❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❘■❖ ●❘❆◆❉❊ ❉❖ ◆❖❘❚❊ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❆ ❚❊❘❘❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲ ✕ P❘❖❋▼❆❚
❈♦♥❥✉♥t♦s ◆✉♠ér✐❝♦s
❈❆❘▲❖❙ ❊❉❯❆❘❉❖ ❉❊ ▲■▼❆ ❉❯❆❘❚❊
❈❆❘▲❖❙ ❊❉❯❆❘❉❖ ❉❊ ▲■▼❆ ❉❯❆❘❚❊
❈♦♥❥✉♥t♦s ◆✉♠ér✐❝♦s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❇❛♥❝❛ ❊①❛♠✐✲ ♥❛❞♦r❛ ❝♦♠♦ ❡①✐❣ê♥❝✐❛ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❊❙❚❘❊ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✕ P❘❖❋▼❆❚ ♣❡❧❛ ❯♥✐✈❡rs✐✲ ❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡✱ s♦❜ ❛ ♦r✐❡♥t❛çã♦ ❞❛ Pr♦❢❛ ❉r❛ ❱✐✈✐❛♥❡ ❙✐♠✐♦❧✐ ▼❡❞❡✐r♦s ❈❛♠♣♦s✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦✱ ❡❧❛❜♦r❛♠♦s ✉♠ t❡①t♦ s♦❜r❡ ♦s ❈♦♥❥✉♥t♦s ◆✉♠ér✐❝♦s✱ ✉t✐❧✐③❛♥❞♦ ❛s ♥❡❝❡s✲ s✐❞❛❞❡s s♦❝✐❛✐s ❤✉♠❛♥❛s ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❝♦♥str✉çã♦ ❞❡ ♥♦✈♦s ♥ú♠❡r♦s✳ ❖ ♣r❡s❡♥t❡ ♠❛t❡r✐❛❧ ✈✐s❛ ❛♣r❡s❡♥t❛r ✉♠ t❡①t♦ q✉❡ ❝♦♥❝✐❧✐❡ ♦ ❡♥s✐♥♦ ❝♦rr❡t♦ ❞❛ ♠❛t❡♠át✐❝❛ ❡ ❛ ❝❧❛r❡③❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ✉♠ ❜♦♠ ❛♣r❡♥❞✐③❛❞♦✳
❆❜str❛❝t
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ❛ ❉❡✉s✱ ♦ ❛✉t♦r ❞❛ ✈✐❞❛✱ q✉❡ ♠❡ ❛♠♦✉ ✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡ ❡ ♠❡ ❞❡✉ s❛ú❞❡✱ ❢♦rç❛ ❡ ❝♦r❛❣❡♠ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❆ ❊❧❡ s❡❥❛ ❛ ❣❧ór✐❛ ❡t❡r♥❛♠❡♥t❡✦
➚ ♠✐♥❤❛ ❡s♣♦s❛✱ ✜❡❧ ❡ ❡t❡r♥❛ ❝♦♠♣❛♥❤❡✐r❛✱ ♣♦r s❡♠♣r❡ ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ✐♠♣♦rt❛♥t❡s ❞❛ ♠✐♥❤❛ ✈✐❞❛✳ ❆ ♣❡ss♦❛ q✉❡ ❉❡✉s ❝♦❧♦❝♦✉ ♥♦ ♠✉♥❞♦ ♣r❛ ♠❡ ❢❛③❡r ❢❡❧✐③✳ ❚❡ ❛♠♦ ♠✉✐t♦✦
❆♦s ♠❡✉s ♣❛✐s✱ ♣♦r ♠❡ ❡♥s✐♥❛r❡♠ ♥♦ ❞✐❛✲❛✲❞✐❛✱ ❛tr❛✈és ❞❡ s✉❛s ✈✐❞❛s✱ q✉❡ ❡❞✉❝❛çã♦ s❡ ❢❛③ ❝♦♠ ♦ ❡①❡♠♣❧♦✳ P❡❧♦ ❡s❢♦rç♦ ✐♠❡♥s✉rá✈❡❧ ♣❛r❛ ♠❡ ❞❛r ✉♠❛ ❡❞✉❝❛çã♦ ❞❡ q✉❛❧✐❞❛❞❡ ❡ ♠❡ ♣♦ss✐❜✐❧✐t❛r t❡r ❝❤❡❣❛❞♦ ❛té ❛q✉✐✳
❆♦s ♠❡✉s ✐r♠ã♦s✱ ❏ú♥✐♦r ❡ ❘❛♣❤❛❡❧✱ ♣❡❧♦s ✐♠♣♦rt❛♥t❡s ♠♦♠❡♥t♦s ❡♠ ❢❛♠í❧✐❛ q✉❡ ♠❡ ❢❡③ s❡r ❛ ♣❡ss♦❛ q✉❡ s♦✉ ❤♦❥❡✳
➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ Pr♦❢❡ss♦r❛ ❱✐✈✐❛♥❡ ❙✐♠✐♦❧✐ ▼❡❞❡✐r♦s ❈❛♠♣♦s✱ ♣❡❧❛s ✈❛❧✐♦s❛s ❝♦♥tr✐❜✉✐çõ❡s ❛ ❡ss❡ tr❛❜❛❧❤♦✱ ❜❡♠ ❝♦♠♦ ♣❡❧❛ s✉❛ ✐♥❝❛♥sá✈❡❧ ❞❡❞✐❝❛çã♦ ❛ ❡ss❡ ♠❡str❛❞♦✳
❆ t♦❞♦s ♦s ❞♦❝❡♥t❡s ❞♦ P❘❖❋▼❆❚ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❡ ♣❡❧❛ ❧✉t❛ ♣❛r❛ ❛ ♠❡❧❤♦r✐❛ ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦ ♥❡st❡ ♣❛ís✳
❆♦s ❝♦❧❡❣❛s ❡ ❛♠✐❣♦s ❞❡ P❘❖❋▼❆❚✱ q✉❡ ♠✉✐t♦ ♠❡ ❛❥✉❞❛r❛♠ ❛♦ ❧♦♥❣♦ ❞❡ss❡s ❞♦✐s ❛♥♦s ❞❡ ❝✉rs♦ ❡✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❛♦ ♥♦ss♦ ❝♦❧❡❣❛ ❏♦❡❧ ♣❡❧❛s ♦r❛çõ❡s ❛♥t❡s ❞❛s ♣r♦✈❛s✳
➚ ❈❛♣❡s✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ ◆ú♠❡r♦s ◆❛t✉r❛✐s ✹
✶✳✶ ❆ ❈♦♥t❛❣❡♠ ❡ ♦s ◆ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❆ ■❞❡✐❛ ❞❡ ❈♦rr❡s♣♦♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❚✐♣♦s ❞❡ ❈♦rr❡s♣♦♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✹ ❊q✉✐✈❛❧ê♥❝✐❛ ❡ Pr❡✈❛❧ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✺ ◆ú♠❡r♦✿ ❯♠ ❈♦♥❝❡✐t♦ ❆❜str❛t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✻ ❖ ■♥✜♥✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✼ ❖ ❈♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✽ ❖ ❩❡r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✾ ❖ ❈♦♥❥✉♥t♦ ❞♦s P♦♥t♦s ❞❛ ❘❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✵ ❈♦rr❡s♣♦♥❞ê♥❝✐❛s ♥♦ ■♥✜♥✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✷ ◆ú♠❡r♦s ■♥t❡✐r♦s ✶✷
✷✳✶ ●r❛♥❞❡③❛s ❚♦♠❛❞❛s ❡♠ ❙❡♥t✐❞♦s ❉✐❢❡r❡♥t❡s ❡ ♦ ❈♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ■♥t❡✐r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✷ ❆ ❈♦rr❡s♣♦♥❞ê♥❝✐❛ N↔Z ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✸ ◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s ✶✹
✸✳✶ ▼❡❞✐çã♦ ❞❡ ❙❡❣♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✷ ❙✉❜❞✐✈✐sã♦ ❞❛ ❯♥✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✸ ❖ ❈♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✹ ❖ Pr♦❜❧❡♠❛ ❞❛ ▼❡❞✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✺ ■♥❝♦♠❡♥s✉r❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✻ ❖ ❈♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s ❡ ♦s P♦♥t♦s ❞❛ ❘❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✻✳✶ ■♥✜♥✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✻✳✷ ❖r❞❡♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✻✳✸ ❉❡♥s✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✻✳✹ ❈♦♠♣❧❡t✉❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✹ ◆ú♠❡r♦s ❘❡❛✐s ✷✺ ✹✳✶ ❉❡✜♥✐çã♦ ❡ ❘❡❧❛çã♦ ❡♥tr❡ ♦s ❈♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✷ ❆❧❣✉♥s ◆ú♠❡r♦s ■rr❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✸ ❈♦rr❡s♣♦♥❞ê♥❝✐❛R↔P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✹ ❖s ❈♦♥❥✉♥t♦s N✱Z✱ Q ❡R ❡ ♦s ❚✐♣♦s ❞❡ ■♥✜♥✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✺ ❊①❡♠♣❧♦s q✉❡ ✐❧✉str❛♠ ❛ ♣r❡✈❛❧ê♥❝✐❛ ❞❡R−Q s♦❜r❡ Q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✸✵
■♥tr♦❞✉çã♦
❆♣❡s❛r ❞❡ ♥♦s ú❧t✐♠♦s ❛♥♦s t❡r ❤❛✈✐❞♦ ✉♠❛ ♠❡❧❤♦r✐❛ ♥♦ ❝❡♥ár✐♦ ❡❞✉❝❛❝✐♦♥❛❧ ❜r❛s✐❧❡✐r♦✱ ♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞❡ ♠❛t❡♠át✐❝❛ ♥ã♦ t❡♠ ❛♣r❡s❡♥t❛❞♦ ✉♠❛ ♠❡❧❤♦r❛ s✐❣♥✐✜❝❛t✐✈❛ ❛ ♣♦♥t♦ ❞❡ s❡ t❡r ❧✐✈r♦s t❡①t♦s s❛t✐s❢❛tór✐♦s ♣❛r❛ ♦ ❡♥s✐♥♦ ❜ás✐❝♦ ❞❡ ♠❛t❡♠át✐❝❛✳
➱ ❝♦♠✉♠ ✈❡r♠♦s ❧✐✈r♦s ❞❡ ♠❛t❡♠át✐❝❛ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❝♦♠ ❡rr♦s ❝♦♥❝❡✐t✉❛✐s✳ ❊ ✐ss♦ é ❣r❛✈❡✱ ♣♦✐s ❞❡✈✐❞♦ às ❢❛❝✉❧❞❛❞❡s ❞❡ ♠❛t❡♠át✐❝❛ ♥ã♦ s❡ ✐♠♣♦rt❛r❡♠ ♠✉✐t♦ ❝♦♠ ❛ ❢♦r♠❛çã♦ ❞♦ ♣r♦❢❡ss♦r ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛♦s ❝♦♥t❡ú❞♦s ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦✱ ♦s ♣r♦❢❡ss♦r❡s ❛❝❛❜❛♠ ✏❛♣r❡♥❞❡♥❞♦✑ ♠❛t❡♠át✐❝❛ ♣❡❧♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✳
❖ ❝♦♥t❡ú❞♦ ✏❈♦♥❥✉♥t♦s ◆✉♠ér✐❝♦s✑ ♥♦r♠❛❧♠❡♥t❡ é ❛♣r❡s❡♥t❛❞♦ ❞❡ ♠❛♥❡✐r❛ ❢r❛❣♠❡♥✲ t❛❞❛ ❡ s❡♠ ❥✉st✐✜❝❛t✐✈❛s✳ Pr❛t✐❝❛♠❡♥t❡ t♦❞♦s ♦s ❧✐✈r♦s ✉t✐❧✐③❛❞♦s ♥❛s ❡s❝♦❧❛s ❜r❛s✐❧❡✐r❛s✱ ❛♦ tr❛t❛r ❞♦s ♥ú♠❡r♦s✱ ❢❛③❡♠ s✐♠♣❧❡s♠❡♥t❡ ❝❧❛ss✐✜❝á✲❧♦s ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ t✐♣♦✿ s❡ ♦ ♥ú♠❡r♦ t❡♠ t❛❧ ❝❛r❛❝t❡ríst✐❝❛ ❝❤❛♠❛♠♦s ❞❡ ♥❛t✉r❛❧✱ s❡ t❡♠ ♦✉tr❛ ❝❛r❛❝t❡ríst✐❝❛ é r❛❝✐♦♥❛❧ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳
