛ r❡t❛✱ ❞✐✈✐❞❡ ❡♠ ❞♦✐s ♣❡❞❛ç♦s
❛ ♣❛rt❡ q✉❡ s❡ ❡♥❝♦♥tr❛✈❛ ✐♥✐❝✐❛❧♠❡♥t❡✳ ❊♠ s❡❣✉✐❞❛✱ ❞✐✈✐❞❡ ❛ ♣❛rt❡ s❡❣✉✐♥t❡✱ ❛ss✐♠ q✉❡ t♦❝❛ ❛ s❡❣✉♥❞❛ r❡t❛ ❡ s✉❝❡ss✐✈❛♠❡♥t❡ ❡❧❛ ✈❛✐ ❞✐✈✐❞✐♥❞♦ ❡♠ ❞♦✐s ❡s♣❛ç♦s ❛s ♣❛rt❡s ❡♠ q✉❡ ♣❛ss❛ ❛♥t❡s ❞❡ ❡♥❝♦♥tr❛r ❝❛❞❛ ✉♠❛ ❞❛s nr❡t❛s✳ ❈♦♠ ✐ss♦ sã♦ ❣❡r❛❞❛s n 1♣❛rt❡s ❛ ♠❛✐s
❞♦ q✉❡ s❡ t✐♥❤❛✳ ❊✱ ♣♦rt❛♥t♦✱
P♣n 1q ✏ P♣nq ♣n 1q
✏ n☎ ♣n 1q
2 1 ♣n 1q
✏ n☎ ♣n 1q 2☎ ♣n 1q
2 1
✏ ♣n 1q ☎ ♣n 2q
✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✶✵
❆ ♣r♦♣r✐❡❞❛❞❡P♣n 1q é ✈❡r❞❛❞❡✐r❛✱ ❡✱ ♣♦rt❛♥t♦P♣nq t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ n P N✳ ❈♦♥❝❧✉✐✲s❡ q✉❡ ♦ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ ♣❛rt❡s ♣♦ssí✈❡❧ q✉❡ ♣♦❞❡♠♦s ❞✐✈✐❞✐r ♦ ♣❧❛♥♦
❝♦♠ n r❡t❛s é ❞❛❞♦ ♣♦r n☎ ♣n 1q
✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✶✶
Pr♦❜❧❡♠❛s
✶✳ ❉❡♠♦♥str❛r ♣♦r ✐♥❞✉çã♦ q✉❡ ♣❛r❛n ➙1♥❛t✉r❛❧
✭❛✮ 12
22 ☎ ☎ ☎
n2 ✏ n♣n 1q♣2n 1q
6 .
✭❜✮ 13
23
☎ ☎ ☎ n3
✏ ♣1 2 ☎ ☎ ☎ nq2
✭❝✮ 1✁22
32
✁ ☎ ☎ ☎ ♣✁1qn✁1
☎n2
✏ ♣✁1qn✁1
☎ n☎ ♣n 1q
2 .
✷✳ ❉❡♠♦♥str❛r q✉❡
✭❛✮ n3✁
n é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ✻ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧n✳
✭❜✮ 5n✁1 é ♠ú❧t✐♣❧♦ ❞❡ ✷✹ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ n ♣❛r✳
✭❝✮ 2n 1 é ♠ú❧t✐♣❧♦ ❞❡ ✸ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧ í♠♣❛r n✳
✸✳ ❉❡♠♦♥str❡ q✉❡ ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s ♥ú♠❡r♦s í♠♣❛r❡s é n2
✱ ♦✉ s❡❥❛
1 3 ☎ ☎ ☎ ♣2n✁1q ✏n2
.
✹✳ ❉❡♠♦♥str❡ q✉❡ ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ♣❛r❡s én☎ ♣n 1q✱ ♦✉ s❡❥❛ 2 4 ☎ ☎ ☎ 2n ✏n☎ ♣n 1q.
