❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠
▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s
❙✉s✐❛♥❡ ❇❡③❡rr❛ ❈❛✐①❡t❛
❙✉s✐❛♥❡ ❇❡③❡rr❛ ❈❛✐①❡t❛
❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢❛✳ ❉r❛✳ ❆❧✐♥❡ ●♦♠❡s ❞❛ ❙✐❧✈❛ P✐♥t♦
❇r❛sí❧✐❛
Ficha catalográfica elaborada automaticamente, com os dados fornecidos pelo(a) autor(a)
CC138a
Caixeta, Susiane Bezerra
Algoritmo da divisão de Euclides / Susiane Bezerra Caixeta; orientador Aline Gomes da Silva Pinto. -- Brasília, 2016.
80 p.
Dissertação (Mestrado - Mestrado Profissional em Matemática) -- Universidade de Brasília, 2016.
❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳ ➱ ♣r♦✐❜✐❞❛ ❛ r❡♣r♦❞✉çã♦ t♦t❛❧ ♦✉ ♣❛r❝✐❛❧ ❞❡st❡ tr❛❜❛❧❤♦ s❡♠ ❛ ❛✉t♦r✐③❛çã♦ ❞❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ❞♦ ❛✉t♦r ❡ ❞♦ ♦r✐❡♥t❛❞♦r✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣♦r t❡r s✐❞♦ ♠✉✐t♦ ❜❡♥❡✈♦❧❡♥t❡ ❝♦♠✐❣♦✳ ❙❡♠♣r❡ ♠❡ ❝♦♥s✐❞❡r❡✐ ✉♠❛ ♣❡ss♦❛ s♦rt✉❞❛✱ ♠❛s s❡✐ q✉❡ t✉❞♦ q✉❡ ❛❝♦♥t❡❝❡✉ ❢♦✐ ♣♦r ♠❡✐♦ ❞❛s ❙✉❛s ✐♥✜♥✐t❛s ❜ê♥çã♦s✳
❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♣❛✐✱ ❖r❝✐✱ ♣❡❧♦ ❛♠♦r ❡ ❝❛r✐♥❤♦✱ ❡ ♣♦r s❡r ♦ ♠❡❧❤♦r ❡①❡♠♣❧♦ ❞❡ ❝❛rát❡r ❡ ✐♥t❡❧✐❣ê♥❝✐❛ q✉❡ ♣♦❞❡r✐❛ ❡①✐st✐r✳ ❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ♠ã❡✱ ❙✉❡❧②✱ q✉❡ s❡♠♣r❡ ♠❡ ♠♦str♦✉ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ tr❛❜❛❧❤♦ ❡ ❞❡ ❝♦♠♦ ❛ s✐♠♣❛t✐❛ ❡ ❛ ❡♠♣❛t✐❛ sã♦ ✐♠♣♦rt❛♥t❡s ♥❛ ♥♦ss❛ ✈✐❞❛✳ ❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ✐r♠ã✱ ❙✉❡❧❡♥✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ❛♣♦✐♦ ❞❛❞♦s ❞❡✲ s✐♥t❡r❡ss❛❞❛♠❡♥t❡✳ ❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✱ q✉❡ ♠❡s♠♦ ♥ã♦ ❝♦♥tr✐❜✉✐♥❞♦ ❞✐r❡t❛♠❡♥t❡ ♥❡ss❡ tr❛❜❛❧❤♦✱ ♣♦r ❢❛③❡r❡♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ✈✐❞❛✳
❆♦s ♠❡✉s ❝♦❧❡❣❛s ❡ ❛♠✐❣♦s ❞♦ P❘❖❋▼❆❚ ✷✵✶✹✳ ◆❡ss❡s ❞♦✐s ❛♥♦s ❡♠ q✉❡ ♥♦s ❡♥✲ ❝♦♥tr❛♠♦s ❞✉r❛♥t❡ ❛s ❛✉❧❛s ♦✉ ♥❛s ❝♦♠❡♠♦r❛çõ❡s ✭❧❛♥❝❤❡s ❡ ❝❤ás ❞❡ ❢r❛❧❞❛s✮ ❢♦r♠❛♠♦s ✉♠❛ ❛♠✐③❛❞❡ tã♦ ❜♦♥✐t❛ ❡ tã♦ s✐♥❝❡r❛ q✉❡ ❥❛♠❛✐s ❡sq✉❡❝❡r❡✐✳ ▼❡ s✐♥t♦ ♣r✐✈✐❧❡❣✐❛❞❛ ♣♦r t❡r ❢❡✐t♦ ♣❛rt❡ ❞❡ss❛ t✉r♠❛✳ ❖❜r✐❣❛❞❛ ❛ t♦❞♦s✦
❯♠ ❣r❛♥❞❡ ❛❣r❛❞❡❝✐♠❡♥t♦ ❛ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ ❉r❛✳ ❆❧✐♥❡✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❞❡❞✐✲ ❝❛çã♦ ❡ ♣♦r ♠❡ ❛❧❡♥t❛r ❡♠ ♠♦♠❡♥t♦s q✉❡ ❛❝❤❡✐ q✉❡ ♥ã♦ ❝♦♥s❡❣✉✐r✐❛✳ ❙❛✐❜❛ q✉❡ s✉❛ ✐♥t❡❧✐❣ê♥❝✐❛ ❡ ❝♦♥❤❡❝✐♠❡♥t♦ s❡♠♣r❡ ♠❡ ✐♥s♣✐r❛r❛♠✱ ❞❡s❞❡ ♦ ♣❡rí♦❞♦ ❞❛ ❣r❛❞✉❛çã♦✳
❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥❇✱ q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♥♦ P❘❖❋▼❆❚✿ ▲✐♥❡✉ ◆❡t♦✱ ❆r② ❱❛s❝♦♥❝❡❧♦s✱ ❉✐❡❣♦ ▼❛rq✉❡s✱ ❆♥❣❡❧ ❘♦❞♦❧❢♦ ❇❛✐❣♦rr✐✱ ❍❡❧❞❡r ▼❛t♦s✱ ❈❛r❧♦s ❆❧❜❡rt♦✱ ▼❛✉r♦ ❘❛❜❡❧♦✱ ❑❡❧❧❝✐♦ ❖❧✐✈❡✐r❛ ❡ ♦ ❝♦♦r❞❡♥❛❞♦r ❘✉✐ ❙❡✐♠❡t③✳ ❖❜r✐❣❛❞❛ ♣♦r ♠❡ ✐♥❝❡♥t✐✈❛r❡♠ ❛ ❜✉s❝❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ s❡♠♣r❡ ♠❛✐s✦
❆❣r❛❞❡ç♦ t❛♠❜é♠ à ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❛ ❡st❡ tr❛❜❛❧❤♦✳
❘❡s✉♠♦
❖ ❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s✱ ❜❡♠ ❝♦♠♦ t♦❞♦s ♦s ❝♦♥t❡ú❞♦s ♠❛t❡♠át✐❝♦s ❛♣r❡s❡♥t❛❞♦s ♥❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛✱ ❞❡✈❡♠ s❡r ❧❡❝✐♦♥❛❞♦s ❞❡ ❢♦r♠❛ ❝♦♥t❡①t✉❛❧✐③❛❞❛✳ ■ss♦ ❢❛✈♦r❡❝❡ ♦ ❡st✉❞❛♥t❡✱ ❞❡ ❢♦r♠❛ q✉❡ ♦ ♠❡s♠♦ t❡♥❤❛ ✉♠ ❛♣r❡♥❞✐③❛❞♦ ♠❛✐s ❡✜❝✐✲ ❡♥t❡✳ ❊st❛ ❞✐ss❡rt❛çã♦ ✈✐s❛ ❢✉♥❞❛♠❡♥t❛r t❡♦r✐❝❛♠❡♥t❡ ❛ ♣❛rt❡ ♠❛t❡♠át✐❝❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ ❞✐s❝✉ssã♦✱ ❛♣❡r❢❡✐ç♦❛♥❞♦ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❞♦ ♣r♦❢❡ss♦r ♥♦ ❛ss✉♥t♦ ❡ ❢❛✈♦r❡❝❡♥❞♦ ❛ s✉❛ ❢♦r♠❛çã♦ ❝♦♥t✐♥✉❛❞❛✳ P❛r❛ ✐ss♦✱ s❡rã♦ ❝♦♥str✉í❞♦s ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❛❧é♠ ❞❡ ❞✐s❝♦rr❡r s♦❜r❡ ❞✐✈✐s✐❜✐✲ ❧✐❞❛❞❡✳ ❚♦❞♦s ❡ss❡s tó♣✐❝♦s s❡rã♦ ❝♦♠♣♦st♦s ❞❡ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ❢♦r♠❛❧✳ ❆❧é♠ ❞✐ss♦✱ ❡st❛ ❞✐ss❡rt❛çã♦ ♣r♦♣õ❡ ❛t✐✈✐❞❛❞❡s q✉❡ r❡❧❛❝✐♦♥❡♠ s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛ ❞♦ ❝♦t✐❞✐❛♥♦ ❝♦♠ ♦ t❡♠❛✱ ❞❡ ❢♦r♠❛ q✉❡ ♦s ❡st✉❞❛♥t❡s ♣♦ss❛♠ ❞❡s❝♦❜r✐r ♣♦r ♠❡✐♦ ❞❡ ❞✐s✲ ❝✉ssõ❡s ❡♠ ❣r✉♣♦ ❛ r❡s♦❧✉çã♦ ❞♦s ♠❡s♠♦s✳ ❉❡ss❛ ❢♦r♠❛✱ sã♦ ♣r♦♣♦st❛s ❛t✐✈✐❞❛❞❡s q✉❡ s❡❣✉❡♠ ✉♠❛ t❡♥❞ê♥❝✐❛ ♠❡t♦❞♦❧ó❣✐❝❛ ❞❡ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠ ❡♠ ❡❞✉❝❛çã♦ ♠❛t❡♠át✐❝❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✳
P❛❧❛✈r❛s✲❝❤❛✈❡
◆ú♠❡r♦s ♥❛t✉r❛✐s✱ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s✱ ❝❛❧❡♥❞ár✐♦✱ ♦ ♣r♦❜❧❡♠❛ ❞♦s três ♠❛r✐♥❤❡✐r♦s✱ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✳
❆❜str❛❝t
❊✉❝❧✐❞✬s ❞✐✈✐s✐♦♥ ❆❧❣♦r✐t❤♠ ❛s ✇❡❧❧ ❛s ❛❧❧ ♠❛t❤❡♠❛t✐❝❛❧ ❝♦♥t❡♥t ♣r❡s❡♥t❡❞ ✐♥ ❜❛s✐❝ ❡❞✉❝❛t✐♦♥ s❤♦✉❧❞ ❜❡ t❛✉❣❤t ✐♥ ❝♦♥t❡①t✳ ❚❤✐s ❢❛✈♦rs t❤❡ st✉❞❡♥t✱ s♦ t❤❛t ✐t ❤❛s ❛ ♠♦r❡ ❡✣❝✐❡♥t ❧❡❛r♥✐♥❣✳ ❚❤✐s ✇♦r❦ ❛✐♠s t♦ ♣r❡s❡♥t t❤❡ t❤❡♦r② ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ ❞✐s❝✉ss✐♦♥ ♦❢ t❤❡ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠✱ ✐♥ ♦t❤❡r t♦ ❣✐✈❡ s✉♣♣♦rt t♦ ▼❛t❤❡♠❛t✐❝ t❡❛❝❤❡rs ♦❢ ❢✉♥❞❛♠❡♥t❛❧ s❝❤♦♦❧ t♦ ✐♠♣r♦✈❡ t❤❡✐r ❦♥♦✇❧❡❞❣❡ ❛❜♦✉t t❤❡ ✐♥t❡❣❡r ♥✉♠❜❡rs✳ ❋♦r t❤✐s✱ ✇❡ ♣r❡s❡♥t t❤❡ ❢♦r♠❛❧ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♥❛t✉r❛❧ ♥✉♠❜❡rs ❛♥❞ ✐♥t❡❣❡r ♥✉♠❜❡rs ❛♥❞ ❛ ❢♦r♠❛❧ ♣r♦♦❢ ♦❢ ❊✉❝❧✐❞✬s ❞✐✈✐s✐♦♥ ❛❧❣♦r✐t♠✳ ■♥ t❤✐s ❞✐ss❡rt❛t✐♦♥✱ ✇❡ ❛❧s♦ ❛✐♠ t♦ ♣r♦♣♦s❡ ❛❝t✐✈✐t✐❡s t❤❛t ❝♦♥t❡①t✉❛❧✐③❡ t❤❡ t❤❡♠❡ ✐♥ ❡✈❡r②❞❛② s✐t✉❛t✐♦♥s✱ s♦ t❤❛t t❡❛❝❤❡rs ❝❛♥ ♠♦t✐✈❛t❡ t❤❡ st✉❞❡♥ts t♦ ❞✐s❝✉ss s♦♠❡ ❡✈❡r②❞❛② ♣r♦❜❧❡♠s ✐♥✈♦❧✈✐♥❣ ❊✉❝❧✐❞✬s ❛❧❣♦r✐t❤♠✱ ✇♦r❦✐♥❣ ✐♥ ❣r♦✉♣s✳ ❚❤❡ ❛❝t✐✈✐t✐❡s ❢♦❧❧♦✇ ❛ ♠❡t❤♦❞♦❧♦❣✐❝❛❧ t❡♥❞❡♥❝② ✐♥ ♠❛t❤ ❡❞✉❝❛t✐♦♥ ❦♥♦✇♥ ❛s ♣r♦❜❧❡♠ s♦❧✈✐♥❣✳