P❛r❛ ✐❧✉str❛r ❡ss❛ ❛✜r♠❛çã♦✱ ✈❡❥❛♠♦s ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✿ P❛✐✈❛ ❬✶✷❪ ❞✐③✿ ✏❈❧❛ss✐✜✲ ❝❛♠♦s ❝♦♠♦ ♥❛t✉r❛✐s ♦s ♥ú♠❡r♦s q✉❡ r❡♣r❡s❡♥t❛♠ q✉❛♥t✐❞❛❞❡s ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ❝♦♥❥✉♥t♦s ✜♥✐t♦s✱ ✐♥❝❧✉s✐✈❡ ♦ ✈❛③✐♦✑ ✭♣✳ ✷✸✮✳ ❊ ❡♠ s❡❣✉✐❞❛✿ ✏❯♠❛ ♣❛rt❡ ❞♦s ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s é ❢♦r♠❛❞❛ ♣❡❧♦s ♥ú♠❡r♦s −1,−2,−3,−4, . . .✱ q✉❡ sã♦ ❝❤❛♠❛❞♦s ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥❡✲ ❣❛t✐✈♦s✑ ✭♣✳ ✷✸✮✳ ❆♣r❡s❡♥t❛ ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❞❡✜♥✐♥❞♦ ❝♦♠♦ ✏t♦❞♦ ❛q✉❡❧❡ q✉❡ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ r❛③ã♦ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ s❡♥❞♦ ♦ s❡❣✉♥❞♦ ♥ã♦ ♥✉❧♦✑ s❡♠ ❢❛③❡r q✉❛❧q✉❡r ❝♦♥t❡①t✉❛❧✐③❛çã♦ ❝♦♠ ♠❡❞✐❞❛s✳ ❙❡❣✉✐♥❞♦ ♥❡ss❛ ♠❡s♠❛ ❧✐♥❤❛✱ ❞❡✜♥❡ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ ❝♦♠♦ ✏t♦❞♦ ♥ú♠❡r♦ q✉❡✱ ❡♠ s✉❛ ❢♦r♠❛ ❞❡❝✐♠❛❧✱ é ✉♠❛ ❞í③✐♠❛ ♥ã♦ ♣❡r✐ó❞✐❝❛✑✳ ❊ ❛♣r❡s❡♥t❛ ♦s ♥ú♠❡r♦s r❡❛✐s ❝♦♠♦ ✏q✉❛❧q✉❡r ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ♦✉ ✐rr❛❝✐♦♥❛❧✑✳
❏á ■❡③③✐ ❡t ❛❧ ❬✾❪ ❝♦♠❡ç❛ ♦ ❛ss✉♥t♦ ❞✐③❡♥❞♦✿ ✏❉❡♥♦♠✐♥❛♠♦s ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s ♦s ❝♦♥❥✉♥t♦s ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ ♥ú♠❡r♦s q✉❡ ❛♣r❡s❡♥t❛♠ ❛❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❝♦♠✉♥s ❡♥tr❡ s✐✑ ✭♣✳ ✷✾✮✳ ❊ ❡♠ s❡❣✉✐❞❛ ❝♦♠❡ç❛ ❛ ❢❛❧❛r s♦❜r❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s ❛ss✐♠✿ ✏❖ ❝♦♥❥✉♥t♦ N é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✿ N = {0,1,2,3,4, ..., n, ...}✱ ❡♠ q✉❡ n r❡♣r❡s❡♥t❛ ♦ ❡❧❡♠❡♥t♦ ❣❡♥ér✐❝♦ ❞♦ ❝♦♥❥✉♥t♦✑✳ ❆♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ Q ❝♦♠♦ ✉♠❛ ❛♠♣❧✐❛çã♦ ❞❡ Z ❞❡✈✐❞♦ à ✐♥❡①✐stê♥❝✐❛ ❞❡ ❛❧❣✉♠❛s ❞✐✈✐sõ❡s ❡♥tr❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❉❡✜♥❡ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ ❝♦♠♦ ✏♥ú♠❡r♦s ❞❡❝✐♠❛✐s ♥ã♦ ❡①❛t♦s✱ q✉❡ ♣♦ss✉❡♠ r❡♣r❡s❡♥t❛çã♦ ✐♥✜♥✐t❛ ♥ã♦ ♣❡r✐ó❞✐❝❛✑✳ ❏á ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s é ❞❡✜♥✐❞♦ ❝♦♠♦ ✏♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧❛ r❡✉♥✐ã♦ ❞♦
❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❡ ♣❡❧❛ r❡✉♥✐ã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✑✳ P❛r❡❝❡ q✉❡ ♦s ♥ú♠❡r♦s sã♦ ❡♥t✐❞❛❞❡s q✉❡ ❡stã♦ ♣♦r ❛í ❡ t❡♠♦s q✉❡ ❝❧❛ss✐✜❝á✲❧♦s ❝♦♠ ❛❧❣✉♠ ♥♦♠❡✳ ◆❛ ✈❡r❞❛❞❡✱ ❤á ✉♠❛ ❝♦♥str✉çã♦ ❧ó❣✐❝❛ ❡ ❤✐stór✐❝❛ ❞❡ss❡s ♥ú♠❡r♦s✳ ❖s ♥ú♠❡r♦s ♥ã♦ ❡①✐st❡♠ ♣♦r s✐ só✳ ❊❧❡s ❡①✐st❡♠ ♣❛r❛ s✉♣r✐r ❛❧❣✉♠❛s ♥❡❝❡ss✐❞❛❞❡s ❤✉♠❛♥❛s✳ ❊ é ✐ss♦ q✉❡ q✉❡r❡♠♦s ♠♦str❛r ❝♦♠ ❡ss❡ tr❛❜❛❧❤♦✳
➱ ❝♦♠✉♠ ✈❡r♠♦s ❡♠ s✐t❡s ❞❡ ♠❛t❡♠át✐❝❛ ♥❛ ✐♥t❡r♥❡t ♦✉ ❛té ❡♠ ❧✐✈r♦s ❞✐❞át✐❝♦s ❞✐❛✲ ❣r❛♠❛s ❝♦♠♦ ❡st❡ ✭❋✐❣✉r❛ ✶✮ ♣❛r❛ r❡♣r❡s❡♥t❛r ♦s ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s✿
❋✐❣✉r❛ ✶✿ ❈♦♥❥✉♥t♦s ◆✉♠ér✐❝♦s ✕ ❋♦♥t❡✿ ❤tt♣✿✴✴✇✇✇✳❛❧✉♥♦s♦♥❧✐♥❡✳❝♦♠✳❜r
❉✐❛❣r❛♠❛s ❞❡ss❡ t✐♣♦ só ✐♥❞✉③❡♠ ♦ ❛❧✉♥♦ ❛ ♣❡♥s❛r q✉❡ ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s sã♦ ❛ ♠❛✐♦r✐❛ ❞❡♥tr❡ ♦s ♥ú♠❡r♦s r❡❛✐s✳ ❖✉tr❛ ❝♦♥❝❧✉sã♦ ♣r❡❝✐♣✐t❛❞❛ q✉❡ ♦ ❛❧✉♥♦ ♣♦❞❡ t✐r❛r é ❛❝❤❛r q✉❡ ❡①✐st❡♠ ♥ú♠❡r♦s r❡❛✐s q✉❡ ♥ã♦ sã♦ r❛❝✐♦♥❛✐s ♥❡♠ ✐rr❛❝✐♦♥❛✐s✳
❆❧é♠ ❞✐ss♦✱ ♠✉✐t❛s q✉❡stõ❡s ✐♠♣♦rt❛♥t❡s ♥ã♦ sã♦ ❞✐s❝✉t✐❞❛s ♥❡♠ s❡q✉❡r ♠❡♥❝✐♦♥❛❞❛s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿
• ❙❡ ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s s❡r✈❡♠ ♣❛r❛ ❢❛③❡r q✉❛❧q✉❡r ♠❡❞✐çã♦ ❝♦♥❝r❡t❛✱ ♣♦r q✉❡ ❛♠✲
♣❧✐❛r♠♦s ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s❄❀
• ❊①✐st❡♠ ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ ✐♥✜♥✐t♦s❄❀
• ◗✉❛❧ ❝♦♥❥✉♥t♦ t❡♠ ✏♠❛✐s ❡❧❡♠❡♥t♦s✑✿ ♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ♦✉ ♦ ❞♦s ♥ú♠❡r♦s
✐rr❛❝✐♦♥❛✐s❄
❉❡✈✐❞♦ ❛ ❡ss❛ ❛✉sê♥❝✐❛ ❞❡ ❡①♣❧✐❝❛çõ❡s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞❡ ❡♥s✐♥♦ ♠é❞✐♦ ❡ ❛♦s ❝♦♥s✲ t❛♥t❡s ❡rr♦s ♣r❡s❡♥t❡s ♥♦s ♠❡s♠♦s✱ ♥♦s ♣r♦♣♦♠♦s ❛ ❡❧❛❜♦r❛r ✉♠ ♠❛t❡r✐❛❧ s♦❜r❡ ❈♦♥❥✉♥t♦s ◆✉♠ér✐❝♦s q✉❡ s❡❥❛ ❛❝❡ssí✈❡❧ ❛ ❡ss❡ ♣ú❜❧✐❝♦ ❡ q✉❡ ❞ê r❡s♣♦st❛s ❝♦♥s✐st❡♥t❡s ❛♦s q✉❡st✐♦♥❛✲ ♠❡♥t♦s ❧❡✈❛♥t❛❞♦s✳ ❯♠ ♠❛t❡r✐❛❧ q✉❡ s❡❥❛ ❝❛♣❛③ ❞❡ ❝♦♥❝✐❧✐❛r ♦ ❡♥s✐♥♦ ❝♦rr❡t♦ ❞♦ ❝♦♥t❡ú❞♦ ❝♦♠ ❛ ❝❧❛r❡③❛ ♥❡❝❡ssár✐❛✳
P♦ré♠✱ ❛♦ ♥♦s ❞❡♣❛r❛r♠♦s ❝♦♠ ♦ ❧✐✈r♦ ❞❡ ❇❡♥t♦ ❞❡ ❏❡s✉s ❈❛r❛ç❛ ❬✸❪✱ ❧✐✈r♦ q✉❡ ❝♦♥✲ t❡♠♣❧❛ t♦❞♦s ❡ss❡s q✉❡st✐♦♥❛♠❡♥t♦s ❡ q✉❡ ❝♦♥stró✐ ♦s ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s ❛ ♣❛rt✐r ❞❛s
✸
♥❡❝❡ss✐❞❛❞❡s s♦❝✐❛✐s✱ ❞❡❝✐❞✐♠♦s r❡❡s❝r❡✈❡r ♦s ❝❛♣ít✉❧♦s ❞❡st❡ ❧✐✈r♦ q✉❡ tr❛t❛♠ s♦❜r❡ ♦s ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s ❞❡ ♠♦❞♦ ❛ ❛❧❡rt❛r ❛♦ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛ s♦❜r❡ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ❞✐s❝✉t✐r ❞❡t❡r♠✐♥❛❞♦s t❡♠❛s q✉❡ sã♦ ✐♥❡r❡♥t❡s ❛♦ ❛ss✉♥t♦ q✉❡ ❡st❛♠♦s tr❛t❛♥❞♦✳
◆❡ss❡ s❡♥t✐❞♦✱ ❡st❡ t❡①t♦ ♥ã♦ é ✉♠ ♠❛t❡r✐❛❧ ❞❡ ❡st✉❞♦ ♣❛r❛ ✉♠ ❛❧✉♥♦ ❞❡ ❡♥s✐♥♦ ♠é❞✐♦✱ ♠❛s ✉♠ s✉♣♦rt❡ ♣❛r❛ ♦ ♣r♦❢❡s♦r ❞❡ ♠❛t❡♠át✐❝❛ ❞❡❝✐❞✐r ❝♦♠♦ ❡❧❡ ✐rá ❛❜♦r❞❛r ♦ ❝♦♥t❡ú❞♦ ♥❡ss❡ ♥í✈❡❧ ❞❡ ❡♥s✐♥♦✳
❆♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❛❞♠✐t✐r❡♠♦s q✉❡ ♦s ❛❧✉♥♦s ❥á tê♠ ❢❛♠✐❧✐❛r✐❞❛❞❡ ❝♦♠ ❛s ♦♣❡✲ r❛çõ❡s ❡❧❡♠❡♥t❛r❡s✿ ❛❞✐çã♦✱ s✉❜tr❛çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❞✐✈✐sã♦✱ ♣♦t❡♥❝✐❛çã♦ ❡ r❛❞✐❝✐❛çã♦✱ ✈✐st♦ q✉❡✱ ❛♦ ❧♦♥❣♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ❥á tr❛❜❛❧❤❛r❛♠ ❝♦♠ t❛✐s ♦♣❡r❛çõ❡s✳
❈❛♣ít✉❧♦ ✶
◆ú♠❡r♦s ◆❛t✉r❛✐s
✶✳✶ ❆ ❈♦♥t❛❣❡♠ ❡ ♦s ◆ú♠❡r♦s ◆❛t✉r❛✐s
❆ t♦❞♦ ♠♦♠❡♥t♦✱ ♣r❡❝✐s❛♠♦s ❢❛③❡r ❝♦♥t❛❣❡♥s✿ ♦ tr❛❜❛❧❤❛❞♦r✱ ♣❛r❛ ✈❡r✐✜❝❛r s❡ ♦ s❛❧ár✐♦ r❡❝❡❜✐❞♦ ❡stá ❝♦rr❡t♦❀ ♦ ♣❛st♦r ❞❡ ♦✈❡❧❤❛s✱ ♣❛r❛ s❛❜❡r s❡ ♣❡r❞❡✉ ❛❧❣✉♠❛ ❞❡❧❛s ❛♦ ❧❡✈á✲❧❛s ♣❛r❛ ♣❛st❛r❀ ♦ ❝✐❡♥t✐st❛✱ ♣❛r❛ s❛❜❡r s❡ t❡♠ ❛ q✉❛♥t✐❞❛❞❡ ❝❡rt❛ ❞❛s s✉❜stâ♥❝✐❛s ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡ ✉♠ ❡①♣❡r✐♠❡♥t♦✳
❙❡r✐❛ ✐♠♣♦ssí✈❡❧ ✐♠❛❣✐♥❛r ✉♠❛ tr❛♥s❛çã♦ ❝♦♠❡r❝✐❛❧ s❡♠ q✉❡ ✉♠ s❛✐❜❛ ❝♦♥t❛r ♦s ✐t❡♥s q✉❡ ❝♦♠♣r❛ ❡ ♦ ♦✉tr♦ ♦ ❞✐♥❤❡✐r♦ q✉❡ r❡❝❡❜❡✳ ❖✉ ❛✐♥❞❛✱ ✉♠ ♠❡r❝❛❞♦ s❡♠ q✉❡ ♥✐♥❣✉é♠ s♦✉❜❡ss❡ ❝♦♥t❛r✳ ❊♥✜♠✱ ❛ t♦❞❛s ❛s ♣❡ss♦❛s✱ ❡♠ ♠❛✐♦r ♦✉ ♠❡♥♦r ❣r❛✉✱ s❡ ✐♠♣õ❡ ❛ ♥❡❝❡s✲ s✐❞❛❞❡ ❞❡ s❡ ❢❛③❡r ❝♦♥t❛❣❡♥s✳
❆ ❝♦♥t❛❣❡♠ é✱ ♣♦rt❛♥t♦✱ ✉♠ ♣r♦❜❧❡♠❛ ❝♦t✐❞✐❛♥♦ ♥❛ ✈✐❞❛ ❞♦ ❤♦♠❡♠ ❞❡s❞❡ ♠✉✐t♦s ❛♥♦s ❛trás✱ ♦ q✉❛❧ ❢♦✐ r❡s♦❧✈✐❞♦ ♣♦r ♠❡✐♦ ❞❛ ❝r✐❛çã♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✿
1,2,3,4,5, . . .