✺✳ ▼♦str❡✱ ✉s❛♥❞♦ ✐♥❞✉çã♦✱ q✉❡ 1 11 ☎ ☎ ☎ 11☎ ☎ ☎1 ✏ 10 81♣10
n✁1q ✁ n
9✱ ♦♥❞❡ n
r❡♣r❡s❡♥t❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✧✉♥s✧ ♣r❡s❡♥t❡s ♥♦ ú❧t✐♠♦ t❡r♠♦ ❞❛ ❡①♣r❡ssã♦✳
✻✳ ▼♦str❡ ♣♦r ✐♥❞✉çã♦ q✉❡ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ ❞❡ ♥ ❧❛❞♦s ♣♦ss✉✐ ❡①❛t❛♠❡♥t❡ n♣n✁3q
2
❞✐❛❣♦♥❛✐s✳
✼✳ Pr♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ ✭P✳❆✳✮ é ✉♠❛ s❡q✉ê♥❝✐❛ ♥✉♠ér✐❝❛ ❡♠ q✉❡ ❝❛❞❛ t❡r♠♦✱ ❛ ♣❛rt✐r ❞♦ s❡❣✉♥❞♦✱ é ♦❜t✐❞♦ s♦♠❛♥❞♦✲s❡ ♦ t❡r♠♦ ❛♥t❡r✐♦r ❝♦♠ ✉♠❛ ❝♦♥st❛♥t❡✱ ❝❤❛♠❛❞❛ r❛③ã♦ ❞❛ P✳❆✳✱ ✐♥❞✐❝❛❞❛ ♣♦r r✳
▼♦str❡✱ ♣♦r ✐♥❞✉çã♦✱ q✉❡ ♦ t❡r♠♦ ❣❡r❛❧ ❞❡ ✉♠❛ P✳❆✳ é ❞❛❞♦ ♣♦r
✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✶✷
✽✳ ❯s❛♥❞♦ ♦ Pr✐❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ ♠♦str❡ q✉❡ ❛ s♦♠❛ ❞♦sn t❡r♠♦s ❞❡ ✉♠❛ P✳❆✳
é ❞❛❞❛ ♣♦r
Sn✏ ♣
a1 anq ☎n
2
✾✳ Pr♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ ✭P✳●✳✮ é ✉♠❛ s❡q✉ê♥❝✐❛ ♥✉♠ér✐❝❛ ❡♠ q✉❡ ❝❛❞❛ t❡r♠♦✱ ❛ ♣❛rt✐r ❞♦ s❡❣✉♥❞♦✱ é ♦❜t✐❞♦ ♠✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ♦ t❡r♠♦ ❛♥t❡r✐♦r ♣♦r ✉♠❛ ❝♦♥st❛♥t❡✱ ❝❤❛♠❛❞❛ r❛③ã♦ ❞❛ P✳●✳✱ ✐♥❞✐❝❛❞❛ ♣♦r q✳
▼♦str❡✱ ♣♦r ✐♥❞✉çã♦✱ q✉❡ ♦ t❡r♠♦ ❣❡r❛❧ ❞❡ ✉♠❛ P✳●✳ é ❞❛❞♦ ♣♦r
an✏a1☎qn✁ 1
✶✵✳ ▼♦str❡✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ q✉❡ ❛ s♦♠❛ ❞♦s n t❡r♠♦s ❞❡ ✉♠❛ P✳●✳ é
❞❛❞❛ ♣♦r
Sn✏
a1♣qn✁1q
q✁1 , ♦♥❞❡ q ✘1.
✶✶✳ ✭❆ ▼♦❡❞❛ ❋❛❧s❛ ❞♦ ❘❡✐✮ ❯♠❛ ❞❛s 3n ♠♦❡❞❛s ❞❡ ✉♠ ❘❡✐ é ❢❛❧s❛ ❡ ♣❡s❛ ♠❡♥♦s
✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✶✸
✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s
◆❡st❛ s❡çã♦ ❡♥✉♥❝✐❛r❡♠♦s ✉♠ ♣r✐♥❝í♣✐♦✱ ❝✉❥❛ ❧ó❣✐❝❛ ♠❛t❡♠át✐❝❛ é ❞❡ ✉♠❛ s✐♠♣❧✐❝✐❞❛❞❡ ❡♥❝❛♥t❛❞♦r❛✱ ♠❛s ♣♦✉❝♦ ❛❜♦r❞❛❞♦ ♣❡❧♦s ♣r♦❢❡ss♦r❡s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ❊st❡ ❜❡❧♦ ♣r✐♥❝í♣✐♦✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s é ❞❡ ❣r❛♥❞❡ ✈❛❧✐❛ q✉❛♥❞♦ ❛t❛❝❛♠♦s ❡①❡♠✲ ♣❧♦s ♦♥❞❡ ♦ ✉s♦ ❞♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦ s❡ ❢❛③ ♥❡❝❡ssár✐♦ s❡♠ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ❢ór♠✉❧❛s ♣❛r❛ r❡s♦❧✉çã♦ ❞♦s ♠❡s♠♦s✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✶✳ ✭P❈P ✲ ✈❡rsã♦ s✐♠♣❧❡s✮ ❙❡ ❞✐str✐❜✉í♠♦s N 1 ♣♦♠❜♦s ❡♠ N ❝❛s❛s✱