❑❡②✇♦r❞s
◆❛t✉r❛❧ ♥✉♠❜❡rs ✱ ✐♥t❡❣❡rs ✱ ❊✉❝❧✐❞✬s ❞✐✈✐s✐♦♥ ❛❧❣♦r✐t❤♠✱ ❝❛❧❡♥❞❛r✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ t❤r❡❡ s❛✐❧♦rs ✱ ♣r♦❜❧❡♠ s♦❧✈✐♥❣✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶✷
✶ ❍✐stór✐❝♦ ✶✺
✶✳✶ Pr✐♠ór❞✐♦s ❞❛ ♦♣❡r❛çã♦ ❞❡ ❞✐✈✐sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷ ❊✉❝❧✐❞❡s ❡ ❖s ❊❧❡♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✸ ◆ú♠❡r♦s ◆❛t✉r❛✐s ❡ ◆ú♠❡r♦s ■♥t❡✐r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷ ❈♦♥str✉çã♦ ❞♦s ◆ú♠❡r♦s ◆❛t✉r❛✐s ✶✾
✷✳✶ ❆①✐♦♠❛s ❞❡ P❡❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ❆❞✐çã♦ ❞❡ ◆ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ◆ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✹ ❘❡❧❛çã♦ ❞❡ ♦r❞❡♠ ❡♠N ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸ ❈♦♥str✉çã♦ ❞♦s ◆ú♠❡r♦s ■♥t❡✐r♦s ✸✷
✸✳✶ ❘❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✷ ❆❞✐çã♦ ❞❡ ◆ú♠❡r♦s ■♥t❡✐r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✸ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ◆ú♠❡r♦s ■♥t❡✐r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✹ ❘❡❧❛çã♦ ❞❡ ♦r❞❡♠ ❡♠Z ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✹ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✹✽
✹✳✶ ❉✐✈✐sã♦ ❞❡ ◆ú♠❡r♦s ■♥t❡✐r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✷ ❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
✺ ❆t✐✈✐❞❛❞❡s Pr♦♣♦st❛s ✺✸
✺✳✶ ❆t✐✈✐❞❛❞❡ ✶✿ ▼❛t❡♠át✐❝❛ ❡ ♦ ❈❛❧❡♥❞ár✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✺✳✶✳✶ ❍✐stór✐❛ ❞♦ ❈❛❧❡♥❞ár✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✺✳✶✳✷ ❆t✐✈✐❞❛❞❡ r❡❧❛❝✐♦♥❛♥❞♦ ♦ ❈❛❧❡♥❞ár✐♦ ●r❡❣♦r✐❛♥♦ ❝♦♠ ♦ ❆❧❣♦✲ r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✺✳✷ ❆t✐✈✐❞❛❞❡ ✷✿ ❆ ❍✐stór✐❛ ❞♦s ❚rês ▼❛r✐♥❤❡✐r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✺✳✷✳✶ ❖ ❍♦♠❡♠ q✉❡ ❈❛❧❝✉❧❛✈❛ ✲ ▼❛❧❜❛ ❚❛❤❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✺✳✷✳✷ ❆t✐✈✐❞❛❞❡ r❡❧❛❝✐♦♥❛♥❞♦ ❛ ❤✐stór✐❛ ❞♦s ❚rês ▼❛r✐♥❤❡✐r♦s ❝♦♠ ♦
❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✼✼
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✾
■♥tr♦❞✉çã♦
❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡ss❛ ❞✐ss❡rt❛çã♦ é ♣r♦♣♦r ❛♦s ♣r♦❢❡ss♦r❡s ✉♠ ♠❛t❡r✐❛❧ ❞❡ ❡s✲ t✉❞♦ ❜❡♠ ❢✉♥❞❛♠❡♥t❛❞♦ s♦❜r❡ ♦ ❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s✱ ❜❡♠ ❝♦♠♦ ❛t✐✈✐❞❛❞❡s ❝♦♥t❡①t✉❛❧✐③❛❞❛s ❡ ❞✐♥â♠✐❝❛s ❛ s❡r❡♠ ❛♣❧✐❝❛❞❛s ❛♦s ❡st✉❞❛♥t❡s✳