◆ã♦ s❛❜❡♠♦s ♣♦r q✉❛♥t♦ t❡♠♣♦ s❡ ❛rr❛st♦✉ ❛ ❝r✐❛çã♦ ❞❡ss❡s ♥ú♠❡r♦s✳ ❖ q✉❡ ♣♦❞❡♠♦s ❞✐③❡r ❝♦♠ ❝❡rt❡③❛ é q✉❡ ♦ ❤♦♠❡♠ ❞❡ ♠✉✐t♦s ❛♥♦s ❛trás ♥ã♦ t✐♥❤❛ ♦ ♠❡s♠♦ ❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ t❡♠♦s ❤♦❥❡ ❞❡ss❡s ♥ú♠❡r♦s✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❈❛r❛ç❛ ❬✸❪✱ ❡st✉❞♦s r❡❛❧✐③❛❞♦s ❝♦♠ ♣♦✈♦s ❛♥t✐❣♦s ❡①✐st❡♥t❡s ♥❛ ➪❢r✐❝❛ ❡ ♥❛ ❆✉strá❧✐❛ ♣❡r♠✐t❡♠✲♥♦s t❡r ✉♠❛ ✐❞❡✐❛ ❞❛ ♠❛♥❡✐r❛ ❝♦♠♦ ♦s ♣♦✈♦s ❞❡ ♠✐❧❤❛r❡s ❞❡ ❛♥♦s ❛trás s❡ ❛❝❤❛✈❛♠ ❡♠ r❡❧❛çã♦ ❛ ❡ss❛ q✉❡stã♦✳ ❖s r❡s✉❧t❛❞♦s ❞❡ss❡s ❡st✉❞♦s ♥♦s ♣❡r♠✐t❡♠ t✐r❛r ❛❧❣✉♠❛s ❝♦♥❝❧✉sõ❡s✿ ❛ ✐❞❡✐❛ ❞❡ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♥ã♦ é ✉♠ ♣r♦❞✉t♦ ♣✉r♦ ❞♦ ♣❡♥s❛♠❡♥t♦✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞❛ ❡①♣❡r✐ê♥❝✐❛✱ ✐st♦ é✱ ♦s ❤♦♠❡♥s ♥ã♦ ❛❞q✉✐r✐r❛♠ ♣r✐♠❡✐r♦ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣❛r❛ ❞❡♣♦✐s ❝♦♥t❛r❡♠✳ P❡❧♦ ❝♦♥trár✐♦✱ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❢♦r❛♠✲s❡ ❢♦r♠❛♥❞♦ ❧❡♥t❛♠❡♥t❡ ♣❡❧❛ ♣rát✐❝❛ ❞✐ár✐❛ ❞❡ ❝♦♥t❛❣❡♥s✳ ◆ã♦ ♣♦❞❡♠♦s ❝♦♥❝❡❜❡r ❛ ✐❞❡✐❛ ❞♦ ❤♦♠❡♠ ❝r✐❛♥❞♦ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣❛r❛ só ❞❡♣♦✐s ❛♣❧✐❝❛r à ♣rát✐❝❛ ❞❡ ❢❛③❡r ❝♦♥t❛❣❡♥s✳
➱✱ ♣♦rt❛♥t♦✱ ✐♠♣♦rt❛♥t❡ ❞❡st❛❝❛r q✉❡✱ ❤✐st♦r✐❝❛♠❡♥t❡✱ ❛ ❝♦♥t❛❣❡♠ ❛♥t❡❝❡❞❡ ♦s ♥ú♠❡r♦s✱
❈❆P❮❚❯▲❖ ✶✳ ◆Ú▼❊❘❖❙ ◆❆❚❯❘❆■❙ ✺
♦✉ s❡❥❛✱ é ♣♦ssí✈❡❧ r❡❛❧✐③❛r♠♦s ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❝♦♥t❛❣❡♠ ❞❡ ♦❜❥❡t♦s s❡♠ ✉s❛r♠♦s ♥ú♠❡r♦s ♣❛r❛ ✐ss♦✳ ❙❡❣✉♥❞♦ ●✐r❛❧❞♦ ❡t ❛❧ ❬✼❪✱ ✏♥♦ s❡♥t✐❞♦ ♠❛t❡♠át✐❝♦✱ ♣❛r❛ ❝♦♥t❛r ❜❛st❛ ❡st❛❜❡❧❡✲ ❝❡r ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ✉♠ ❛ ✉♠ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s✑✳
✶✳✷ ❆ ■❞❡✐❛ ❞❡ ❈♦rr❡s♣♦♥❞ê♥❝✐❛
▼❛s ❡♠ ♣♦ss❡ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❝♦♠♦ ✉♠❛ ♣❡ss♦❛ ❞❡✈❡ ♣r♦❝❡❞❡r ♣❛r❛ ❢❛③❡r ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❝♦♥t❛❣❡♠❄ ❆ ❝❛❞❛ ✉♠ ❞♦s ♦❜❥❡t♦s ❡❧❛ ❢❛③ ❝♦rr❡s♣♦♥❞❡r ✉♠ ♥ú♠❡r♦ ❞❛ s✉❝❡ssã♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✿ ❛♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ ❡❧❛ ❛ss♦❝✐❛ ♦ ♥ú♠❡r♦ ✉♠✱ ❛♦ s❡❣✉♥❞♦✱ ♦ ♥ú♠❡r♦ ❞♦✐s ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❙❡ ❛♦ ú❧t✐♠♦ ❡❧❡♠❡♥t♦ é ❛ss♦❝✐❛❞♦ ♦ ♥ú♠❡r♦ ❝✐♥❝♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❞✐③❡♠♦s q✉❡ t❡♠ ❝✐♥❝♦ ❡❧❡♠❡♥t♦s✳
❆ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❞❡ ❞♦✐s ❡♥t❡s ✭♥♦ ❡①❡♠♣❧♦ ❛❝✐♠❛✱ ♦❜❥❡t♦s ❡ ♥ú♠❡r♦s✮ ❡①✐❣❡ q✉❡ ❤❛❥❛ ✉♠ ❛♥t❡❝❡❞❡♥t❡ ❡ ✉♠ ❝♦♥s❡q✉❡♥t❡✳ ❆ ♠❛♥❡✐r❛ ♣❡❧❛ q✉❛❧ ♦ ♣❡♥s❛r ♥♦ ❛♥t❡❝❡❞❡♥t❡ ❞❡s♣❡rt❛ ♦ ♣❡♥s❛r ♥♦ ❝♦♥s❡q✉❡♥t❡ ❝❤❛♠❛✲s❡ ❧❡✐ ❞❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ✕ ♥♦ ♥♦ss♦ ❝❛s♦✱ ♦ ❛♥t❡❝❡❞❡♥t❡ sã♦ ♦s ♦❜❥❡t♦s ❡ ♦ ❝♦♥s❡q✉❡♥t❡✱ ♦s ♥ú♠❡r♦s✳
✶✳✸ ❚✐♣♦s ❞❡ ❈♦rr❡s♣♦♥❞ê♥❝✐❛
❉❡ ❛❝♦r❞♦ ❝♦♠ ❈❛r❛ç❛ ❬✸❪✱ ♣♦❞❡♠♦s ❡♥t❡♥❞❡r ♦s t✐♣♦s ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛s ❝♦♠ ❛ s❡❣✉✐♥t❡ ✐❧✉str❛çã♦✿
■♠❛❣✐♥❡♠♦s ❛ s✐t✉❛çã♦ ❛ s❡❣✉✐r✿ ◆✉♠❛ s❛❧❛ ❤á três ♣❡ss♦❛s ✕ ❞✉❛s ❞❡❧❛s ❝❤❛♠❛❞❛s ✏▼❛r✐❛✑ ❡ ✉♠❛ ❝❤❛♠❛❞❛ ✏❏♦ã♦✑✳ ❙❡ ♣❡♥s❛r♠♦s ❡♠ ✉♠❛ ❞❡ss❛s ♣❡ss♦❛s✱ ♥❛t✉r❛❧♠❡♥t❡ ❛s✲ s♦❝✐❛♠♦s ❛ ❡❧❛ ♦ s❡✉ ♥♦♠❡✳ ❚❡♠♦s✱ ♣♦rt❛♥t♦✱ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛✿ P❡ss♦❛ ✭❛♥t❡❝❡❞❡♥t❡✮
→◆♦♠❡ ✭❝♦♥s❡q✉❡♥t❡✮✳
❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ ♣❡♥s❛r♠♦s ♥✉♠ ♥♦♠❡✱ ♥♦s r❡♠❡t❡ ❛ ♣❡♥s❛r ♥❛ ♣❡ss♦❛ ✭♦✉ ♣❡ss♦❛s✮ ❝♦♠ ❡ss❡ ♥♦♠❡✱ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ t❡♠♦s ♦✉tr❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛✿ ◆♦♠❡ ✭❛♥t❡❝❡❞❡♥t❡✮
→P❡ss♦❛ ✭❝♦♥s❡q✉❡♥t❡✮✳
❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡ss❛s ❞✉❛s ❝♦rr❡s♣♦♥❞ê♥❝✐❛s ❡stá ♥♦ ❢❛t♦ ❞♦ ❛♥t❡❝❡❞❡♥t❡ ❡ ❞♦ ❝♦♥✲ s❡q✉❡♥t❡ ❡st❛r❡♠ ♥❛ ♦r❞❡♠ ✐♥✈❡rs❛ ✉♠ ❞♦ ♦✉tr♦✳ ◗✉❛♥❞♦ ❞✉❛s ❝♦rr❡s♣♦♥❞ê♥❝✐❛s ❡stã♦ ♥❡ss❛s ❝♦♥❞✐çõ❡s✱ ❞✐③❡♠♦s q✉❡ ❡❧❛s sã♦ r❡❝í♣r♦❝❛s ✉♠❛ ❞❛ ♦✉tr❛✳
❈♦♥s✐❞❡r❡♠♦s ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ P❡ss♦❛ → ◆♦♠❡ ❞❛❞❛ ❛❝✐♠❛✳ ❱❡❥❛ q✉❡ t♦❞♦ ❛♥✲
t❡❝❡❞❡♥t❡ t❡♠ ❝♦♥s❡q✉❡♥t❡✱ ✐st♦ é✱ t♦❞❛s ❛s ♣❡ss♦❛s q✉❡ ❡stã♦ ♥❛ s❛❧❛ tê♠ ✉♠ ♥♦♠❡✳ ◗✉❛♥❞♦ t❡♠♦s ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❝♦♠ ❡ss❛ ❝❛r❛❝t❡ríst✐❝❛ ❞✐③❡♠♦s q✉❡ ❡❧❛ é ❝♦♠♣❧❡t❛✳
❆✐♥❞❛ s♦❜r❡ ❡ss❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ❝❛❞❛ ❛♥t❡❝❡❞❡♥t❡ ✭♣❡ss♦❛✮ t❡♠ ✉♠ ú♥✐❝♦ ❝♦♥s❡q✉❡♥t❡ ✭♥♦♠❡✮✳ ◗✉❛♥❞♦ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❝♦♠♣❧❡t❛ t❡♠ ❡ss❛ ❝❛r❛❝t❡ríst✐❝❛ ❞✐③❡♠♦s q✉❡ ❡❧❛ é ✉♥í✈♦❝❛✳
❈❆P❮❚❯▲❖ ✶✳ ◆Ú▼❊❘❖❙ ◆❆❚❯❘❆■❙ ✻
é ♦❜✈✐❛♠❡♥t❡ ❝♦♠♣❧❡t❛✱ ✐st♦ é✱ ❛ ❝❛❞❛ ♥♦♠❡ ❞❛❞♦✱ ❡①✐st❡ ✉♠❛ ♣❡ss♦❛ ❝♦♠ ❡ss❡ ♥♦♠❡✳ P♦ré♠✱ ❡❧❛ ♥ã♦ é ✉♥í✈♦❝❛✱ ♣♦✐s ♦ ♥♦♠❡ ✏▼❛r✐❛✑ ❡stá ❛ss♦❝✐❛❞♦ ❛ ❞✉❛s ♣❡ss♦❛s✱ ♦✉ s❡❥❛✱ ✉♠ ❛♥t❡❝❡❞❡♥t❡ t❡♠ ♠❛✐s ❞❡ ✉♠ ❝♦♥s❡q✉❡♥t❡✳
❙❡✱ ♣♦ré♠✱ t✐✈❡r♠♦s ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ✉♥í✈♦❝❛ ❝✉❥❛ r❡❝í♣r♦❝❛ t❛♠❜é♠ s❡❥❛ ✉♥í✈♦❝❛✱ ❞✐③❡♠♦s q✉❡ t❛❧ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ é ❜✐✉♥í✈♦❝❛✳ ◆♦ ♥♦ss♦ ❡①❡♠♣❧♦✱ ♣❛r❛ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ P❡ss♦❛ → ◆♦♠❡ s❡r ❜✐✉♥í✈♦❝❛✱ ❜❛st❛r✐❛ q✉❡ ❛s ♣❡ss♦❛s q✉❡ ❡st✐✈❡ss❡♠ ♥❛ s❛❧❛ t✐✈❡ss❡♠
♥♦♠❡s ❞✐st✐♥t♦s✳