❡♥tã♦ ❛❧❣✉♠❛ ❞❛s ❝❛s❛s ❝♦♥té♠ ❞♦✐s ♦✉ ♠❛✐s ♣♦♠❜♦s✳
❉❡♠♦♥str❛çã♦✿ ◗✉❛♥❞♦ ❞✐str✐❜✉í♠♦s ✉♠ ♣♦♠❜♦ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ❝❛s❛s ❡①✐st❡♥t❡s✱ ♥♦t❛♠♦s q✉❡ s♦❜r❛ ✉♠ ♣♦♠❜♦ ♣❛r❛ s❡r ❞✐str✐❜✉í❞♦ ❡♠ ✉♠❛ ❞❛s ❝❛s❛s q✉❡ ❥á ❡①✐st❡ ✉♠ ♣♦♠❜♦ ♥❡❧❛✳
P♦r ♦✉tr♦ ❧❛❞♦✱ s✉♣♦♥❤❛♠♦s q✉❡ ❡♠ ❝❛❞❛ ❝❛s❛ ♥ã♦ ❡①✐st❡ ♠❛✐s ❞♦ q✉❡ ✉♠ ♣♦♠❜♦✱ ❡♥tã♦ q✉❛♥❞♦ ❝♦♥t❛♠♦s t♦❞♦s ♦s ♣♦♠❜♦s ❝♦♥t✐❞♦s ♥❛s N ❝❛s❛s✱ ♥ã♦ t❡r❡♠♦s ♠❛✐s ❞♦ q✉❡ N ♣♦♠❜♦s✳ ❖ q✉❡ ❝♦♥tr❛r✐❛ ❛ ❤✐♣ót❡s❡ ❞❡ t❡r♠♦s N 1 ♣♦♠❜♦s✳ ▲♦❣♦ t❡r❡♠♦s ♣❡❧♦
♠❡♥♦s ✉♠❛ ❞❛s ❝❛s❛s ❝♦♠ ♠❛✐s ❞❡ ✉♠ ♣♦♠❜♦✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✷✳ ✭P❈P ✲ ✈❡rsã♦ ❣❡r❛❧✮ ❙❡ ❞✐str✐❜✉í♠♦s N k 1 ♣♦♠❜♦s ❡♠ N ❝❛s❛s✱
❡♥tã♦ ❛❧❣✉♠❛ ❞❛s ❝❛s❛s ❝♦♥té♠ ♣❡❧♦ ♠❡♥♦s k 1 ♣♦♠❜♦s✳
❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ ❡♠ ❝❛❞❛ ❝❛s❛ ♥ã♦ ❡①✐st❡ ♠❛✐s ❞♦ q✉❡ k ♣♦♠❜♦s✱ ❡♥tã♦
❝♦♥t❛♥❞♦ t♦❞♦s ♦s ♣♦♠❜♦s ❝♦♥t✐❞♦s ♥❛s N ❝❛s❛s ♥ã♦ t❡r❡♠♦s ♠❛✐s ❞♦ q✉❡ N k ♣♦♠❜♦s✱
❝♦♥tr❛❞✐③❡♥❞♦ ❛ ❤✐♣ót❡s❡ ❞❡ t❡r♠♦s N k 1 ♣♦♠❜♦s ❞✐str✐❜✉í❞♦s ♥❛s N ❝❛s❛s✳
❙❡♥❞♦ k✏1✱ t❡r❡♠♦s ❛ ✈❡rsã♦ s✐♠♣❧❡s ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛s ❞♦s P♦♠❜♦s✳
❯♠❛ ✈❡rsã♦ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s q✉❡ ♣♦❞❡ s❡r ✉s❛❞❛ ❝♦♠♦ ✉♠❛ ❛♣❧✐❝❛çã♦ é✿
❙❡ ❛ s♦♠❛ ❞♦s n♥ú♠❡r♦s ♥❛t✉r❛✐s é ✐❣✉❛❧ ❛ S✱ ❡♥tã♦ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞❡❧❡s q✉❡
♥ã♦ é ♠❛✐♦r q✉❡ S
n✱ ❛ss✐♠ ❝♦♠♦ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞❡❧❡s q✉❡ ♥ã♦ é ♠❡♥♦r q✉❡ S n✳
❖❜s❡r✈❡ ✉♠ ❜♦♠ ❡①❡♠♣❧♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ❡♥✉♥❝✐❛❞♦ ♥❛ ♣r♦♣♦s✐çã♦ ✶✳✷✳✶✳
✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✶✹
◆♦t❡ ❛ ❣r❛❝✐♦s✐❞❛❞❡ ❞❡ss❛ s♦❧✉çã♦✱ ♣❛r❛ ❡♥t❡♥❞❡r ❢❛❝✐❧♠❡♥t❡ ❡ss❡ ♣r✐♥❝í♣✐♦✳
❖s ✸✶✵ ❝❛❥✉❡✐r♦s r❡♣r❡s❡♥t❛♠ ♦s ♣♦♠❜♦s ♥❡ss❡ ♣r✐♥❝í♣✐♦ ❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❢r✉t♦s ❡♠ ❝❛❞❛ ❝❛❥✉❡✐r♦ s❡rá r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ ❝❛s❛ ❞♦s ♣♦♠❜♦s✳ ❚❡r❡♠♦s ❝❛❥✉❡✐r♦ q✉❡ ♣♦❞❡ ♥ã♦ ❝♦❧♦❝❛r ❢r✉t♦s✱ ❛ss✐♠ ❝♦♠♦ ♣♦❞❡ ❝♦❧♦❝❛r ❛♣❡♥❛s ✉♠ ❢r✉t♦✱ ♦✉ ❞♦✐s✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡ ❛té ✉♠❛ q✉❛♥t✐❞❛❞❡ ♠á①✐♠❛ ❞❡ ✸✵✵ ❢r✉t♦s ♥♦ ♣é✳ ❊♥✉♠❡r❡♠♦s ❛s ❝❛s❛s ❞♦s ♣♦♠❜♦s ❞❡ ✵ ❛ ✸✵✵✱ ♦♥❞❡ ✵ r❡♣r❡s❡♥t❛ ♦ ❝❛❥✉❡✐r♦ q✉❡ ♥ã♦ ❝♦❧♦❝❛ ❢r✉t♦ ❡ ✸✵✵ r❡♣r❡s❡♥t❛ ♦ ❝❛❥✉❡✐r♦ q✉❡ ❝♦❧♦❝❛ ✸✵✵ ❢r✉t♦s ❡ ♦r❞❡♥❡♠♦s ♦s ♣♦♠❜♦s✱ q✉❡ r❡♣r❡s❡♥t❛ ♦s ❝❛❥✉❡✐r♦s✱ ❞❡ ✶♦ ❛ ✸✶✵♦✳
❙✉♣♦♥❤❛ q✉❡ ♦ ✶♦ ❝❛❥✉❡✐r♦ ♥ã♦ ♣♦♥❤❛ ❢r✉t♦✱ ♦ ✷♦ ♣♦♥❤❛ ❛♣❡♥❛s ✉♠ ❢r✉t♦✱ ❡ ❛ss✐♠ ♣♦r
❞✐❛♥t❡✳ ❖❜s❡r✈❡ ♦ ❡sq✉❡♠❛ ❛ s❡❣✉✐r✳
◗✉❛♥t✐❞❛❞❡ ❞❡ ❢r✉t♦s ♥♦ ♣é ✭❝❛s❛ ❞♦s ♣♦♠❜♦s✮ ✵ ✶ ✷ ☎ ☎ ☎ ✸✵✵
❈❛❥✉❡✐r♦ ✭♣♦♠❜♦s✮ 1♦ 2♦ 3♦ ☎ ☎ ☎ 301♦
◆♦t❡ q✉❡✱ ❛✐♥❞❛ s♦❜r❛♠ ♥♦✈❡ ❝❛❥✉❡✐r♦s ♣❛r❛ ❝♦❧♦❝❛r ❡♠ ✉♠❛ ❞❛s ♣♦s✐çõ❡s ❥á ♦❝✉♣❛❞❛s ♣♦r ❛❧❣✉♠ ♦✉tr♦ ❝❛❥✉❡✐r♦✳ P♦rt❛♥t♦✱ t❡r❡♠♦s ❞♦✐s ❝❛❥✉❡✐r♦s ♦✉ ♠❛✐s q✉❡ t❡♥❤❛♠ ❛ ♠❡s♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❢r✉t♦s✳
❖❜s❡r✈❡ ♦✉tr♦ ❡①❡♠♣❧♦✳
❊①❡♠♣❧♦ ✶✳✷✳✷✳ ❊♠ ✉♠❛ r❡✉♥✐ã♦ ❤á ✷✵ ♣❡ss♦❛s✳ ▼♦str❡ q✉❡ ♥❡ss❛ r❡✉♥✐ã♦ ❡①✐st❡♠ ❞✉❛s ♣❡ss♦❛s q✉❡ ❝♦♥❤❡❝❡♠ ❡①❛t❛♠❡♥t❡ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s✳
◆❡st❡ ❝❛s♦✱ ❛s ✷✵ ♣❡ss♦❛s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦s ♣♦♠❜♦s ❡ ❛s ❝❛s❛s ❞♦s ♣♦♠❜♦s✱ q✉❡ r❡♣r❡s❡♥t❛♠ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♣❡ss♦❛s ❝♦♥❤❡❝✐❞❛s ♣♦r ❝❛❞❛ ♣❡ss♦❛✱ sã♦ ❡♥✉♠❡r❛❞❛s ❞❡ ✵ ❛ ✶✾✱ ♦♥❞❡ ③❡r♦ ✐♥❞✐❝❛ q✉❡ ❛ ♣❡ss♦❛ ♥ã♦ ❝♦♥❤❡❝❡ ♥✐♥❣✉é♠ ❡ ✶✾ ✐♥❞✐❝❛ q✉❡ ❛ ♣❡ss♦❛ ❝♦✲ ♥❤❡❝❡ t♦❞♦s ♦s ♦✉tr♦s ❞❛ r❡✉♥✐ã♦✳ ◆❡ss❛ r❡✉♥✐ã♦✱ ♥ã♦ ❡①✐st❡ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rr❡r✱ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ♥ã♦ ❝♦♥❤❡❝❡r ♥✐♥❣✉é♠ ❡ ❝♦♥❤❡❝❡r ❛ t♦❞♦s✳ ▲♦❣♦✱ ✉♠❛ ❞❛s ❝❛s❛s✱ ✵ ♦✉ ✶✾✱ ♣❡r♠❛♥❡❝❡ ❞❡s♦❝✉♣❛❞❛✳ ❊♥✉♠❡r❛♥❞♦ ❞❡ ✵ ❛ ✶✾ ♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s ❝♦♥❤❡❝✐❞❛s ❡ ♦r❞❡♥❛♥❞♦ ❛s ✷✵ ♣❡ss♦❛s ❞❛ r❡✉♥✐ã♦✱ t❡♠♦s✿
◗✉❛♥t✐❞❛❞❡ ❞❡ ♣❡ss♦❛s ❝♦♥❤❡❝✐❞❛s ✭❝❛s❛ ❞♦s ♣♦♠❜♦s✮ ✵ ✶ ✷ ☎ ☎ ☎ ✶✾
P❡ss♦❛s ✭♣♦♠❜♦s✮ ✶❛ ✷❛ ✸❛ ☎ ☎ ☎ ❳
♦✉
◗✉❛♥t✐❞❛❞❡ ❞❡ ♣❡ss♦❛s ❝♦♥❤❡❝✐❞❛s ✭❝❛s❛ ❞♦s ♣♦♠❜♦s✮ ✵ ✶ ✷ ☎ ☎ ☎ ✶✾
P❡ss♦❛s ✭♣♦♠❜♦s✮ ❳ ✶❛ ✷❛ ☎ ☎ ☎ ✶✾❛
✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✶✺