■♥❢❡❧✐③♠❡♥t❡✱ ♠✉✐t♦s ❡st✉❞❛♥t❡s ❛❧❝❛♥ç❛♠ ♦ s❡①t♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ s❡♠ ♦ ❞♦♠í♥✐♦ s❛t✐s❢❛tór✐♦ ❞❛ ♦♣❡r❛çã♦ ❞❡ ❞✐✈✐sã♦✳ P♦✉❝♦s ❛♣❧✐❝❛♠ ♦ ❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐✲ sã♦✱ ♠❛s ♦ ❢❛③❡♠ ❞❡ ❢♦r♠❛ ♠❡❝❛♥✐③❛❞❛✱ s❡♠ ❝♦♠♣r❡❡♥❞❡r ♦ s❡✉ r❡❛❧ s✐❣♥✐✜❝❛❞♦✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱ ❛ ♠❛✐♦r✐❛ ❞♦s ❡st✉❞❛♥t❡s ♥ã♦ ❝♦♥s❡❣✉❡♠ r❡s♦❧✈❡r s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛ ❞♦ ❝♦t✐❞✐❛♥♦ q✉❡ ❡♥✈♦❧✈❡♠ ♦ t❡♠❛✳
❙♦✉ ♣r♦❢❡ss♦r❛ ❞❡ ♠❛t❡♠át✐❝❛ ❞❛ ❙❡❝r❡t❛r✐❛ ❞❡ ❊st❛❞♦ ❞❡ ❊❞✉❝❛çã♦ ❞♦ ❉✐str✐t♦ ❋❡❞❡r❛❧ ✭❙❊❊❉❋✮✱ ❡ ❞✉r❛♥t❡ ❛ ♠✐♥❤❛ ❡①♣❡r✐ê♥❝✐❛ ❡♠ s❛❧❛✱ ❛♦ ❧❡❝✐♦♥❛r s♦❜r❡ ♦ t❡♠❛✱ ♣✉❞❡ ♥♦t❛r ❡ss❛ ❞✐✜❝✉❧❞❛❞❡ ♣♦r ♣❛rt❡ ❞♦s ❡st✉❞❛♥t❡s✳ P♦r ✈❡③❡s✱ ❡❧❡s ♠❡s♠♦s ❢❛❧❛✈❛♠ s♦❜r❡ ✐ss♦✱ ❛✜r♠❛♥❞♦ q✉❡ ♥ã♦ t✐♥❤❛♠ ❡♥t❡♥❞✐❞♦ ❝♦♠♦ ❡r❛ ❢❡✐t❛ ❛ ♦♣❡r❛çã♦ ❞❡ ❞✐✈✐sã♦ ❡✴♦✉ ♥ã♦ s❛❜✐❛♠ ♦ ♣♦rq✉ê ❞❛ ✉t✐❧✐③❛çã♦ ❞♦s ♣❛ss♦s ❡♥s✐♥❛❞♦s ❡♠ s❛❧❛ ♣❛r❛ ❡ss❡ ✜♠✳ ❊ss❛ s❡q✉ê♥❝✐❛ ❞❡ ♣❛ss♦s é ♦ q✉❡ ❝❤❛♠❛♠♦s ❞❡ ❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s✱ ❡ ❡ss❛ ♥♦çã♦ ❣❡r❛❧♠❡♥t❡ é ❛♣r❡s❡♥t❛❞❛ ❛♦s ❡st✉❞❛♥t❡s ♥♦ ✹♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✳
▼❡s♠♦ ❝♦♠ ❡ss❛ ❛♣r❡s❡♥t❛çã♦✱ q✉❛♥❞♦ ♦s ❡st✉❞❛♥t❡s ❛❧❝❛♥ç❛♠ ♦ ✻♦ ❛♥♦ ❞♦ ❡♥s✐♥♦
❢✉♥❞❛♠❡♥t❛❧✱ ♥♦t❛✲s❡ ❛ ❣r❛♥❞❡ ❞✐✜❝✉❧❞❛❞❡ ❝♦♠ ♦ ❞♦♠í♥✐♦ ❡ ❛ ✉t✐❧✐③❛çã♦ ❞❡ss❡s ❝♦♥❝❡✐t♦s ❡ r❡❣r❛s✳ ■ss♦ ♣♦❞❡ ❡st❛r ♦❝♦rr❡♥❞♦ ❞❡✈✐❞♦ às ❣r❛♥❞❡s ♠✉❞❛♥ç❛s q✉❡ ♦❝♦rr❡♠ ❞♦ ✺♦
♣❛r❛ ♦ ✻♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✳ ◆❡ss❛ ❢❛s❡✱ ♦s ❡st✉❞❛♥t❡s s❛❡♠ ❞♦s ❛♥♦s ✐♥✐❝✐❛✐s✱
❝✉❥♦s ♣r♦❢❡ss♦r❡s sã♦ ♣❡❞❛❣♦❣♦s✱ ♣❛r❛ ✐♥❣r❡ss❛r❡♠ ♥♦s ❛♥♦s ✜♥❛✐s✱ ❝✉❥♦s ♣r♦❢❡ss♦r❡s sã♦ ❧✐❝❡♥❝✐❛❞♦s ❡♠ s✉❛s r❡s♣❡❝t✐✈❛s ár❡❛s✳ ❈♦♠ ✐ss♦✱ ❤á ✉♠ ❣r❛♥❞❡ ❞❡s❛✜♦ ♣♦st♦ ❛♦s ♣r♦❢❡ss♦r❡s q✉❡ r❡❝❡❜❡♠ ❡ss❡s ❡st✉❞❛♥t❡s ♥♦ ✻♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ♣♦✐s ❛
❛❞❛♣t❛çã♦ ❞❡✈❡ ♦❝♦rr❡r ❞❡ ♠♦❞♦ ♣r♦❣r❡ss✐✈♦ ❡ s❛t✐s❢❛tór✐♦✱ t❛♥t♦ ♣❛r❛ ♦s ❡st✉❞❛♥t❡s q✉❛♥t♦ ♣❛r❛ ♦ ♣r♦❢❡ss♦r✳ ❖ ♦❜❥❡t✐✈♦ ❞❛s ❛t✐✈✐❞❛❞❡s ♣r♦♣♦st❛s é ❢❛❝✐❧✐t❛r ❡ss❛ ❛❞❛♣t❛çã♦✳ P❛r❛ ❝✉♠♣r✐r s❡✉s ♣r♦♣ós✐t♦s ❞❡ ❢♦r♠❛r ❝✐❞❛❞ã♦s ❝♦♥s❝✐❡♥t❡s ❡ ♣r❡♣❛r❛❞♦s ♣❛r❛ ♦ ♠✉♥❞♦ ❞♦ tr❛❜❛❧❤♦ ❡ ❛ ❝♦♥✈✐✈ê♥❝✐❛ s♦❝✐❛❧✱ ♦s P❛râ♠❡tr♦s ◆❛❝✐♦♥❛✐s ❈✉rr✐❝✉❧❛r❡s ❡♠
▼❛t❡♠át✐❝❛ ✭❬✻❪✱ ♣✳ ✻✵✮✱ ✏✐♥❞✐❝❛♠ ❛s♣❡❝t♦s ♥♦✈♦s ♥♦ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s ❡ ♦♣❡r❛çõ❡s✱ ♣r✐✈✐❧❡❣✐❛♥❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ s❡♥t✐❞♦ ♥✉♠ér✐❝♦ ❡ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ❞✐❢❡r❡♥t❡s s✐❣♥✐✜❝❛❞♦s ❞❛s ♦♣❡r❛çõ❡s✑✳ ❉❡ss❛ ❢♦r♠❛✱ s❡rã♦ ♣r♦♣♦st❛s ❛t✐✈✐❞❛❞❡s ❝♦♠ ❡ss❡ ❡♥❢♦q✉❡✱ ♥❛s q✉❛✐s ♦ ❡st✉❞❛♥t❡ ✉s❛rá ♦ ❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛✳ ❆❧é♠ ❞✐ss♦✱ ♣♦r ♠❡✐♦ ❞❛ ♠❡❞✐❛çã♦ ❞♦ ♣r♦❢❡ss♦r✱ ♦ ❡st✉❞❛♥t❡ ♣♦❞❡rá ❡♥t❡♥❞❡r ♦s ♠♦t✐✈♦s ❞❡ss❛ ✉t✐❧✐③❛çã♦ ❡ ❛✐♥❞❛ r❡❧❛❝✐♦♥á✲❧♦s ❝♦♠ ♦✉tr❛s s✐t✉❛çõ❡s✳