❖❜s❡r✈❡ q✉❡ ❞✐③❡r q✉❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ é ❝♦♠♣❧❡t❛ ❡ ✉♥í✈♦❝❛ é ♦ ♠❡s♠♦ q✉❡ ❞✐③❡r q✉❡ s❡ tr❛t❛ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ❡ ❞✐③❡r q✉❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ é ❜✐✉♥í✈♦❝❛ é ♦ ♠❡s♠♦ q✉❡ ❞✐③❡r q✉❡ t❡♠♦s ✉♠❛ ❢✉♥çã♦ ❜✐❥❡t♦r❛✳ P♦ré♠✱ ❡ss❡s t❡r♠♦s ♥ã♦ ❢♦r❛♠ ✉s❛❞♦s ♣❡❧♦ ❢❛t♦ ❞♦ ❝♦♥t❡ú❞♦ ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s ❛♥t❡❝❡❞❡r ♦ ❝♦♥t❡ú❞♦ ❢✉♥çõ❡s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳
✶✳✹ ❊q✉✐✈❛❧ê♥❝✐❛ ❡ Pr❡✈❛❧ê♥❝✐❛
❉❛❞❛s ❞✉❛s ❝♦❧❡çõ❡s ❞❡ ♦❜❥❡t♦s✱ s❡ ♣✉❞❡r♠♦s ♣ô✲❧❛s ❡♠ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❞✐③❡♠♦s q✉❡ ❡❧❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s✳ ❱❡❥❛ ♦ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r ✭❋✐❣✉r❛ ✶✳✶✮✿
❋✐❣✉r❛ ✶✳✶✿ ❈♦❧❡çõ❡s ❞❡ ❖❜❥❡t♦s
P♦❞❡♠♦s ❝♦rr❡s♣♦♥❞❡r ❜✐✉♥✐✈♦❝❛♠❡♥t❡ ♦s ♦❜❥❡t♦s ❞❛s ❝♦❧❡çõ❡sA ❡B ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✷ ❛❜❛✐①♦✮✳
❋✐❣✉r❛ ✶✳✷✿ ❈♦❧❡çõ❡s ❊q✉✐✈❛❧❡♥t❡s
❈❆P❮❚❯▲❖ ✶✳ ◆Ú▼❊❘❖❙ ◆❆❚❯❘❆■❙ ✼
é✱ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❞✉❛s ❝♦❧❡çõ❡s ❞❡ ♦❜❥❡t♦s s✐❣♥✐✜❝❛ ✐❣✉❛❧❞❛❞❡ ❞❡ ♥ú♠❡r♦ ❞❡ ♦❜❥❡t♦s✳ ❱❡❥❛♠♦s✱ ❛❣♦r❛✱ ✉♠ ♦✉tr♦ ❡①❡♠♣❧♦ ✭❋✐❣✉r❛ ✶✳✸✮✿
❋✐❣✉r❛ ✶✳✸✿ ❈♦❧❡çõ❡s ❝♦♠ ♥ú♠❡r♦ ❞❡ ♦❜❥❡t♦s ❞✐st✐♥t♦s
❖❜✈✐❛♠❡♥t❡✱ ♥ã♦ ❤á ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ♦s ♦❜❥❡t♦s ❞❛s ❝♦❧❡çõ❡sA ❡C✱ ✐st♦ é✱ ♦ ♥ú♠❡r♦ ❞❡ ♦❜❥❡t♦s ❞❡A ❡ C sã♦ ❞✐st✐♥t♦s ✭❋✐❣✉r❛ ✶✳✹✮✳
❋✐❣✉r❛ ✶✳✹✿ ❈♦❧❡çõ❡s ♥ã♦ ❡q✉✐✈❛❧❡♥t❡s
▼❛s ♣♦❞❡♠♦s t❡r ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ♦s ♦❜❥❡t♦s ❞❡A❡ ✉♠❛ ♣❛rt❡ ❞❡C✳ ◆❡ss❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡C é ♣r❡✈❛❧❡♥t❡ ❛ A✳ P♦rt❛♥t♦✱ ❡q✉✐✈❛❧ê♥❝✐❛ s✐❣♥✐✜❝❛ ✐❣✉❛❧❞❛❞❡ ❡ ♣r❡✈❛❧ê♥❝✐❛✱ ❞❡s✐❣✉❛❧❞❛❞❡✳
✶✳✺ ◆ú♠❡r♦✿ ❯♠ ❈♦♥❝❡✐t♦ ❆❜str❛t♦
❈♦♥❢♦r♠❡ ●✐r❛❧❞♦ ❡t ❛❧ ❬✼❪✱ ✏✉♠ ♥ú♠❡r♦ ✭♥❛t✉r❛❧✮ é ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❡♠ ❝♦♠✉♠ ❛ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ❝♦♠ ❛ ♠❡s♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s✱ ✐st♦ é✱ ✉♠ ❵rót✉❧♦✬ q✉❡ s❡ ❞á ❛ ✉♠❛ ❞❛❞❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉♥t♦s ❝♦♠ q✉❛♥t✐❞❛❞❡s ❡q✉✐✈❛❧❡♥t❡s ❞❡ ❡❧❡♠❡♥t♦s✑✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ ♥ú♠❡r♦ ✸ é ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❛❜str❛t❛ q✉❡ três ❧á♣✐s✱ três ❜♦rr❛❝❤❛s✱ três ❜♦❧❛s✱ ❡t❝✳ tê♠ ❡♠ ❝♦♠✉♠✳ ❖❜s❡r✈❡ q✉❡ ♦ ✸ ♥ã♦ é ♥❡♠ ♦s três ❧á♣✐s✱ ♥❡♠ ❛s três ❜♦rr❛❝❤❛s✱ ♥❡♠ ❛s três ❜♦❧❛s✱ ♠❛s ✉♠❛ ❛❜str❛çã♦ ❞❛ ✐❞❡✐❛ ❝♦♥❝r❡t❛ ❞❛ ❝♦♥t❛❣❡♠✳
✶✳✻ ❖ ■♥✜♥✐t♦
❈❆P❮❚❯▲❖ ✶✳ ◆Ú▼❊❘❖❙ ◆❆❚❯❘❆■❙ ✽
❆ r❡t✐❝ê♥❝✐❛ ❝♦❧♦❝❛❞❛ ❛♣ós ❛ ú❧t✐♠❛ ✈ír❣✉❧❛ s✐❣♥✐✜❝❛ q✉❡ ♥ã♦ ❡stã♦ ❡s❝r✐t♦s t♦❞♦s ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❋❛❧t❛♠ ♥ú♠❡r♦s✳ ▼❛s q✉❛♥t♦s❄ ❊①✐st❡ ♦ ♠❛✐♦r ♥ú♠❡r♦ ♥❛t✉r❛❧❄
P♦❞❡♠♦s r❡s♣♦♥❞❡r ❛ ❡ss❛ ♣❡r❣✉♥t❛ ♣❡♥s❛♥❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ♥❛ s❡q✉ê♥❝✐❛ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ♣❛ss❛♠♦s ❞❡ ✉♠ ♥ú♠❡r♦ ♣❛r❛ ♦ s❡❣✉✐♥t❡ ❛❝r❡s❝❡♥t❛♥❞♦✲❧❤❡ ✉♠❛ ✉♥✐❞❛❞❡✳ P♦r ♠❡✐♦ ❞❡ss❛ ♦♣❡r❛çã♦ ❡❧❡♠❡♥t❛r ♣❛ss❛♠♦s ❞♦ ✶ ♣❛r❛ ♦ ✷✱ ❞♦ ✷ ♣❛r❛ ♦ ✸ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❖✉ s❡❥❛✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ n ❞❛❞♦✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✕ ✉s❛♥❞♦ ❛ ♠❡s♠❛ ♦♣❡r❛çã♦ ❡❧❡♠❡♥t❛r ✕ ♦ ♣ró①✐♠♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❞❛ s❡q✉ê♥❝✐❛✱ ♦n+ 1✳ ▲♦❣♦✱ ♥ã♦ ❡①✐st❡ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♠❛✐♦r q✉❡ t♦❞♦s ♦s ♦✉tr♦s✳ ❉✐③❡♠♦s✱ ❡♥tã♦✱ q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s é ✐❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡ ✕ ✐st♦ é✱ ♣♦r ♠❛✐♦r q✉❡ s❡❥❛ ♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ n q✉❡ t♦♠❡♠♦s s❡♠♣r❡ ♣♦❞❡r❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ♠❛✐♦r q✉❡ ❡❧❡ ✭n+ 1✮ ✕ ❡ ✐♥✜♥✐t❛✱ ♦✉ s❡❥❛✱ ❡ss❡ ♣r♦❝❡ss♦ ❞❡ ❡♥❝♦♥tr❛r ♦ ♣ró①✐♠♦ ♥ú♠❡r♦ ♣♦❞❡ s❡r r❡♣❡t✐❞♦ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✱ ♦ q✉❡ q✉❡r ❞✐③❡r q✉❡ ♥ã♦ ❡①✐st❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s✳
✶✳✼ ❖ ❈♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ◆❛t✉r❛✐s
❉✐③❡♠♦s q✉❡ t❡♠♦s ✉♠ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r ❝❡rt♦s ❡❧❡♠❡♥t♦s q✉❛♥❞♦✿ ✭✐✮ t❛✐s ❡❧❡✲ ♠❡♥t♦s tê♠ ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❡♠ ❝♦♠✉♠ ❡ ✭✐✐✮ ❞❛❞♦ ✉♠ ❡❧❡♠❡♥t♦ q✉❛❧q✉❡r✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r s❡ ❡❧❡ ♣❡rt❡♥❝❡ ♦✉ ♥ã♦ ❛♦ ❝♦♥❥✉♥t♦✳
❙❡rá✱ ❡♥tã♦✱ q✉❡ ♣♦❞❡♠♦s ♥♦s r❡❢❡r✐r ❛♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s❄ ◗✉❛❧ ❛ ❝❛r❛❝t❡ríst✐❝❛ q✉❡ t♦❞♦s ❡❧❡s tê♠ ❡♠ ❝♦♠✉♠❄ ❆ ❞❡ q✉❡✱ ❝♦♠ ❡①❝❡çã♦ ❞♦ ✶ ✭q✉❡ é ♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦✮✱ t♦❞♦s ♦s ♦✉tr♦s sã♦ ♦❜t✐❞♦s ♣❡❧❛ ♦♣❡r❛çã♦ ❡❧❡♠❡♥t❛r ❞❡ ❡♥❝♦♥tr❛r ♦ ♣ró①✐♠♦ ❡❧❡♠❡♥t♦ ✕ é ♦ q✉❡ ❝❤❛♠❛♠♦s ❞❡ s✉❝❡ss♦r✳ P♦r ❡①❡♠♣❧♦✱ ❞✐③❡♠♦s q✉❡ ✷ é ♦ s✉❝❡ss♦r ❞❡ ✶✱ ✸ é ♦ s✉❝❡ss♦r ❞❡ ✷✱ ❡t❝✳✱ ♦✉ s❡❥❛✱ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s sã♦ s✉❝❡ss♦r❡s ❞❡ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ✭❛ ♣❛rt✐r ❞♦ ✶✮✳ ❆❧é♠ ❞✐ss♦✱ ❞❛❞♦ ✉♠ ❡❧❡♠❡♥t♦ q✉❛❧q✉❡r✱ é ❢á❝✐❧ ✈❡r✐✜❝❛r s❡ ❡❧❡ ❡stá ♥❡ss❡ ❝♦♥❥✉♥t♦✱ ❜❛st❛♥❞♦ ♣❛r❛ ✐ss♦ ✈❡r✐✜❝❛r s❡ ❡❧❡ é ♦ s✉❝❡ss♦r ❞❡ ❛❧❣✉é♠ q✉❡ ❥á ❡stá ♥❡ss❡ ❝♦♥❥✉♥t♦✳