P❡r❝❡❜❛ q✉❡ ✉s❛♠♦s ❛r❣✉♠❡♥t♦s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ss❡ ❡①❡♠♣❧♦✱ s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❢ór♠✉❧❛s✳ ■ss♦ ♥♦s ♦❜r✐❣❛ ❛ ✉s❛r ♦ r❛❝✐♦❝í♥✐♦ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❡❧❡✈❛❞❛✱ ♦ q✉❡ é ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♦ ♥♦ss♦ ❝❛♠✐♥❤❛r ♣r♦✜ss✐♦♥❛❧✳ ❉❡ss❛ ❢♦r♠❛ ♣♦❞❡r❡♠♦s ❡♥❝❛r❛r ♦s ♣r♦❜❧❡♠❛s ❝♦♠ ♠❛✐s s❡r✐❡❞❛❞❡ ❡ tr❛♥q✉✐❧✐❞❛❞❡ ♣❛r❛ s♦❧✉❝✐♦♥á✲❧♦s✳
❊①❡♠♣❧♦ ✶✳✷✳✸✳ ◆✉♠ ❝♦❧é❣✐♦ ❝♦♠ ✶✻ s❛❧❛s sã♦ ❞✐str✐❜✉í❞❛s ❝❛♥❡t❛s ♥❛s ❝♦r❡s ♣r❡t❛✱ ❛③✉❧ ❡ ✈❡r♠❡❧❤❛ ♣❛r❛ r❡❛❧✐③❛r ✉♠❛ ♣r♦✈❛ ❞❡ ❝♦♥❝✉rs♦✳ ❙❡ ❝❛❞❛ s❛❧❛ r❡❝❡❜❡ ❝❛♥❡t❛s ❞❛ ♠❡s♠❛ ❝♦r✱ ❡♥tã♦ ♣r♦✈❡ q✉❡ ❡①✐st❡♠ ♣❡❧♦ ♠❡♥♦s s❡✐s s❛❧❛s q✉❡ r❡❝❡❜❡r❛♠ ❝❛♥❡t❛s ❞❡ ♠❡s♠❛ ❝♦r✳
❉❡ss❛ ❢♦r♠❛✱ ❛s ✶✻ s❛❧❛s s❡rã♦ r❡♣r❡s❡♥t❛❞❛s ♣❡❧♦s ♣♦♠❜♦s ❡ ❛s ❝♦r❡s ❞❛s ❝❛♥❡t❛s s❡rã♦ r❡♣r❡s❡♥t❛❞❛s ♣❡❧❛s ❝❛s❛s✳ ❊♥✉♠❡r❡♠♦s ❛s s❛❧❛s ❞❡ ✶ ❛ ✶✻✳ ❙✉♣♦♥❤❛ q✉❡ ❛s s❛❧❛s ❞❡ ✶ ❛ ✺ r❡❝❡❜❡r❛♠ ❝❛♥❡t❛s ❞❡ ❝♦r ♣r❡t❛✱ ❛s s❛❧❛s ❞❡ ✻ ❛ ✶✵ r❡❝❡❜❡r❛♠ ❝❛♥❡t❛s ❞❡ ❝♦r ❛③✉❧✱ ❡ ❛s s❛❧❛s ❞❡ ✶✶ ❛ ✶✺ r❡❝❡❜❡r❛♠ ❝❛♥❡t❛s ❞❡ ❝♦r ✈❡r♠❡❧❤❛✱ r❡st❛♥❞♦ ✉♠❛ s❛❧❛ q✉❡ r❡❝❡❜❡✉ ❝❛♥❡t❛s ❞❡ ✉♠❛ ❞❡ss❛s ❝♦r❡s ❥á ❞✐str✐❜✉í❞❛s✳ ▲♦❣♦✱ t❡r❡♠♦s s❡✐s s❛❧❛s q✉❡ r❡❝❡❜❡r❛♠ ❝❛♥❡t❛s ❞❡ ♠❡s♠❛ ❝♦r✳
✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✶✻
Pr♦❜❧❡♠❛s
✶✳ ◗✉❛❧ é ♦ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ♣❡ss♦❛s q✉❡ ❞❡✈❡♠♦s r❡✉♥✐r ♣❛r❛ q✉❡ t❡♥❤❛♠♦s ❝❡rt❡③❛ ❞❡ q✉❡ ❡♥tr❡ ❡❧❛s ❤á ♣❡❧♦ ♠❡♥♦s ❞✉❛s q✉❡ ❢❛③❡♠ ❛♥✐✈❡rsár✐♦ ♥♦ ♠❡s♠♦ ♠ês❄
✷✳ ▼♦str❡ q✉❡✱ ❡♠ ✉♠ ❣r✉♣♦ ❞❡ ✺✵ ♣❡ss♦❛s✱ ❤á s❡♠♣r❡ ♣❡❧♦ ♠❡♥♦s ❝✐♥❝♦ q✉❡ ♥❛s❝❡r❛♠ ♥♦ ♠❡s♠♦ ♠ês✳
✸✳ ❊♠ ✉♠❛ r❡✉♥✐ã♦ ❤á n ♣❡ss♦❛s✳ ▼♦str❡ q✉❡ ❡①✐st❡♠ ❞✉❛s ♣❡ss♦❛s q✉❡ ❝♦♥❤❡❝❡♠
❡①❛t❛♠❡♥t❡ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ♦✉tr♦s ♣❛rt✐❝✐♣❛♥t❡s ❛❞♠✐t✐♠♦s q✉❡ ❝♦♥❤❡❝❡r s❡❥❛ ✉♠❛ r❡❧❛çã♦ s✐♠étr✐❝❛✱ ♦✉ s❡❥❛✱ s❡ a ❝♦♥❤❡❝❡ b✱ ❡♥tã♦ b ❝♦♥❤❡❝❡ a✳
✹✳ ✭Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s ❞❡ ❉✐r✐❝❤❧❡t✶✮ ❙❡ n 1 ♦✉ ♠❛✐s ♦❜❥❡t♦s sã♦ ❝♦❧♦❝❛❞♦s ❡♠n
♦✉ ♠❡♥♦s ❣❛✈❡t❛s✱ ❡♥tã♦✱ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❣❛✈❡t❛ r❡❝❡❜❡ ♠❛✐s ❞❡ ✉♠ ♦❜❥❡t♦✳
✺✳ ❈✐♥❝♦ ♣♦♥t♦s sã♦ t♦♠❛❞♦s s♦❜r❡ ❛ s✉♣❡r❢í❡ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ 2✳ ▼♦str❡ q✉❡