❊ss❛ ❞✐ss❡rt❛çã♦ s❡rá ❝♦♠♣♦st❛ ❞❡ ❝✐♥❝♦ ❝❛♣ít✉❧♦s✳ ❖ ♣r✐♠❡✐r♦ ❝♦♥s✐st❡ ❡♠ ✉♠❛ ❡①♣♦s✐çã♦ ❞♦ ❝♦♥t❡①t♦ ❤✐stór✐❝♦ q✉❡ ❛❜r❛♥❣❡ ♦ ❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s✳ ❖ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❢❛rá ❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ s♦❜r❡ ♦ t❡♠❛✱ ❝♦♠ ❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣♦r ♠❡✐♦ ❞♦s ❆①✐♦♠❛s ❞❡ P❡❛♥♦✳ ❉❛♥❞♦ s❡q✉ê♥❝✐❛✱ ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ s❡rá tr❛❜❛❧❤❛❞❛ ❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❖ q✉❛rt♦ ❝❛♣ít✉❧♦ ✜♥❛❧✐③❛rá ❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ ❛r❣✉♠❡♥t❛♥❞♦ ❛ r❡s♣❡✐t♦ ❞❛ ❞✐✲ ✈✐s✐❜✐❧✐❞❛❞❡✳ ❚♦❞♦s ♦s ❝❛♣ít✉❧♦s q✉❡ ❛❜r❛♥❣❡♠ ❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ t❡rã♦ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ❢♦r♠❛❧✱ ❝♦♠♣♦st❛ ❞❡ ❛①✐♦♠❛s ❡ ❞❡✜♥✐çõ❡s✱ ❛❧é♠ ❞❡ t❡♦r❡♠❛s ❡ ♣r♦♣♦s✐çõ❡s ❝♦♠ s✉❛s r❡s♣❡❝t✐✈❛s ❞❡♠♦♥str❛çõ❡s✳ ❊ ♣♦r ✜♠✱ ♥♦ q✉✐♥t♦ ❝❛♣ít✉❧♦ s❡rã♦ ♣r♦♣♦st❛s ❛t✐✈✐❞❛❞❡s q✉❡ ♦❜❥❡t✐✈❛♠ q✉❡ ♦s ❡st✉❞❛♥t❡s ❝♦♠♣r❡❡♥❞❛♠ ❛ ♦♣❡r❛çã♦ ❞❡ ❞✐✈✐sã♦ ❞❡ ❢♦r♠❛ s❛t✐s❢❛tór✐❛✳
❆ ✜♥❛❧✐❞❛❞❡ ❞❛s ❛t✐✈✐❞❛❞❡s ♣r♦♣♦st❛s é ♦ tr❛❜❛❧❤♦ ❝♦♦♣❡r❛t✐✈♦ ❡♥tr❡ ♦s ❡st✉❞❛♥t❡s✱ ♦✉ s❡❥❛✱ ❛s r❡s♦❧✉çõ❡s ❞❡✈❡♠ s❡r ❞✐s❝✉t✐❞❛s ❡ ❛✈❛❧✐❛❞❛s ♣♦r t♦❞♦s q✉❡ ❝♦♠♣õ❡♠ ♦ ❣r✉♣♦✳ ❆s r❡✢❡①õ❡s s✉❣❡r✐❞❛s ✈✐s❛♠ ✐♥st✐❣❛r ❞✐s❝✉ssõ❡s ♣❛r❛ ❢❛❝✐❧✐t❛r ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦s ♣r♦❜❧❡♠❛s ♣r♦♣♦st♦s✱ ❜✉s❝❛♥❞♦ ✉t✐❧✐③❛r ♦ ❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s ♥❛ r❡s♦❧✉çã♦✳ ❯♠❛ ✈❡③ ❞✐s❝✉t✐❞♦ ❡ ✉t✐❧✐③❛❞♦ ♥❛ r❡s♦❧✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s ♣r♦♣♦st♦s✱ ♦ t❡♠❛ ♣♦❞❡rá s❡r r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦✉tr❛s s✐t✉❛çõ❡s ❝♦t✐❞✐❛♥❛s ❞❡ ♠♦❞♦ ♠❛✐s ❝♦♥✈❡♥✐❡♥t❡✳
Pr♦❝❡❞❡♥t❡s ❞❡ ❛t✐✈✐❞❛❞❡s ❛♣❧✐❝❛❞❛s ♣❛r❛ ❡♥s✐♥♦ ♠é❞✐♦✱ ❛s ❛t✐✈✐❞❛❞❡s ♣r♦♣♦st❛s ♥❡ss❛ ❞✐ss❡rt❛çã♦ ❢♦r❛♠ r❡❢♦r♠✉❧❛❞❛s ♣❛r❛ s❡r❡♠ ❛♣❧✐❝❛❞❛s ♥♦ ✻♦ ❡ ✼♦ ❛♥♦s ❞♦ ❡♥s✐♥♦
❢✉♥❞❛♠❡♥t❛❧✳ ❯♠❛ ❞❛s ♣r♦♣♦st❛s s✉❣❡r✐❞❛s ❢♦✐ ❛❞❛♣t❛❞❛ ❞❛ ❛t✐✈✐❞❛❞❡ ❉❡s✈❡♥❞❛♥❞♦ ♦ ❈❛❧❡♥❞ár✐♦ ❞❡ P❛q✉❡s ✭❬✶✽❪✮✱ q✉❡ ❡♥✈♦❧✈❡ ♦ ❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s ❡ ♦ ❝❛❧❡♥❞ár✐♦✱ ✉t✐❧✐③❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❆ ❛t✐✈✐❞❛❞❡ ❢♦✐ ❛❞❛♣t❛❞❛ ♣❛r❛ q✉❡ ❛♣❡♥❛s ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❢♦ss❡ ♥❡❝❡ssár✐♦✳ ❆ ♦✉tr❛ ♣r♦♣♦st❛ s✉❣❡r✐❞❛ ❢♦✐ ✐♥s♣✐r❛❞❛ ❡♠ ✉♠❛ ❤✐stór✐❛ ❞♦ ❧✐✈r♦ ❖ ❍♦♠❡♠ q✉❡ ❈❛❧❝✉❧❛✈❛ ✭❬✷✷❪✱ ♣✳ ✶✹✼✲✶✹✽✮✳ ❊ss❛ ❛t✐✈✐❞❛❞❡ ❢♦✐ ❜❛s❡❛❞❛ ♥❛ ❍✐stór✐❛ ❞♦s ❚rês ▼❛r✐♥❤❡✐r♦s q✉❡ ❣❡r❛❧♠❡♥t❡ é tr❛❜❛❧❤❛❞❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡ ✉t✐❧✐③❛ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ♥❛ r❡s♦❧✉çã♦✳ ❆ ❛t✐✈✐❞❛❞❡ ❢♦✐ ❛❞❛♣t❛❞❛ ♣❛r❛ q✉❡ ❛♣❡♥❛s ♦ ❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s ❢♦ss❡ ✉t✐❧✐③❛❞♦✱ ✈✐st♦ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❡♥✈♦❧✈❡ ✈ár✐❛s ❞✐✈✐sõ❡s ❝♦♠ r❡st♦s ❞✐❢❡r❡♥t❡s ❞❡ ③❡r♦✳
❆ss✐♠ s❡♥❞♦✱ ❡ss❛ ❞✐ss❡rt❛çã♦ ♣r♦♣♦r❝✐♦♥❛ ✉♠ ❡st✉❞♦ ❛♣r♦❢✉♥❞❛❞♦ ❞♦ t❡♠❛ ♣❛r❛ ♦s
♣r♦❢❡ss♦r❡s✱ ❡ ❛ ♣❛rt✐r ❞❡ss❡ ❡st✉❞♦ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❛ ♠❡❞✐❛çã♦ ❞♦ ♠❡s♠♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❝♦♠ ♦s ❡st✉❞❛♥t❡s✱ r❡❛❧✐③❛♥❞♦ ❛s ♠♦❞✐✜❝❛çõ❡s ♥❡❝❡ssár✐❛s q✉❛♥t♦ ❛ ❧✐♥❣✉❛❣❡♠✳
❈❛♣ít✉❧♦ ✶
❍✐stór✐❝♦
❊♠ q✉❛❧q✉❡r ❝♦♥t❡ú❞♦ ♠❛t❡♠át✐❝♦ ❛ ❛❜♦r❞❛❣❡♠ ❤✐stór✐❝❛ s❡ ❢❛③ ✐♠♣♦rt❛♥t❡✳ ❊♥t❡♥✲ ❞❡r ❛s ♦r✐❣❡♥s ❡ ❛ ❡✈♦❧✉çã♦ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❛ss✉♥t♦ ♣♦❞❡ ❢❛❝✐❧✐t❛r ❛ s✉❛ ❡❧✉❝✐❞❛çã♦✱ t♦r♥❛♥❞♦✲♦ ♠❛✐s ✐♥t❡r❡ss❛♥t❡✳ ❆❧é♠ ❞✐ss♦✱ ❡ss❛ ❛❜♦r❞❛❣❡♠ ❛❥✉❞❛ ❛ ❡♥t❡♥❞❡r q✉❡ ❛ ♠❛t❡♠át✐❝❛ ♥ã♦ é ❛❧❣♦ ❛❝❛❜❛❞♦✱ ❧♦❣♦✱ é ✉♠❛ ❝♦♥str✉çã♦ ❤✉♠❛♥❛ ✭só❝✐♦✲❤✐stór✐❝❛✮✳
❆ s❡❣✉✐r s❡rã♦ ❞❡s❝r✐t♦s ❛❧❣✉♥s ❛ss✉♥t♦s ✐♠♣♦rt❛♥t❡s ❞❛ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛ q✉❡ ❡♥✈♦❧✈❡♠ ♦s ❛ss✉♥t♦s ♣❡rt✐♥❡♥t❡s ❛♦ t❡♠❛✳ ❊ss❡s ❛ss✉♥t♦s s❡rã♦ ❞✐✈✐❞✐❞♦s ❡♠ três tó♣✐❝♦s✳ ❖ ♣r✐♠❡✐r♦ tó♣✐❝♦ s❡rá s♦❜r❡ ♦s ♣r✐♠ór❞✐♦s ❞❛ ♦♣❡r❛çã♦ ❞❡ ❞✐✈✐sã♦✱ q✉❡ ❞❡s❝r❡✈❡rá ❝♦♠♦ ❡r❛ ❢❡✐t❛ ❡ss❛ ♦♣❡r❛çã♦ ❛♥t❡s ❞♦ ❛❧❣♦r✐t♠♦ ✉t✐❧✐③❛❞♦ ❛t✉❛❧♠❡♥t❡✳ ❖ s❡❣✉♥❞♦ tó♣✐❝♦ t❡rá ❛❧❣✉♥s ♣♦♥t♦s s♦❜r❡ ❊✉❝❧✐❞❡s ❡ ✉♠❛ ❞❡ s✉❛s ♦❜r❛s✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ✉♠❛ ❞❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❛ ♠❛t❡♠át✐❝❛✱ ❖s ❊❧❡♠❡♥t♦s✳ ❖ t❡r❝❡✐r♦ tó♣✐❝♦ ♣♦♥t✉❛rá ❛❧❣✉♥s r❡❣✐str♦s ❛♥t✐❣♦s ❛❝❡r❝❛ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ ✐♥t❡✐r♦s✳ ❊ss❡ ❝❛♣ít✉❧♦ ❢♦✐ ❜❛s❡❛❞♦ ❡♠ ❆❛❜♦❡ ✭❬✶❪✮✱ ❡♠ ❇♦②❡r ✭❬✺❪✮ ❡ ❡♠ ❊✈❡s ✭❬✾❪✮✳
✶✳✶ Pr✐♠ór❞✐♦s ❞❛ ♦♣❡r❛çã♦ ❞❡ ❞✐✈✐sã♦
❙❡❣✉♥❞♦ ❇♦②❡r ✭❬✺❪✱ ♣✳ ✾✲✶✶✮✱ ✉♠ ❞♦s r❡❣✐str♦s ♠❛✐s ❡①t❡♥s♦s ❞❡ ♥❛t✉r❡③❛ ♠❛t❡♠á✲ t✐❝❛ q✉❡ r❡s✐st✐✉ ❛♦ ❞❡s❣❛st❡ ❞♦ t❡♠♣♦ é ♦ P❛♣✐r♦ ❘❤✐♥❞ ✭✷✵✵✵ ❛ ✶✽✵✵ ❛✳❝✳✮✳ ❚❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ P❛♣✐r♦ ❆❤♠❡s ✭❞❡✈✐❞♦ ❛ ✉♠ ❡s❝r✐❜❛ q✉❡ ♦ ❝♦♣✐♦✉ ♣♦r ✈♦❧t❛ ❞❡ ✶✻✺✵ ❛✳❝✳✮ ❡ss❡ ❞♦❝✉♠❡♥t♦ ❡❣í♣❝✐♦ ♠♦str❛ ❝♦♠♦ ❡r❛ r❡❛❧✐③❛❞❛ ✉♠❛ ♦♣❡r❛çã♦ ❞❡ ❞✐✈✐sã♦ ♥❛ é♣♦❝❛✳ ❆ ♦♣❡r❛çã♦ ❛r✐t♠ét✐❝❛ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ ❊❣✐t♦ ❡r❛ ❛ ❛❞✐çã♦✱ ❡ ❝♦♠♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❛ ❞✐✈✐sã♦ t❛♠❜é♠ ❡r❛ ♦♣❡r❛❞❛ ❛tr❛✈és ❞❡ s✉❝❡ss✐✈❛s ✏❞✉♣❧❛çõ❡s✑✳