❆ss✐♠✱ ♣♦❞❡♠♦s✱ ❞❡ ❢❛t♦✱ ♥♦s r❡❢❡r✐r ❛♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ♦ q✉❛❧ ✜❝❛ ❝❛r❛❝t❡r✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ✭❛✮ ♦ ✶ ♣❡rt❡♥❝❡ ❛ ❡ss❡ ❝♦♥❥✉♥t♦ ❡ ✭❜✮ ❝❛❞❛ ✉♠ ❞♦s ❡❧❡♠❡♥t♦s s❡❣✉✐♥t❡s é ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛ ♦♣❡r❛çã♦ ❡❧❡♠❡♥t❛r ❞❡ ❛❞✐❝✐♦♥❛r ✉♠❛ ✉♥✐❞❛❞❡✳
❖❜s❡r✈❡ q✉❡ ❛ ♠❛♥❡✐r❛ q✉❡ ❝❛r❛❝t❡r✐③❛♠♦s ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❝♦✐♥❝✐❞❡ ❝♦♠ ♦s ❆①✐♦♠❛s ❞❡ P❡❛♥♦✱ s❡♥❞♦✱ ♣♦ré♠✱ ❛♣r❡s❡♥t❛❞♦ ❝♦♠ ♠❡♥♦s ❢♦r♠❛❧✐❞❛❞❡✳
❈❆P❮❚❯▲❖ ✶✳ ◆Ú▼❊❘❖❙ ◆❆❚❯❘❆■❙ ✾
✶✳✽ ❖ ❩❡r♦
➱ ♠✉✐t♦ ❝♦♠✉♠✱ ❡♠ ❧✐✈r♦s ❞✐❞át✐❝♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❛ s✉❝❡ssã♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❝♦♠❡ç❛r ♣❡❧♦ ③❡r♦ ❡ ♥ã♦ ♣❡❧♦ ✉♠✳ ❆ss✐♠✱ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s s❡r✐❛♠✿
0,1,2,3,4,5, . . .
◆❡st❡ tr❛❜❛❧❤♦✱ ♥ã♦ ✉t✐❧✐③❛r❡♠♦s ♦ ③❡r♦ ❝♦♠♦ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ◆ã♦ ❡♥tr❛r❡♠♦s ♥♦ ♠ér✐t♦ ❞❛ q✉❡stã♦✱ ♣♦✐s ❢♦❣❡ à ♥♦ss❛ ♣r♦♣♦st❛✳ ▼❛s ❡♥t❡♥❞❡♠♦s q✉❡ ♥ã♦ ❤á ❡q✉í✈♦❝♦ ❛❧❣✉♠ ❝♦♥s✐❞❡r❛r ♦ ③❡r♦ ❝♦♠♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♦✉ ♦ ❝♦♥trár✐♦✳ ➱ ❛♣❡♥❛s ✉♠❛ q✉❡stã♦ ❞❡ ❡s❝♦❧❤❛ ❡ ❝♦♠♦❞✐❞❛❞❡✳
✶✳✾ ❖ ❈♦♥❥✉♥t♦ ❞♦s P♦♥t♦s ❞❛ ❘❡t❛
❙❡rá q✉❡✱ ❛❧é♠ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❡①✐st❡♠ ♦✉tr♦s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s❄ ❱❛♠♦s ♣❡♥s❛r ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛✳ Pr✐♠❡✐r❛♠❡♥t❡✱ t❡♠♦s q✉❡ ✈❡r✐✜❝❛r s❡ ♦ ❝♦♥❥✉♥t♦ ✭r❡t❛ ❞❡ ✉♠ ♣❧❛♥♦✮ ❡stá ❜❡♠ ❞❡t❡r♠✐♥❛❞♦✳ ❙❛❜❡♠♦s ❞❛ ❣❡♦♠❡tr✐❛ q✉❡ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s A ❡ B ❞❡t❡r♠✐♥❛♠ ✉♠❛ ú♥✐❝❛ r❡t❛✱ ✐st♦ é✱ q✉❛❧q✉❡r ♣♦♥t♦ ❞❛ r❡t❛ q✉❡ ❝♦♥té♠ A ❡ B ❡stá ❛❧✐♥❤❛❞♦ ❝♦♠ ❡ss❡s ❞♦✐s ♣♦♥t♦s✳ P♦✐s ❜❡♠✱ ❡♥tã♦ ❞❛❞♦ ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞♦ ♣❧❛♥♦ ✭s♦❜r❡ ♦ q✉❛❧ ❡ss❛ r❡t❛ ❡stá✮✱ ♣♦❞❡♠♦s s❡♠♣r❡ ✈❡r✐✜❝❛r s❡ ❡❧❡ ❡stá ♦✉ ♥ã♦ ❛❧✐♥❤❛❞♦ ❝♦♠A❡B✳ ❊♠ ❝❛s♦ ❛✜r♠❛t✐✈♦✱ ♦ ♣♦♥t♦ ♣❡rt❡♥❝❡ à r❡t❛ ✭❛♦ ❝♦♥❥✉♥t♦✮❀ ❝❛s♦ ❝♦♥trár✐♦✱ ♥ã♦ ♣❡rt❡♥❝❡✳
P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❢❛❧❛r ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛ ❞♦ ♣❧❛♥♦✳ ❆ ♥♦ss❛ ❞ú✈✐❞❛ ❛❣♦r❛ é✿ ❡ss❡ ❝♦♥❥✉♥t♦ é ✐♥✜♥✐t♦❄ P❛r❛ r❡s♣♦♥❞❡r à ❡ss❛ ♣❡r❣✉♥t❛✱ ❝♦♥s✐❞❡r❡♠♦s ♦ s❡❣✲ ♠❡♥t♦ AB✱ ✐st♦ é✱ ❛♣❡♥❛s ❛ ♣❛rt❡ ❞❛ r❡t❛ q✉❡ ✈❛✐ ❞❡ A ❛té B ✭❋✐❣✉r❛ ✶✳✺✮✳
❋✐❣✉r❛ ✶✳✺✿ ❙❡❣♠❡♥t♦AB
❙❡ ❞✐✈✐❞✐r♠♦s ❡ss❡ s❡❣♠❡♥t♦ ❛♦ ♠❡✐♦✱ ♦❜t❡♠♦s ♦ ♣♦♥t♦ A1❀ s❡ ❞✐✈✐❞✐r♠♦s ❛❣♦r❛ ♦
s❡❣♠❡♥t♦A1B ❛♦ ♠❡✐♦✱ ♦❜t❡♠♦s ♦ ♣♦♥t♦A2❀ ❞✐✈✐❞✐♥❞♦A2B ❛♦ ♠❡✐♦✱ ♦❜t❡♠♦s A3 ❡ ❛ss✐♠
♣♦r ❞✐❛♥t❡✳ ❙❡ ✉s❛r♠♦s ♦ ❢❛t♦ ❞❡ q✉❡ ♦ ♣♦♥t♦ ♥ã♦ t❡♠ ❞✐♠❡♥sã♦✱ ♣♦❞❡♠♦s r❡❛❧✐③❛r ❡ss❛ ♦♣❡r❛çã♦ ✐♥❞❡✜♥✐❞❛♠❡♥t❡ ✭❋✐❣✉r❛ ✶✳✻✮✳ ❊✱ ❡♠ ❝♦♥s❡q✉ê♥❝✐❛ ❞✐ss♦✱ t❡r❡♠♦s ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♣♦♥t♦sAn, n= 1,2,3, . . .q✉❡ ♣❡rt❡♥❝❡♠ ❛♦ s❡❣♠❡♥t♦AB ✭❡ ♣♦rt❛♥t♦ ♣❡rt❡♥❝❡♠ à r❡t❛
q✉❡ ❝♦♥té♠ A ❡ B✮✳ ▲♦❣♦✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛✱ ❛ss✐♠ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ é ✉♠ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦✶✳
✶❖ ❝♦♥❝❡✐t♦ ❞❡ ✐♥✜♥✐t♦ q✉❡ ❡st❛♠♦s ✉s❛♥❞♦ ♥❡ss❡ t❡①t♦ ❡stá ❜❛s❡❛❞♦ ♥❛ ✐♥❥❡çã♦ ❞❡N♥♦ ❝♦♥❥✉♥t♦✱ ✐st♦
❈❆P❮❚❯▲❖ ✶✳ ◆Ú▼❊❘❖❙ ◆❆❚❯❘❆■❙ ✶✵
❋✐❣✉r❛ ✶✳✻✿ ❙❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s An
❖❜s❡r✈❡ q✉❡✱ ❛❧é♠ ❞❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♣♦♥t♦s An, n = 1,2,3, . . . q✉❡ ❝♦♥s✐❞❡r❛♠♦s✱ ❤á
❛✐♥❞❛ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♣♦♥t♦s ♥♦s s❡❣♠❡♥t♦s AA1✱ A1A2✱ A2A3✱ ❡t❝✳ ✭q✉❡ ♥ã♦ ❢♦r❛♠
❝♦♥t❛❞♦s ❛q✉✐✮✱ ♣♦✐s ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ❛♥t❡r✐♦r✳
❊st❛♠♦s✱ ❛ss✐♠✱ ❞✐❛♥t❡ ❞❡ ✐♥✜♥✐t♦s ❝♦♠ ❝❛r❝t❡ríst✐❝❛s ❞✐❢❡r❡♥t❡s✿ ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❞❛❞♦s ❞♦✐s ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s t❡♠♦s ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ♥ú♠❡r♦s ❡♥tr❡ ❡❧❡s❀ ❥á ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛✱ ❞❛❞♦s ❞♦✐s ♣♦♥t♦s✱ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♣♦♥t♦s ❡♥tr❡ ❡❧❡s✳ ❙❡rá✱ ❡♥tã♦✱ q✉❡ ♣♦❞❡♠♦s ❝♦♠♣❛r❛r ❡ss❡s ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ ✐♥✜♥✐t♦s❄ ❱❡r❡♠♦s ♠❛✐s ❛❞✐❛♥t❡✳
✶✳✶✵ ❈♦rr❡s♣♦♥❞ê♥❝✐❛s ♥♦ ■♥✜♥✐t♦
❆ ♦♣❡r❛çã♦ ❞❛ ❝♦♥t❛❣❡♠ ✈❛✐ ♥♦s ❢♦r♥❡❝❡r ♦ ♠♦❞❡❧♦ ❞❡ ❝♦♠♦ ✈❛♠♦s ♣r♦❝❡❞❡r ♣❛r❛ ❝♦♠♣❛r❛r ♦s ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ ✐♥✜♥✐t♦s✳ ❱✐♠♦s q✉❡✱ ♣❛r❛ r❡❛❧✐③❛r ✉♠❛ ❝♦♥t❛❣❡♠✱ ❢❛③❡♠♦s ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦❜❥❡t♦s ❡ ♥ú♠❡r♦s✳ ❖ q✉❡ ✈❛♠♦s ❢❛③❡r é ✉s❛r ❡ss❛ ♠❡s♠❛ ✐❞❡✐❛ ✭❞❡ ❢❛③❡r ❝♦rr❡s♣♦♥❞❡r✮ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✱ ✐st♦ é✱ ✈❛♠♦s ✈❡r✐✜❝❛r s❡ ♣♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s✳
❱✐♠♦s ✭s❡çã♦ ✶✳✺✮ q✉❡✱ q✉❛♥❞♦ s❡ tr❛t❛ ❞❡ ❝♦♥❥✉♥t♦s ✜♥✐t♦s✱ ✉♠❛ ♣❛rt❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❥❛♠❛✐s ♣♦❞❡✲s❡ ♣♦r ❡♠ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❝♦♠ ♦ t♦❞♦✳ ❏á ♥♦ ✐♥✜♥✐t♦✱ ✈❛♠♦s ✈❡r ❛tr❛✈és ❞❡ ❞♦✐s ❡①❡♠♣❧♦s q✉❡ ♥♦ss❛ ✐♥t✉✐çã♦ ❢❛❧❤❛✳
❊❳❊▼P▲❖ ✶✳ ❈♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✿ ❙❡❥❛ N ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ P ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♣❛r❡s ✭♣♦s✐t✐✈♦s✮✳