❤á ❞♦✐s ❞❡ss❡s ♣♦♥t♦s t❛✐s q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡❧❡s é ♠❡♥♦r q✉❡✱ ♦✉ ✐❣✉❛❧ ❛✱ ❄2✳
✻✳ ▼♦str❡ q✉❡ ❡♠ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❞❡ ♦✐t♦ ✐♥t❡✐r♦s ❤á s❡♠♣r❡ ❞♦✐s ❞❡❧❡s ❝✉❥❛ ❞✐❢❡r❡♥ç❛ é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ s❡t❡✳
✼✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r ✶✵ ♥ú♠❡r♦s ♥❛t✉r❛✐s ❞✐❢❡r❡♥t❡s✳ ❙❡ ❝❛❧❝✉✲ ❧❛r♠♦s t♦❞❛s ❛s ❞✐❢❡r❡♥ç❛s ❡♥tr❡ ❡ss❡s ♥ú♠❡r♦s✱ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❡ss❛s ❞✐❢❡r❡♥ç❛s é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ✾❄
✽✳ ❊♠ ✉♠ ❝❛♠♣❡♦♥❛t♦✱ ❝❛❞❛ ❞♦✐s t✐♠❡s ❥♦❣❛♠ ❡♥tr❡ s✐ ✉♠❛ ú♥✐❝❛ ✈❡③✳ ▼♦str❡ q✉❡✱ ❡♠ q✉❛❧q✉❡r ♠♦♠❡♥t♦✱ ❤á s❡♠♣r❡ ❞♦✐s t✐♠❡s q✉❡ ❞✐s♣✉t❛r❛♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ♣❛rt✐❞❛s✳
✾✳ ❉♦ ❝♦♥❥✉♥t♦A✏ t1, 2,☎ ☎ ☎ , 99, 100✉✱ ❡s❝♦❧❤❡♠♦s ❛♦ ❛❝❛s♦ ✺✶ ♥ú♠❡r♦s✳ ❉❡♠♦♥str❛r
q✉❡ ❡♥tr❡ ♦s ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s s❡♠♣r❡ ❡①✐st❡♠ ❞♦✐s q✉❡ sã♦ ❝♦♥s❡❝✉t✐✈♦s✳
✶✵✳ ❊s❝♦❧❤❡♠✲s❡ ✼ ♣♦♥t♦s ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❞✐♠❡♥sõ❡s 2✂3✳ ❉❡♠♦♥str❛r
q✉❡ s❡♠♣r❡ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ❞♦✐s ♣♦♥t♦s t❛❧ q✉❡ s✉❛ ❞✐stâ♥❝✐❛ é ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ ❄2✳
✶✶✳ ❉❛❞❛s s❡✐s ♣❡ss♦❛s ♥✉♠❛ ❢❡st❛✱ ❞❡♠♦♥str❛r q✉❡ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❡①✐st❡♠ três ♣❡ss♦❛s q✉❡ s❡ ❝♦♥❤❡❝❡♠ ♠✉t✉❛♠❡♥t❡ ♦✉ três ♣❡ss♦❛s q✉❡ ♥ã♦ s❡ ❝♦♥❤❡❝❡♠ ♠✉t✉❛♠❡♥t❡✳ ❙✉♣♦♥❤❛ q✉❡ ❛ r❡❧❛çã♦ ❞❡ ❝♦♥❤❡❝❡r é s✐♠étr✐❝❛✳
✶P❡t❡r ●✉st❛✈ ▲❡❥❡✉♥❡ ❉✐r✐❝❤❧❡t ✭✶✽✵✺ ✲ ✶✽✺✾✮✱ ♠❛t❡♠át✐❝♦ ❛❧❡♠ã♦ ❛ q✉❡♠ s❡ ❛tr✐❜✉✐ ❛ ♠♦❞❡r♥❛
✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✶✼
✶✷✳ ✭■▼❖✶✾✽✺✮✳ ❉❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ M ❝♦♠ ✶✾✽✺ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ❞✐st✐♥t♦s✱ ♥❡♥❤✉♠
❞♦s q✉❛✐s t❡♠ ❞✐✈✐s♦r❡s ♣r✐♠♦s ♠❛✐♦r❡s ❞♦ q✉❡ ✷✸✱ ♠♦str❡ q✉❡ ❤á ✹ ❡❧❡♠❡♥t♦s ❡♠
M ❝✉❥♦ ♣r♦❞✉t♦ é ✉♠❛ q✉❛rt❛ ♣♦tê♥❝✐❛✳