❙❡❣✉♥❞♦ ❇♦②❡r ✭❬✺❪✱ ♣✳ ✶✻✮✱ ♦ P❛♣✐r♦ ❘❤✐♥❞ ❡stá ❡♥tr❡ ✉♠ ❞♦s ♠❛✐s ❛♥t✐❣♦s r❡✲ ❣✐str♦s ♠❛t❡♠át✐❝♦s✱ ❝❡r❝❛ ❞❡ ♠✐❧ ❛♥♦s ❛♥t❡s ❞♦ ♥❛s❝✐♠❡♥t♦ ❞❛ ♠❛t❡♠át✐❝❛ ❣r❡❣❛✳ ❊ ❝♦♠♦ t♦❞♦s ♦s ❡stá❣✐♦s ❞❛ ♠❛t❡♠át✐❝❛ ❡❣í♣❝✐❛ ❡r❛♠ ❜❛s❡❛❞♦s ♥❛ ♦♣❡r❛çã♦ ❞❡ ❛❞✐çã♦✱ ✐ss♦ s❡ t♦r♥♦✉ ✏✉♠❛ ❞❡s✈❛♥t❛❣❡♠ q✉❡ ❝♦♥❢❡r✐❛ ❛♦s ❝á❧❝✉❧♦s ❞♦s ❡❣í♣❝✐♦s ✉♠ ♣❡❝✉❧✐❛r ♣r✐♠✐t✐✈✐s♠♦✱ ❝♦♠❜✐♥❛❞♦ ❝♦♠ ✉♠❛ ♦❝❛s✐♦♥❛❧ ❡ ❛ss♦♠❜r♦s❛ ❝♦♠♣❧❡①✐❞❛❞❡✑✳
❆ s❡❣✉✐r t❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❡ ❝♦♠♦ ❛ ♦♣❡r❛çã♦ ❞❡ ❞✐✈✐sã♦ ❡r❛ ❡❢❡t✐✈❛♠❡♥t❡ r❡❛❧✐✲ ③❛❞❛ ♣❡❧♦s ❡❣í♣❝✐♦s✱ s❡❣✉♥❞♦ ❊✈❡s ✭❬✾❪✱ ♣✳ ✼✷✲✼✸✮✳
❊ ♣❛r❛✱ ❞✐❣❛♠♦s✱ ❞✐✈✐❞✐r ✼✺✸ ♣♦r ✷✻✱ ❞♦❜r❛♠♦s s✉❝❡ss✐✈❛♠❡♥t❡ ♦ ❞✐✈✐s♦r ✷✻ ❛té ♦ ♣♦♥t♦ ❡♠ q✉❡ ♦ ♣ró①✐♠♦ ❞♦❜r♦ ❡①❝❡❞❛ ♦ ❞✐✈✐❞❡♥❞♦ ✼✺✸✳ ❖ ♣r♦❝❡❞✐♠❡♥t♦ ❡stá ❡①♣♦st♦ ❛❜❛✐①♦✳ 1 2 ∗4 ∗8 ∗16 28 26 52 104 208 416 ❖r❛✱ ❝♦♠♦
753 = 416 + 337 = 416 + 208 + 129 = 416 + 208 + 104 + 25
✈❡♠♦s✱ ♦❜s❡r✈❛♥❞♦ ❛s ❧✐♥❤❛s ❝♦♠ ❛st❡r✐s❝♦s ♥❛ ❝♦❧✉♥❛ ❛❝✐♠❛✱ q✉❡ ♦ q✉♦❝✐❡♥t❡ é
16 + 8 + 4 = 28 ❡ ♦ r❡st♦ é25✳ ❖ ♣r♦❝❡ss♦ ❡❣í♣❝✐♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❞✐✈✐sã♦ ♥ã♦
só ❡❧✐♠✐♥❛ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❛♣r❡♥❞❡r ❛ t❛❜✉❛❞❛ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❝♦♠♦ t❛♠❜é♠ s❡ ❛♠♦❧❞❛ t❛♥t♦ ❛♦ á❜❛❝♦ q✉❡ ♣❡r❞✉r♦✉ ❡♥q✉❛♥t♦ ❡ss❡ ✐♥str✉♠❡♥t♦ ❡st❡✈❡ ❡♠ ✉s♦ ❡ ♠❡s♠♦ ❞❡♣♦✐s✳
◆♦ ❡①❡♠♣❧♦ ❛❝✐♠❛ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ❛ ❝♦♥str✉çã♦ ❞❛ t❛❜❡❧❛ ❝♦♠ ❛s s✉❝❡ss✐✈❛s ❞✉♣❧❛çõ❡s ❡r❛ ❝♦♠♣♦st❛ ♣♦r ❞✉❛s ❝♦❧✉♥❛s✳ ❆ ♣r✐♠❡✐r❛ ✐♥❞✐❝❛✈❛ ♦ ♥ú♠❡r♦ ❞❡ ✈❡③❡s q✉❡ ♦ ❞✐✈✐s♦r✱ ♥♦ ❝❛s♦ ♦ ✷✻✱ ❡st❛✈❛ s❡♥❞♦ ♠✉❧t✐♣❧✐❝❛❞♦✳ ❉♦❜r❛r ✉♠ ♥ú♠❡r♦ é ♠✉❧t✐♣❧✐❝á✲❧♦ ♣♦r ✷✳ ❙❡ ❞♦❜r❛r♠♦s ♥♦✈❛♠❡♥t❡✱ ❡st❛r❡♠♦s ♠✉❧t✐♣❧✐❝❛♥❞♦ ❡ss❡ ♥ú♠❡r♦ ♣♦r ✹✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ ❆ s❡❣✉♥❞❛ ❝♦❧✉♥❛ ✐♥❞✐❝❛✈❛ ♦ ✈❛❧♦r ❞❡ss❛ ♠✉❧t✐♣❧✐❝❛çã♦✳ ❆ss✐♠✱ ♦s ♥ú♠❡r♦s ❝♦♠ ♦s ❛st❡r✐s❝♦s ♥❛ ♣r✐♠❡✐r❛ ❝♦❧✉♥❛ ✐♥❞✐❝❛✈❛ q✉❛♥t❛s ✈❡③❡s ♦ ✷✻ ✏❝❛❜✐❛✑ ♥♦ ❞✐✈✐❞❡♥❞♦✱ ♥♦ ❝❛s♦ ♦s ✼✺✸✳ ❆ss✐♠✱ ♦ r❡s✉❧t❛❞♦ ❞❛ ❞✐✈✐sã♦ ❢♦✐ ✷✽✱ ❝♦♠ r❡st♦ ✷✺✳
❙❡❣✉♥❞♦ ❆❛❜♦❡ ✭❬✶❪✱ ♣✳ ✾✲✷✶✮✱ ❛❧é♠ ❞♦ P❛♣✐r♦ ❘❤✐♥❞✱ ❢♦r❛♠ ❡♥❝♦♥tr❛❞♦s ♥♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ❞❡③❡♥♦✈❡ ❞✉r❛♥t❡ ❡s❝❛✈❛çõ❡s ❡♠ ❝♦❧✐♥❛s ♥❛ ▼❡s♦♣♦tâ♠✐❛✱ t❛❜❧❡t❡s ✭tá❜✉❛s✮
❞❡ ❛r❣✐❧❛ ❝♦♠ ❛ ❡s❝r✐t❛ ❝✉♥❡✐❢♦r♠❡✱ ❛❧❣✉♥s ❞❛t❛♥❞♦ ❞❡ ✶✼✵✵ ❛✳❝✳✳ ❆tr❛✈és ❞❡ ❡st✉❞♦s ❢❡✐t♦s ♥❡ss❡s t❛❜❧❡t❡s ❝♦♥st❛t♦✉✲s❡ q✉❡ ❡r❛♠ r❡❣✐str♦s ❞❛ ♠❛t❡♠át✐❝❛ ❜❛❜✐❧ô♥✐❝❛✳ ❈♦♠ ✐ss♦✱ ❞❡s❝♦❜r✐✉✲s❡ q✉❡ ♦ s✐st❡♠❛ ❞❡ ♥✉♠❡r❛çã♦ ❜❛❜✐❧ô♥✐❝♦ ❡r❛ ♣♦s✐❝✐♦♥❛❧ ❝♦♠♦ ♦ ❛t✉❛❧✱ ♣♦ré♠✱ ❛♦ ✐♥✈és ❞❡ s❡r ✉♠ s✐st❡♠❛ ❞❡ ♥✉♠❡r❛çã♦ ❞❡❝✐♠❛❧ ✭♥❛ ❜❛s❡ ✶✵✮✱ ♦ ♠❡s♠♦ ❡r❛ s❡①❛❣❡s✐♠❛❧ ✭♥❛ ❜❛s❡ ✻✵✮✳ ❋♦r❛♠ ❡♥❝♦♥tr❛❞❛s t❛♠❜é♠ ❧✐st❛s ❞❡ ♠✉❧t✐♣❧✐❝❛çõ❡s ❡ ❧✐st❛s ❞❡ ♥ú♠❡r♦s ❝♦♠ ♦s s❡✉s r❡s♣❡❝t✐✈♦s r❡❝í♣r♦❝♦s ✭✐♥✈❡rs♦s✮✳ ❙❡❣✉♥❞♦ ❇♦②❡r ✭❬✺❪✱ ♣✳ ✷✷✮❀