■♥t✉✐t✐✈❛♠❡♥t❡✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♣❛r❡s tê♠ ❜❡♠ ♠❡♥♦s ❡❧❡♠❡♥t♦s q✉❡ ♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ✭❥á q✉❡ ❡st❡ t❡♠ ❛❧é♠ ❞❛q✉❡❧❡s✱ ♦s ♥ú♠❡r♦s í♠♣❛r❡s✮✳ P♦ré♠✱ s❡ ❝♦r✲ r❡s♣♦♥❞❡r♠♦s ♦ ✶ ❛♦ ♣r✐♠❡✐r♦ ♥ú♠❡r♦ ♣❛r ✭♦ ♥ú♠❡r♦ ✷✮✱ ♦ ♥ú♠❡r♦ ✷ ❛♦ s❡❣✉♥❞♦ ✭✹✮✱ ♦ ✸ ❛♦ t❡r❝❡✐r♦ ✭✻✮ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱ t❡r❡♠♦s ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ N ❡ P✿ 1 ↔ 2,2 ↔ 4,3 ↔ 6, . . .✳ ■st♦ ♥♦s ❧❡✈❛ ❛ ❝♦♥❝❧✉✐r q✉❡✱ ❡♠ s❡ tr❛t❛♥❞♦ ❞❡ ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s✱ ♦ t♦❞♦ ❡ ❛ ♣❛rt❡ ♣♦❞❡♠ s❡r ❡q✉✐✈❛❧❡♥t❡s✱ ♦ q✉❡ ♥ã♦ ❢❛③✐❛ s❡♥t✐❞♦ ♥♦ ✜♥✐t♦✷✳
✷❊♠ s✉❛ ♦❜r❛ ❉✐á❧♦❣♦s ❞❡ ❉✉❛s ◆♦✈❛s ❈✐ê♥❝✐❛s✱ ❡s❝r✐t❛ ❞♦✐s sé❝✉❧♦s ❡ ♠❡✐♦ ❛♥t❡s ❞❡ ❈❛♥t♦r✱ ●❛❧✐❧❡✉
❈❆P❮❚❯▲❖ ✶✳ ◆Ú▼❊❘❖❙ ◆❆❚❯❘❆■❙ ✶✶
❊❳❊▼P▲❖ ✷✳ ❙❡❥❛♠ ABC ✉♠ tr✐â♥❣✉❧♦✱ M ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡AB ❡ N ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ BC ✭❋✐❣✉r❛ ✶✳✼✮✳ ❙❛❜❡♠♦s q✉❡ ♦ s❡❣♠❡♥t♦ M N t❡♠ ❝♦♠♣r✐♠❡♥t♦ ✐❣✉❛❧ ❛ ♠❡t❛❞❡ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ AC✳ P♦ré♠✱ ♣♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦s ♣♦♥t♦s ❞❡M N ❡ ♦s ♣♦♥t♦s ❞❡ AC✳
P❛r❛ ✐ss♦✱ ❜❛st❛ ❝♦rr❡s♣♦♥❞❡r♠♦s ♦s ♣♦♥t♦s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ❞❛❞♦ ✉♠ ♣♦♥t♦ P q✉❛❧q✉❡r ❡♠ M N✱ tr❛ç❛♠♦s ♦ s❡❣♠❡♥t♦ q✉❡ ♣❛ss❛ ♣♦r B ❡ ♣❡❧♦ ♣♦♥t♦ ❞❛❞♦ ❛té t♦❝❛r ♦ s❡❣♠❡♥t♦ AC ❡♠ P′❀ ❡st❡ é ♦ ♣♦♥t♦ q✉❡ ❝♦rr❡s♣♦♥❞❡♠♦s ❛♦ ♣♦♥t♦
P ❞❛❞♦✳ ❊ss❛ ❝♦rr❡s✲ ♣♦♥❞ê♥❝✐❛ é ❧♦❣✐❝❛♠❡♥t❡ ❜✐✉♥í✈♦❝❛ ❡ t❡♠♦s✱ ❛ss✐♠✱ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ s❡❣♠❡♥t♦s ❞❡ t❛♠❛♥❤♦s ❞✐❢❡r❡♥t❡s✳
❋✐❣✉r❛ ✶✳✼✿ ❈♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ s❡❣♠❡♥t♦s
❈♦♠♦ ✈✐♠♦s✱ t❡♠♦s ❞✉❛s ❝❛r❛❝t❡r✐③❛çõ❡s ❞♦ ✐♥✜♥✐t♦✿ ✉♠❛ ♣♦r ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡ ✭❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✮✱ ❡ ♦✉tr❛ ♣♦r ❞❡♥s✐❞❛❞❡ ✭❞♦s ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛✮✳ ❆ q✉❡stã♦ q✉❡ s❡ ❝♦❧♦❝❛ é✿ s❡rá q✉❡ ❡ss❛s ❞✉❛s ❝❧❛ss❡s ❞♦ ✐♥✜♥✐t♦ sã♦ ❞✐st✐♥t❛s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛ ❡q✉✐✈❛❧ê♥❝✐❛❄ ■st♦ é✱ ❡①✐st❡ ♦✉ ♥ã♦ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ss❡s ❞♦✐s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s❄ ❉✐s❝✉t✐r❡♠♦s ❡ss❡ ❛ss✉♥t♦ ♥❛ s❡çã♦ ✹✳✹✳
❈❛♣ít✉❧♦ ✷
◆ú♠❡r♦s ■♥t❡✐r♦s
✷✳✶ ●r❛♥❞❡③❛s ❚♦♠❛❞❛s ❡♠ ❙❡♥t✐❞♦s ❉✐❢❡r❡♥t❡s ❡ ♦ ❈♦♥✲
❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ■♥t❡✐r♦s
❊①✐st❡♠ ❣r❛♥❞❡③❛s q✉❡ ♣♦❞❡♠ s❡r t♦♠❛❞❛s ❡♠ s❡♥t✐❞♦s ♦♣♦st♦s✳ ❆♦ ❝♦♥str✉✐r ✉♠❛ ❡s❝❛❧❛ ❞❡ t❡♠♣♦✱ ♣♦r ❡①❡♠♣❧♦✱ t♦♠❛♠♦s ❝♦♠♦ ♦r✐❣❡♠ ✉♠ ❢❛t♦ ❤✐stór✐❝♦ ♠❛r❝❛♥t❡ ✭♦ ♥❛s❝✐✲ ♠❡♥t♦ ❞❡ ❈r✐st♦✮ ❡ t♦❞❛s ❛s ❞❡♠❛✐s ❞❛t❛s sã♦ ❝♦♥t❛❞❛s ❛♥t❡s ❡ ❛♣ós ❡ss❡ ❛❝♦♥t❡❝✐♠❡♥t♦✳ ❆ss✐♠✱ ✉♠ ❝❡rt♦ ❛♥♦ é ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡❧❡ ❡ ❛ ♦r✐❣❡♠ ❡ ♣♦r ❡st❛r ❛♥t❡s ♦✉ ❛♣ós ❛ ♦r✐❣❡♠✳ P♦r ❡①❡♠♣❧♦✱ ❞✐③❡r q✉❡ ❛❧❣✉é♠ ♠♦rr❡✉ ❡♠ ✸✺✵ ❛✳❈✳ s✐❣♥✐✜❝❛ q✉❡ ❡❧❡ ♠♦rr❡✉ ✸✺✵ ❛♥♦s ❛♥t❡s ❞♦ ♥❛s❝✐♠❡♥t♦ ❞❡ ❈r✐st♦ ❡ ❞✐③❡r q✉❡ ❛❧❣✉é♠ ♥❛s❝❡✉ ❡♠ ✶✺✺✵ ❞✳❈✳ s✐❣♥✐✜❝❛ q✉❡ ❡❧❡ ♥❛s❝❡✉ ✶✺✺✵ ❛♥♦s ❞❡♣♦✐s ❞♦ ♥❛s❝✐♠❡♥t♦ ❞❡ ❈r✐st♦✳ ❖❜s❡r✈❡ q✉❡ ❡♥tr❡ ♦s ❞♦✐s ❛❝♦♥t❡❝✐♠❡♥t♦s ♣❛ss♦✉✲s❡ ✉♠ ♣❡rí♦❞♦ ❞❡ ✶✾✵✵ ❛♥♦s ✭1550 + 350✮✳
❆♥❛❧♦❣❛♠❡♥t❡✱ s✉♣♦♥❤❛♠♦s q✉❡ ✉♠ ❝♦r♣♦ s❡ ❞❡s❧♦q✉❡ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ r❡t❛ ❛ ♣❛rt✐r ❞❡ ✉♠ ♣♦♥t♦ O ✭♦ q✉❛❧ ❝❤❛♠❛r❡♠♦s ❞❡ ♦r✐❣❡♠✮✳ ❙❡ ❡ss❡ ❝♦r♣♦ ♣❡r❝♦rr❡ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❡s♣❛ç♦ ♣♦r s❡❣✉♥❞♦✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ❡❧❡ ♣❡r❝♦rr❡rá ✸ ✉♥✐❞❛❞❡s ❞❡ ❡s♣❛ç♦ ❛♣ós ✸ s❡❣✉♥❞♦s✳ P♦ré♠✱ ❡ss❛ ✐♥❢♦r♠❛çã♦✱ ♣♦r s✐ só✱ ♥ã♦ ♥♦s ❞✐③ q✉❛❧ ❛ ♣♦s✐çã♦ ❡①❛t❛ ❞♦ ❝♦r♣♦✱ ♣♦✐s ♥ã♦ s❛❜❡♠♦s ❡♠ q✉❛❧ s❡♥t✐❞♦ ❢♦✐ ♦ s❡✉ ❞❡s❧♦❝❛♠❡♥t♦✳ P♦rt❛♥t♦✱ ❝♦♥✈❡♥❝✐♦♥❛r❡♠♦s ❝❤❛♠❛r ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞♦ ❝♦r♣♦ ♣❛r❛ ❛ ❞✐r❡✐t❛ ❞❡ s❡♥t✐❞♦ ♣♦s✐t✐✈♦ ❡ ♦ ❡sq✉❡r❞♦ ❞❡ ♥❡❣❛t✐✈♦✳ ❆ss✐♠✱ s❡ ✉♠ ❝♦r♣♦ s❡ ❞❡s❧♦❝❛ ✽ ✉♥✐❞❛❞❡s ♥♦ s❡♥t✐❞♦ ♣♦s✐t✐✈♦ ❡ ✺ ✉♥✐❞❛❞❡s ♥♦ s❡♥t✐❞♦ ♥❡❣❛t✐✈♦✱ t❡♠♦s q✉❡ ♦ ♠❡s♠♦ s❡ ❡♥❝♦♥tr❛ ♥❛ ♣♦s✐çã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ✸ ✉♥✐❞❛❞❡s ♥♦ s❡♥t✐❞♦ ♣♦s✐t✐✈♦✳ ❙❡✱ ♣♦ré♠✱ ♦ ❝♦r♣♦ s❡ ❞❡s❧♦❝❛ ✽ ✉♥✐❞❛❞❡s ♥♦ s❡♥t✐❞♦ ♣♦s✐t✐✈♦ ❡ ✶✵ ✉♥✐❞❛❞❡s ♥♦ s❡♥t✐❞♦ ♥❡❣❛t✐✈♦✱ ❡❧❡ s❡ ❡♥❝♦♥tr❛rá ♥❛ ♣♦s✐çã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ✷ ✉♥✐❞❛❞❡s ♥♦ s❡♥t✐❞♦ ♥❡❣❛t✐✈♦✳ ❆r✐t♠❡t✐❝❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛s ❞✉❛s s✐t✉❛çõ❡s ❛❝✐♠❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ 8−5 = 3 ❡ 8−10 =−2✳
❖❜s❡r✈❡ q✉❡ ✉s❛♠♦s ♦ sí♠❜♦❧♦ −2♣❛r❛ r❡♣r❡s❡♥t❛r ✷ ✉♥✐❞❛❞❡s à ❡sq✉❡r❞❛ ❞❛ ♦r✐❣❡♠✳ ❆ss✐♠✱ s❡♠♣r❡ q✉❡ q✉✐s❡r♠♦s r❡♣r❡s❡♥t❛r ❛❧❣✉♠ ♥ú♠❡r♦ à ❡sq✉❡r❞❛ ❞❛ ♦r✐❣❡♠ ✭♦ ③❡r♦✮✱ ♦ ♣r❡❝❡❞❡r❡♠♦s ♣❡❧♦ s✐♥❛❧ ❞❡ ✏✕✑✳
❈❆P❮❚❯▲❖ ✷✳ ◆Ú▼❊❘❖❙ ■◆❚❊■❘❖❙ ✶✸
P♦rt❛♥t♦✱ s❡❣✉♥❞♦ ❈❛r❛ç❛ ❬✸❪✱ é ♣♦ssí✈❡❧ ❡st❡♥❞❡r ♦ ❝♦♥❥✉♥t♦ N ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣❛r❛ ❝r✐❛r ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ♦ q✉❛❧ s❡rá r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❧❡tr❛Z✳ ❖ ③❡r♦ t❛♠❜é♠ s❡rá ✉♠ ❡❧❡♠❡♥t♦ ✭❛ ♦r✐❣❡♠✮ ❞❡ss❡ ❝♦♥❥✉♥t♦✳ ❆ss✐♠✱ ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s sã♦✿
. . . ,−3,−2,−1,0,1,2,3, . . .