✶✸✳ ❊♠ ✉♠❛ ♣❡sq✉✐s❛ ❞❡ r✉❛✱ ❝❛❞❛ ❡♥tr❡✈✐st❛❞♦ r❡s♣♦♥❞❡✉ ❛ q✉❛tr♦ ♣❡r❣✉♥t❛s✱ ♣♦❞❡♥❞♦ s✉❛ r❡s♣♦st❛ s❡r s✐♠ ♦✉ ♥ã♦✱ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ♣❡r❣✉♥t❛s✳ ◗✉❛❧ ♦ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ❡♥tr❡✈✐st❛❞♦s ♣❛r❛ ❣❛r❛♥t✐r♠♦s q✉❡ ❞✉❛s ♣❡ss♦❛s r❡s♣♦♥❞❛♠ ✐❣✉❛❧♠❡♥t❡ ❛ t♦❞❛s ❛s ♣❡r❣✉♥t❛s❄
✶✹✳ ❙❡ n ❡m sã♦ ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ A✏ tm 1, m 2, ☎ ☎ ☎ , m n✉
♣♦ss✉✐ ❛❧❣✉♠ ❞✐✈✐s♦r ❞❡ n✳
✶✺✳ ❉❡♠♦♥str❛r q✉❡ t♦❞♦ ✐♥t❡✐r♦ t❡♠ ✉♠ ♠ú❧t✐♣❧♦ ❝✉❥❛ r❡♣r❡s❡♥t❛çã♦ ❞❡❝✐♠❛❧ ❝♦♠❡ç❛ ❝♦♠ ♦ ❜❧♦❝♦ ❞❡ ❞í❣✐t♦s 1234567890✳
✶✻✳ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n✱ ♠♦str❡ q✉❡ ❡①✐st❡ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ n q✉❡ s❡
❡s❝r❡✈❡ ❝♦♠ ♦s ❛❧❣❛r✐s♠♦s ✵ ❡ ✶ ❛♣❡♥❛s✳ ✭P♦r ❡①❡♠♣❧♦✱ s❡ n ✏ 3✱ t❡♠♦s ✶✶✶ ♦✉
✶✳✶✵✶✱ ❡t❝✳✮
✶✼✳ Pr♦✈❡ q✉❡ ❡♥tr❡n 1❡❧❡♠❡♥t♦s ❡s❝♦❧❤✐❞♦s ♥♦ ❝♦♥❥✉♥t♦ t1, 2, 3, ☎ ☎ ☎, 2n✉ ❡①✐st❡♠
❈❛♣ít✉❧♦
2
❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛
➪❧❣❡❜r❛ ❞♦s ◆ú♠❡r♦s ■♥t❡✐r♦s
◆❡st❡ ❝❛♣ít✉❧♦✱ ❡st✉❞❛r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ♥♦s ❝♦♥❥✉♥t♦s ❞♦s ✐♥t❡✐r♦s✳ ❙❡♠ ❞✐♠✐♥✉✐r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦✱ s✉❜tr❛çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦✱ t❡♠♦s ♥❛ ❞✐✈✐sã♦ ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s ✉♠❛ ❢♦♥t❡ ✐♥❡s❣♦tá✈❡❧ ❞❡ ♣r♦❜❧❡♠❛s ✐♥t❡r❡ss❛♥t❡s ❞❡♥tr♦ ❞❛ ▼❛t❡♠át✐❝❛✳ ❆♣❡s❛r ❞❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞♦ ❝♦♥❝❡✐t♦✱ ❛ ❝♦♥str✉çã♦ ❞♦ ❢♦r♠❛❧✐s♠♦ ♠❛t❡♠át✐❝♦ ❞❡st❡ ❝♦♥❝❡✐t♦ é ❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ q✉❡ ♦ ♣r♦❢❡ss♦r ♣♦ss❛ ❞❡s❡♥✈♦❧✈❡r ❛ t❡♦r✐❛ ❞❡ ❢♦r♠❛ ♥❛t✉r❛❧ ❡ ❝♦♥s✐st❡♥t❡✳
✷✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡
❉❡✜♥✐çã♦ ✷✳✶✳✶✳ ❉❛❞♦s a, bPZ✱ ❝♦♠ b✘0✱ ❞✐③❡♠♦s q✉❡ b ❞✐✈✐❞❡ a✱ ❡ ❡s❝r❡✈❡♠♦s b⑤a✱
s❡ ❡①✐st✐r cPZ t❛❧ q✉❡ a✏bc✳ ❈❛s♦ b ♥ã♦ ❞✐✈✐❞❛ a✱ ❡s❝r❡✈❡♠♦s b∤a✳
❉❛❞♦ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ b ✘0✱ s❡ b ❞✐✈✐❞✐r a✱ ❞✐③❡♠♦s q✉❡ b é ✉♠ ❞✐✈✐s♦r ❞❡ a✱ q✉❡ a é ❞✐✈✐sí✈❡❧ ♣♦r b ♦✉ ❛✐♥❞❛ q✉❡ a é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ b✳ ❙❡ b ⑤ a ❡ b → 0✱ ❡♥tã♦ b é ✉♠
❞✐✈✐s♦r ♣♦s✐t✐✈♦ ❞❡ a✳