◆♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ✈✐♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ N ❢♦✐ ❝❛r❛❝t❡r✐③❛❞♦ ♣❡❧❛ ❡①✐stê♥❝✐❛ ❞♦ s✉❝❡ss♦r✱ ✐st♦ é✱ t♦❞♦s ♦s ♥ú♠❡r♦s ❞♦ ❝♦♥❥✉♥t♦✱ ❝♦♠ ❡①❝❡çã♦ ❞♦ ♥ú♠❡r♦ ✶✱ ❡r❛♠ s✉❝❡ss♦r ❞❡ ❛❧❣✉é♠✳ ◆♦ ❝♦♥❥✉♥t♦Z✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ t♦❞♦ ❡❧❡♠❡♥t♦ t❡♠ s✉❝❡ss♦r ❡ ❛♥t❡❝❡ss♦r✱ ✐st♦ é✱ ❞❛❞♦ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦n✱ s❡♠♣r❡ ❡①✐st❡♠ s❡✉ s✉❝❡ss♦rn+1❡ s❡✉ ❛♥t❡❝❡ss♦rn−1✳ ❖❜s❡r✈❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ Z é ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ✭q✉❡ ❞❛q✉✐ ♣♦r ❞✐❛♥t❡ t❛♠❜é♠ s❡rã♦ ❝❤❛♠❛❞♦s ❞❡ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✮✱ ♦ ③❡r♦ ❡ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❛❝r❡s❝✐❞♦s ❞♦ s✐♥❛❧ ♥❡❣❛t✐✈♦ ✭♦s q✉❛✐s ❝❤❛♠❛r❡♠♦s ❞❡ ✐♥t❡✐r♦s ♥❡❣❛t✐✈♦s✮✳
✷✳✷ ❆ ❈♦rr❡s♣♦♥❞ê♥❝✐❛
N
↔
Z
❆♦ ♦❜s❡r✈❛r♠♦s ♦s ❡❧❡♠❡♥t♦s ❞♦s ❝♦♥❥✉♥t♦s N ❡ Z✱ ♣♦❞❡♠♦s ✐♥❢❡r✐r✱ ✐♥t✉✐t✐✈❛♠❡♥t❡✱ q✉❡ ♦ ❝♦♥❥✉♥t♦ Z t❡♠ ♠❛✐s ❡❧❡♠❡♥t♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ N✱ ✈✐st♦ q✉❡ ❛❧é♠ ❞♦s ♥ú♠❡r♦s ♣♦s✐t✐✈♦s ❡❧❡ ❛✐♥❞❛ ❝♦♥té♠ ♦s ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s ❡ ♦ ③❡r♦✳ P♦ré♠✱ ❛♦ ❝♦♥trár✐♦ ❞♦ q✉❡ ❛ ♥♦ss❛ ✐♥t✉✐çã♦ ♥♦s ❞✐③✱ ♣♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ss❡s ❞♦✐s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s✳
P❛r❛ ✐ss♦✱ ❜❛st❛ ❝♦rr❡s♣♦♥❞❡r♠♦s ♦s ♥ú♠❡r♦s ♣❛r❡s ❛♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ✭❡ ♦ ③❡r♦✮ ❡ ♦s ♥ú♠❡r♦s í♠♣❛r❡s ❛♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥❡❣❛t✐✈♦s✿ 1 ↔ −1✱ 2 ↔ 0✱ 3↔ −2✱ 4↔1✱ 5↔ −3✱ 6↔2✱ 7↔ −4✱ 8↔3✱. . .✳
❈❛♣ít✉❧♦ ✸
◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s
✸✳✶ ▼❡❞✐çã♦ ❞❡ ❙❡❣♠❡♥t♦s
❆ss✐♠ ❝♦♠♦ ❝♦♥t❛r✱ ♠❡❞✐r é ✉♠❛ ♦♣❡r❛çã♦ q✉❡ ✉t✐❧✐③❛♠♦s ❜❛st❛♥t❡ ♥♦ ♥♦ss♦ ❞✐❛✲❛✲❞✐❛✳ ❊♠ ♠✉✐t❛s s✐t✉❛çõ❡s✱ ❤á ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❢❛③❡r ♠❡❞✐❞❛s✿ ♦ ❝♦♠❡r❝✐❛♥t❡✱ q✉❛♥❞♦ ♣r❡❝✐s❛ s❛❜❡r ❛ q✉❛♥t✐❞❛❞❡ ❝❡rt❛ ❞❡ ✉♠ ♣r♦❞✉t♦ q✉❡ ❡stá ✈❡♥❞❡♥❞♦❀ ♦ ❛❣r✐❝✉❧t♦r✱ ❛♦ ❝❛❧❝✉❧❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ s❡♠❡♥t❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ❧❛♥ç❛r à t❡rr❛ ❞❡ q✉❡ ❞✐s♣õ❡❀ ♦ ♦♣❡rár✐♦✱ ♣❛r❛ s❛❜❡r s❡ ❡stá ❡①❡❝✉t❛♥❞♦ ❝♦rr❡t❛♠❡♥t❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ s❡r✈✐ç♦❀ ❛ ❞♦♥❛✲❞❡✲❝❛s❛✱ ♣❛r❛ s❛❜❡r ❛ ❞♦s❛❣❡♠ ❞❡ r❡♠é❞✐♦ q✉❡ ♦ ♠é❞✐❝♦ ♣❛ss♦✉ ♣❛r❛ s❡✉ ✜❧❤♦✳ ❊♥✜♠✱ ❡♠ ♠✉✐t❛s s✐t✉❛çõ❡s ❞✐ár✐❛s✱ ♣r❡❝✐s❛♠♦s r❡❛❧✐③❛r ✉♠❛ ♠❡❞✐çã♦✳ ▼❛s ♦ q✉❡ é ♠❡❞✐r❄ ▼❡❞✐r s✐❣♥✐✜❝❛ ❝♦♠♣❛r❛r ❣r❛♥❞❡③❛s ❞❡ ♠❡s♠❛ ❡s♣é❝✐❡✳ ◆ã♦ s❡ tr❛t❛ ❞❡ ✉♠❛ s✐♠♣❧❡s ❝♦♠♣❛r❛çã♦ ♣❛r❛ ✈❡r q✉❡♠ é ♠❛✐♦r✳ ▼❛s ❞❡ ✈❡r q✉❛♥t❛s ✈❡③❡s ✉♠ ✏❝❛❜❡✑ ♥♦ ♦✉tr♦✳
❙❡ q✉❡r❡♠♦s ❝♦♠♣❛r❛r✱ ♣♦r ❡①❡♠♣❧♦✱ ♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s s❡❣♠❡♥t♦sAB❡CD❛❜❛✐①♦✱ ✈❡r✐✜❝❛♠♦s q✉❛♥t❛s ✈❡③❡s CD ❝❛❜❡ ❡♠ AB✳ ❆ss✐♠✱ s❡ ❞✐ss❡r♠♦s q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ CD ✭♦ q✉❛❧ ❞❡♥♦t❛r❡♠♦s ♣♦rCD✮ é ❞❡ ✶ ✉♥✐❞❛❞❡ ✭1u✮✱ ♦ s❡❣♠❡♥t♦AB t❡♠ ❝♦♠♣r✐♠❡♥t♦ ✐❣✉❛❧ ❛3u ✭❋✐❣✉r❛ ✸✳✶✮✳
❋✐❣✉r❛ ✸✳✶✿ ▼❡❞✐❞❛ ❞❡ ✉♠ s❡❣♠❡♥t♦
P♦rt❛♥t♦✱ é ♥❡❝❡ssár✐♦ ❡st❛❜❡❧❡❝❡r♠♦s ✉♠❛ ✉♥✐❞❛❞❡ ♣❛❞rã♦ ❞❡ ♠❡❞✐❞❛ ♣❛r❛ t♦❞❛s ❛s ❣r❛♥❞❡③❛s ❞❡ ♠❡s♠❛ ❡s♣é❝✐❡ ✭❝♦♠♦ ♦ ❝❡♥tí♠❡tr♦✱ ♠❡tr♦✱ ❧✐tr♦✱ ❡t❝✳✮ ❡✱ ❞❡♣♦✐s✱ ✈❡r✐✜❝❛r q✉❛♥t❛s ✈❡③❡s ❛ ✉♥✐❞❛❞❡ ❝❛❜❡ ♥♦ ♦❜❥❡t♦ ❛ s❡r ♠❡❞✐❞♦✳ ❊st❡ ♥ú♠❡r♦ ✭q✉❛♥t✐❞❛❞❡ ❞❡ ✈❡③❡s q✉❡ ❛ ✉♥✐❞❛❞❡ ❝❛❜❡ ♥♦ ♦❜❥❡t♦✮ ❝❤❛♠❛✲s❡ ♠❡❞✐❞❛ ❞❛ ❣r❛♥❞❡③❛ ❡♠ r❡❧❛çã♦ ❛ ❡ss❛ ✉♥✐❞❛❞❡✳ ◆♦t❡ q✉❡✱ ❛♣❡s❛r ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ s❡❣♠❡♥t♦ s❡r ❝♦♥st❛♥t❡✱ s✉❛ ♠❡❞✐❞❛ ❞❡♣❡♥❞❡ ❞❛ ✉♥✐❞❛❞❡ ❛❞♦t❛❞❛✳ ◆♦ ❡①❡♠♣❧♦ ❞❛ ✜❣✉r❛ ❛❝✐♠❛✱ s❡ ❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ❛❞♦t❛❞❛✱ ❛♦
❈❆P❮❚❯▲❖ ✸✳ ◆Ú▼❊❘❖❙ ❘❆❈■❖◆❆■❙ ✶✺
✐♥✈és ❞❡ s❡r ♦ s❡❣♠❡♥t♦ CD✱ ❢♦ss❡ ✉♠ s❡❣♠❡♥t♦ ❝✉❥♦ ❝♦♠♣r✐♠❡♥t♦ é ❛ ♠❡t❛❞❡ ❞❡CD✱ ❛ ♠❡❞✐❞❛ ❞❡AB s❡r✐❛ 6u❡ ♥ã♦ 3u✳
❊♠ r❡s✉♠♦✱ ♣❛r❛ r❡❛❧✐③❛r♠♦s ✉♠❛ ♠❡❞✐❞❛✱ ❢❛③❡♠♦s ❡♠ três ❡t❛♣❛s✿ ❡s❝♦❧❤❛ ❞❛ ✉♥✐❞❛❞❡❀ ❝♦♠♣❛r❛çã♦ ❞♦ ♦❜❥❡t♦ ❝♦♠ ❛ ✉♥✐❞❛❞❡❀ ❡①♣r❡ssã♦ ❞♦ r❡s✉❧t❛❞♦ ❞❡ss❛ ❝♦♠♣❛r❛çã♦ ♣♦r ♠❡✐♦ ❞❡ ✉♠ ♥ú♠❡r♦✳
➱ ✐♠♣♦rt❛♥t❡ ♦❜s❡r✈❛r♠♦s q✉❡✱ ❛♣❡s❛r ❞❛ ❡s❝♦❧❤❛ ❞❛ ✉♥✐❞❛❞❡ s❡r ❢❡✐t❛ ❞❛ ♠❛♥❡✐r❛ q✉❡ q✉✐s❡r♠♦s✱ é ✐♥t❡r❡ss❛♥t❡ q✉❡ ❡❧❛ s❡❥❛ ❢❡✐t❛ ✈✐s❛♥❞♦ s✐♠♣❧✐✜❝❛r ♦ r❡s✉❧t❛❞♦✳ P♦r ❡①❡♠♣❧♦✱ s❡r✐❛ ❡①tr❡♠❛♠❡♥t❡ ✐♥❝ô♠♦❞♦ ❝❛❧❝✉❧❛r ❛ ♠❡❞✐❞❛ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞✉❛s ❝✐❞❛❞❡s ✉s❛♥❞♦ ♦ ♠✐❧í♠❡tr♦ ❝♦♠♦ ✉♥✐❞❛❞❡✳