❆ ♣r♦♣♦s✐çã♦ q✉❡ s❡❣✉❡ ❡st❛❜❡❧❡❝❡ ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞❛ r❡❧❛çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✳ ❙❛❧✐❡♥t❛♠♦s q✉❡ ❡st❛s ♣r♦♣r✐❡❞❛❞❡s sã♦ ♠✉✐t♦ r❡❧❡✈❛♥t❡s ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❡✱ ♣♦rt❛♥t♦✱ ❞❡✈❡♠♦s ❞❛r ❛ ❡❧❛s ❛ ❛t❡♥çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ✉♠ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♠❡s♠❛✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✶✳ ❙❡❥❛♠ a, b ❡ c ✐♥t❡✐r♦s ♥ã♦ ♥✉❧♦s ❡ x, y ✐♥t❡✐r♦s q✉❛✐sq✉❡r✳
✭❛✮ ❙❡b ⑤a ❡ a⑤b✱ ❡♥tã♦ a✏ ✟b✳ ✭❞✮ ❙❡b ⑤a✱ ❡♥tã♦ ⑤b⑤ ↕ ⑤a⑤✳
✭❜✮ ❙❡c⑤b ❡ b⑤a✱ ❡♥tã♦ c⑤a✳ ✭❡✮ ❙❡ c⑤b✱ ❡♥tã♦ c⑤ab✳
✷✳✷ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛ ✶✾
❉❡♠♦♥str❛çã♦✿
✭❛✮ ❙❡m ❡n sã♦ ✐♥t❡✐r♦s t❛✐s q✉❡a✏bm ❡b✏an✱ ❡♥tã♦a ✏ ♣anq ☎m✏a☎ ♣mnq ❡✱ ❞❛í✱ m☎n✏1ñ
★
m ✏1 ❡ n✏1
m ✏ ✁1 ❡ n✏ ✁1 , m, nPZ.
❙❡❣✉❡ q✉❡ a✏ ✟b✳
✭❜✮ ❙❡ b ✏ cn ❡ a ✏ bm✱ ❝♦♠ m, n P Z✱ ❡♥tã♦ a ✏ ♣cnq ☎m ✏ c☎ ♣mnq✱ ❝♦♠ mn P Z✳
▲♦❣♦✱ c⑤a✳
✭❝✮ ❙❡a ✏cm❡ b✏cn✱ ❝♦♠m, nP Z✱ ❡♥tã♦ax by✏cmx cny✏ ♣mx nyq ☎c✱ ❝♦♠ x ❡ yPZ✳ ▲♦❣♦✱ c⑤ ♣ax byq✳
✭❞✮ ❙❡ a✏bm✱ ❝♦♠ mPZ✳ ❚♦♠❛♥❞♦ ♦s ♠ó❞✉❧♦s✱ t❡♠♦s q✉❡ ⑤a⑤ ✏ ⑤b⑤⑤m⑤✳ ❈♦♠♦ a ✘0✱
t❡♠♦s q✉❡ m✘0✱ ❧♦❣♦1↕ ⑤m⑤ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ⑤b⑤ ↕ ⑤b⑤⑤m⑤ ✏ ⑤a⑤✳
✭❡✮ ❙❡ b✏cn✱ ❝♦♠ nPZ✱ ❡♥tã♦ ab✏acn✱ ❧♦❣♦ ab✏ ♣anq ☎c❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ c⑤ab✳
✭❢✮ ❙❡ a✏bm✱ ❝♦♠ m PZ✱ ❡♥tã♦ ac✏cbm ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ bc⑤ac✳
✷✳✷ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛
▼❡s♠♦ q✉❛♥❞♦ ✉♠ ♥ú♠❡r♦ b ✐♥t❡✐r♦ ♥ã♦ ♥✉❧♦ ♥ã♦ ❞✐✈✐❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ a✱ é
♣♦ssí✈❡❧ ❡❢❡t✉❛r ❛ ❞✐✈✐sã♦ ❞❡ a ♣♦rb ❝♦♠ r❡st♦✳
❚❡♦r❡♠❛ ✷✳✷✳✶✳ ❙❡❥❛♠ a ❡ b ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♠ b ✘ 0✳ ❊①✐st❡♠ ❞♦✐s ú♥✐❝♦s
♥ú♠❡r♦s ✐♥t❡✐r♦s q ❡ r t❛✐s q✉❡
a✏b☎q r, ❝♦♠ 0↕r➔ ⑤b⑤
❉❡♠♦♥str❛çã♦✿ ❊①✐stê♥❝✐❛✿
❙✉♣♦♥❤❛b →0✱ ❡ s❡❥❛ q ♦ ♠❛✐♦r ✐♥t❡✐r♦ t❛❧ q✉❡q ↕ a
b✳ ❊♥tã♦✱ ♣❡❧❛ ♦❜s❡r✈❛çã♦ ❛❝✐♠❛✱ q ↕ a
b ➔q 1✱ ♦✉ s❡❥❛✱b☎q ↕a ➔b☎ ♣q 1q✳ ❙♦♠❛♥❞♦ ✁b☎q ❛ ❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ ♦❜t❡♠♦s
q✉❡ 0 ↕ a✁b☎q ➔ b✳ P♦rt❛♥t♦✱ ❞❡✜♥✐♥❞♦ r ✏ a✁b☎q✱ s❡❣✉❡ q✉❡ a ✏ b☎q r✱ ❝♦♠