✸✳✷ ❙✉❜❞✐✈✐sã♦ ❞❛ ❯♥✐❞❛❞❡
➚s ✈❡③❡s✱ é ♥❡❝❡ssár✐♦ s✉❜❞✐✈✐❞✐r ❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ♥✉♠ ❝❡rt♦ ♥ú♠❡r♦ ❞❡ ♣❛rt❡s ✐❣✉❛✐s✳ ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ ✭❋✐❣✉r❛ ✸✳✷✮✿
❊❳❊▼P▲❖ ✶✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ✉♠ s❡❣♠❡♥t♦ AB ♠❡❞✐❞♦ ❝♦♠ ❛ ✉♥✐❞❛❞❡ CD = u✱ ♠❡❞❡ ✸✳ ❙❡ ❞✐✈✐❞✐r♠♦s ❛ ✉♥✐❞❛❞❡CD ❡♠ ✹ ♣❛rt❡s ✐❣✉❛✐s ❡ t♦♠❛r♠♦s ♣❛r❛ ❛ ♥♦✈❛ ✉♥✐❞❛❞❡ ♦ s❡❣♠❡♥t♦ CE✱ ❞❡ ♠❡❞✐❞❛ u′✱ t❡r❡♠♦s q✉❡ ❛ ♠❡❞✐❞❛ ❞❡
AB s❡rá ✐❣✉❛❧ ❛ ✶✷✳ ❆❧é♠ ❞✐ss♦✱ ♦ r❡s✉❧t❛❞♦ ❞❛ ♠❡❞✐çã♦ ❝♦♠ ❛ ✉♥✐❞❛❞❡ u t❛♥t♦ ♣♦❞❡ s❡r ❡①♣r❡ss♦ ♣❡❧♦ ♥ú♠❡r♦ ✸ ❝♦♠♦ ♣❡❧❛ r❛③ã♦ ❞♦s ❞♦✐s ♥ú♠❡r♦s ✶✷ ❡ ✹✱ ✐st♦ é✱ ♣❡❧♦ q✉♦❝✐❡♥t❡ 12 : 4 ♦✉ 12
4 ✳
❋✐❣✉r❛ ✸✳✷✿ ▼❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦ AB
❉❡ ♠❛♥❡✐r❛ ❣❡r❛❧✱ s❡ ✉♠❛ ❣r❛♥❞❡③❛✱ ♠❡❞✐❞❛ ❝♦♠ ❛ ✉♥✐❞❛❞❡u✱ ♠❡❞❡m✱ ❡ s✉❜❞✐✈✐❞✐r♠♦s u❡♠ n ♣❛rt❡s ✐❣✉❛✐s✱ ❛ ♠❡❞✐❞❛ ❞❛ ♠❡s♠❛ ❣r❛♥❞❡③❛✱ ❝♦♠ ❛ ♠❡s♠❛ ✉♥✐❞❛❞❡u✱ ❡①♣r✐♠❡✲s❡ ♣❡❧❛ r❛③ã♦ ❞♦s ❞♦✐s ♥ú♠❡r♦sM =m.n❡ n✳ ❆r✐t♠❡t✐❝❛♠❡♥t❡✱ t❡♠♦s m = M
n ✳
❖ ❝❛s♦ q✉❡ ❛♥❛❧✐s❛♠♦s✱ ♣♦ré♠✱ é ✉♠ ❝❛s♦ ♠✉✐t♦ r❛r♦ ❞❡ ❛❝♦♥t❡❝❡r✿ ♦ ❢❛t♦ ❞❛ ✉♥✐❞❛❞❡ ❝❛❜❡r ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❞❡ ✈❡③❡s ♥❛ ❣r❛♥❞❡③❛ ❛ s❡ ♠❡❞✐r✳ ❖ ❝❛s♦ ♠❛✐s ❝♦♠✉♠ é ♦ ❞♦ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r✱ ♦♥❞❡ ❛ ✉♥✐❞❛❞❡ ♥ã♦ ❝❛❜❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❞❡ ✈❡③❡s ❡♠AB✳
❊❳❊▼P▲❖ ✷✳ ❙✉♣♦♥❤❛♠♦s ♦s s❡❣♠❡♥t♦sAF ❡ CD ❝♦♥❢♦r♠❡ ❋✐❣✉r❛ ✸✳✸✳
❈❆P❮❚❯▲❖ ✸✳ ◆Ú▼❊❘❖❙ ❘❆❈■❖◆❆■❙ ✶✻
❋✐❣✉r❛ ✸✳✸✿ ▼❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦AF
❙❡ ❞✐✈✐❞✐r♠♦s ♦ s❡❣♠❡♥t♦ CD ✭❛ ✉♥✐❞❛❞❡✮ ❡♠ ✹ ♣❛rt❡s ✐❣✉❛✐s ❛ ♥♦✈❛ ✉♥✐❞❛❞❡ ❝❛❜❡rá ✶✶ ✈❡③❡s ❡♠ AF✳ ◆❡ss❡ ❝❛s♦✱ ❛ ♠❡❞✐❞❛ ❞❡ AF ❡♠ r❡❧❛çã♦ à ♥♦✈❛ ✉♥✐❞❛❞❡ é ✶✶✳ ▼❛s ❡♠ r❡❧❛çã♦ à ❛♥t✐❣❛ ✉♥✐❞❛❞❡ CD✱ ❝♦♠♦ ♣♦❞❡♠♦s ❡①♣r✐♠✐r ❛ ♠❡❞✐❞❛ ❞❡ AF❄ ❙❡r✐❛ ❛ r❛③ã♦ ❡♥tr❡ ♦s ♥ú♠❡r♦s ✶✶ ❡ ✹✱ ♠❛s ❡♠N ❡ss❛ r❛③ã♦ ♥ã♦ ❡①✐st❡✳
❆ r❡s♣♦st❛ ❛ ❡ss❛ ♣❡r❣✉♥t❛✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❈❛r❛ç❛ ❬✸❪✱ ♥♦s ❧❡✈❛ à ❝r✐❛çã♦ ❞❡ ✉♠ ♥♦✈♦ t✐♣♦ ❞❡ ♥ú♠❡r♦✳ ❊ss❛ ❝♦♥str✉çã♦ ❞❡✈❡ s❡r ❢❡✐t❛ ❞❡ ♠♦❞♦ q✉❡ ❝♦♠ ♦s ♥♦✈♦s ♥ú♠❡r♦s✱ s❡❥❛♠ ❛❜r❛♥❣✐❞❛s t♦❞❛s ❛s ❤✐♣ót❡s❡s ❞❡ ♠❡❞✐çã♦✱ s❡❥❛♠ ❡❧❛s ❞♦ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦ ♦✉ ❞♦ s❡❣✉♥❞♦ ❡①❡♠♣❧♦❀ ❛❧é♠ ❞✐ss♦✱ ♦s ♥♦✈♦s ♥ú♠❡r♦s s❡ r❡❞✉③❡♠ ❛♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s q✉❛♥❞♦ ❛ ♠❡❞✐çã♦ s❡ ❞❡r ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❛♦ ♥♦ss♦ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦✳
✸✳✸ ❖ ❈♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s
❙❡❥❛♠ ♦s ❞♦✐s s❡❣♠❡♥t♦s ❞❡ r❡t❛ AB ❡ CD✱ ♦♥❞❡ ❡♠ ❝❛❞❛ ✉♠ ❞❡❧❡s ❝❛❜❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❞❡ ✈❡③❡s ♦ s❡❣♠❡♥t♦ ❞❡ ♠❡❞✐❞❛ u ✕ AB ❝♦♥té♠ m ✈❡③❡s ❡ CD ❝♦♥té♠ n ✈❡③❡s✳ ❉✐③❡♠♦s q✉❡ ❛ ♠❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦ AB✱ t♦♠❛♥❞♦ CD ❝♦♠♦ ✉♥✐❞❛❞❡✱ é ♦ ♥ú♠❡r♦ m
n ❡ ❡s❝r❡✈❡♠♦s
AB = m n ·CD,
q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦sm❡n✭❞❡s❞❡ q✉❡n♥ã♦ s❡❥❛ ♥✉❧♦✮✳ ◆♦t❡ q✉❡ s❡m ❢♦r ❞✐✈✐sí✈❡❧ ♣♦rn✱ ♦ ♥ú♠❡r♦ m
n ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ q✉❡ é q✉♦❝✐❡♥t❡ ❞❛ ❞✐✈✐sã♦❀ s❡m ♥ã♦ ❢♦r ❞✐✈✐sí✈❡❧ ♣♦r n✱ ♦ ♥ú♠❡r♦ m
n é ❞✐t♦ ❢r❛❝✐♦♥ár✐♦✳ ❊♠ q✉❛❧q✉❡r ✉♠ ❞♦s ❝❛s♦s✱ ♦ ♥ú♠❡r♦ m
n é ❞✐t♦ r❛❝✐♦♥❛❧ ✭❛♦ ♥ú♠❡r♦ m ❝❤❛♠❛♠♦s ♥✉♠❡r❛❞♦r ❡ ❛♦ n✱ ❞❡♥♦♠✐♥❛❞♦r✮✳ ❊st❛♠♦s✱ ❛❣♦r❛✱ ❞✐❛♥t❡ ❞❡ ✉♠ ♥♦✈♦ ❝♦♥❥✉♥t♦ ♥✉♠ér✐❝♦ ✕ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ✕ q✉❡ ❝♦♠♣r❡❡♥❞❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡ ♠❛✐s ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s ♥ú♠❡r♦s ❢r❛❝✐♦♥ár✐♦s✶✳ ❈♦♠ ❛ ❝r✐❛çã♦ ❞❡ss❡s ♥ú♠❡r♦s✱ ♥♦s ♣❛r❡❝❡ q✉❡ é s❡♠♣r❡ ♣♦ssí✈❡❧ ❡①♣r✐♠✐r ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ s❡❣♠❡♥t♦ t♦♠❛♥❞♦ ♦✉tr♦ ❝♦♠♦ ✉♥✐❞❛❞❡✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❞✐✈✐sã♦ ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s m ❡ n ♣♦❞❡ ❛❣♦r❛ s❡♠♣r❡ ❡①♣r✐♠✐r✲s❡ s✐♠❜♦❧✐❝❛♠❡♥t❡ ♣❡❧♦
✶❆♣❡s❛r ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❝♦♠♣r❡❡♥❞❡r t♦❞♦s ❡ss❡s ♥ú♠❡r♦s ✕ ♣♦s✐t✐✈♦s ❡ ♥❡❣❛t✐✈♦s