➱ ❝❧❛r♦ q✉❡ ❛s ♦♣❡r❛çõ❡s ❛r✐t♠ét✐❝❛s ❢✉♥❞❛♠❡♥t❛✐s ❡r❛♠ tr❛t❛❞❛s ♣❡❧♦s ❜❛❜✐❧ô♥✐✲ ❝♦s ❞❡ ♠♦❞♦ ♥ã♦ ♠✉✐t♦ ❞✐❢❡r❡♥t❡ ❞♦ ✉s❛❞♦ ❤♦❥❡✱ ❡ ❝♦♠ ❢❛❝✐❧✐❞❛❞❡ ❝♦♠♣❛rá✈❡❧✳ ❆ ❞✐✈✐sã♦ ♥ã♦ ❡r❛ ❡❢❡t✉❛❞❛ ♣❡❧♦ ✐♥❝ô♠♦❞♦ ♣r♦❝❡ss♦ ❞❡ ❞✉♣❧✐❝❛çã♦ ❞♦s ❡❣í♣❝✐♦s✱ ♠❛s ♣♦r ✉♠❛ ❢á❝✐❧ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦ ❞✐✈✐❞❡♥❞♦ ♣❡❧♦ ✐♥✈❡rs♦ ❞♦ ❞✐✈✐s♦r✱ ✉s❛♥❞♦ ♦s ✐t❡♥s ❛♣r♦♣r✐❛❞♦s ♥❛s t❛❜❡❧❛s✳
▼❛✐s ❞❡t❛❧❤❡s ❡ ❝✉r✐♦s✐❞❛❞❡s ❛ r❡s♣❡✐t♦ ❞❡ ❝♦♠♦ ♦❝♦rr❡r❛♠ ❡ss❛s ❞❡s❝♦❜❡rt❛s ❞❛ ♠❛t❡♠át✐❝❛ ❜❛❜✐❧ô♥✐❝❛ ♣♦r ♠❡✐♦ ❞♦s t❛❜❧❡t❡s ❝✐t❛❞♦s ♣♦❞❡♠ s❡r ❛♣r❡❝✐❛❞❛s ❡♠ ❆❛❜♦❡ ✭❬✶❪✱ ♣✳ ✾✲✹✷✮✳
❖✉tr♦s ♠♦❞♦s ❞✐❢❡r❡♥t❡s ❞❡ ❝♦♠♦ ❡r❛ r❡❛❧✐③❛❞❛ ❛ ♦♣❡r❛çã♦ ❞❡ ❞✐✈✐sã♦ ♥❛ ❛♥t✐❣✉✐❞❛❞❡ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ◆❡✈❡s ✭❬✶✼❪✱ ♣✳ ✷✹✲✸✹✮✳
❈♦♠ ✐ss♦✱ s❡❣✉♥❞♦ ◆❡✈❡s ✭❬✶✼❪✱ ♣✳ ✸✹✮✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ❛ ♣rát✐❝❛ ❞❛ ❞✐✈✐sã♦ ♥❛s ❝✐✈✐❧✐③❛çõ❡s ❛♥t✐❣❛s✱ ❝♦♥❢♦r♠❡ r❡❣✐str♦s ❤✐stór✐❝♦s✱ ✏❛♣r❡s❡♥t❛♠ ✐♥❞í❝✐♦s ❞♦ r❛❝✐♦❝í♥✐♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❡ ♣r♦♣♦r❝✐♦♥❛❧ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ♠✉✐t❛s s✐t✉❛çõ❡s✑✱ ❛❧é♠ ❞❛ ✏♣r❡s❡♥ç❛ ❞❡ ♣r♦❜❧❡♠❛s ✭❡♠ ❝❛♠♣♦✮ ♦✉ ❝r✐❛❞♦s ♣❛r❛ ✜♥s ❞❡ ✐♥str✉çã♦✑ ❡✱ ❞❡ss❛ ❢♦r♠❛✱ s✉r❣✐✉ ✏❛ ❝r✐❛çã♦ ❞❡ ♣r♦❝❡❞✐♠❡♥t♦s ✭❛❧❣♦r✐t♠♦s✮ ♣❛r❛ ❛ s✉❛ r❡s♦❧✉çã♦✑✳
✶✳✷ ❊✉❝❧✐❞❡s ❡ ❖s ❊❧❡♠❡♥t♦s
❙❡❣✉♥❞♦ ❇♦②❡r✱ ✭❬✺❪✱ ♣✳ ✼✹✲✼✺✮✱ ♣♦r ✈♦❧t❛ ❞❡ ✸✵✻ ❛✳❝✳✱ Pt♦❧♦♠❡✉ ■ ❝r✐♦✉ ❡♠ ❆❧❡①❛♥✲ ❞r✐❛ ✉♠❛ ❡s❝♦❧❛ ✭♦✉ ✐♥st✐t✉t♦✮ ❝❤❛♠❛❞♦ ❞❡ ▼✉s❡✉✱ ✏✐♥s✉♣❡r❛❞♦ ❡♠ s❡✉ t❡♠♣♦✑✳ ❊✉❝❧✐❞❡s ❢♦✐ ✉♠ ❞♦s ♣r♦❢❡ss♦r❡s ❞♦ ❣r✉♣♦ ❞❡ sá❜✐♦s ✭♦s ♠❛✐s r❡❝♦♥❤❡❝✐❞♦s ❞❛ é♣♦❝❛✮ ❝♦♥tr❛t❛❞♦s ♣❛r❛ ❧❡❝✐♦♥❛r ♥♦ ▼✉s❡✉✳ P♦r ✐ss♦✱ ✜❝♦✉ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛✱ ❛♣❡s❛r ❞❡ ❛❧❣✉♠❛s ✈❡③❡s s❡r ❝❤❛♠❛❞♦ ❡rr♦♥❡❛♠❡♥t❡ ❞❡ ❊✉❝❧✐❞❡s ❞❡ ▼❡❣❛r❛ ✭✉♠ ❞✐s❝í♣✉❧♦ ❞❡ ❙ó❝r❛t❡s✮✳ ❆♣❡s❛r ❞♦ ❞❡s❝♦♥❤❡❝✐♠❡♥t♦ s♦❜r❡ s✉❛ ✈✐❞❛✱ ❊✉❝❧✐❞❡s é ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ♦ ❛✉t♦r ❞♦ t❡①t♦ ❞❡ ♠❛t❡♠át✐❝❛ ♠❛✐s ❜❡♠ s✉❝❡❞✐❞♦ ❞❡ t♦❞♦s ♦s t❡♠♣♦s✱ ❖s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✳
❆♦ ❝♦♥trár✐♦ ❞♦ q✉❡ ♠✉✐t♦s ✐♠❛❣✐♥❛♠ ❖s ❊❧❡♠❡♥t♦s ♥ã♦ tr❛t❛ ❛♣❡♥❛s ❞❡ ❣❡♦♠❡tr✐❛✱ ♠❛s ❤á t❛♠❜é♠ ✉♠❛ ❡①♣♦s✐çã♦ ❡♠ ♦r❞❡♠ ❧ó❣✐❝❛ ❞♦s ❛ss✉♥t♦s ❜ás✐❝♦s ❞❛ ♠❛t❡♠át✐❝❛
❡❧❡♠❡♥t❛r ✭❛r✐t♠ét✐❝❛ ♦✉ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ❣❡♦♠❡tr✐❛ ❡ á❧❣❡❜r❛✮✳ ❉♦s tr❡③❡ ❧✐✈r♦s ♦✉ ❝❛♣ít✉❧♦s q✉❡ ❝♦♠♣õ❡ ❡ss❛ ♦❜r❛ três ❞❡❧❡s ❢❛❧❛♠ s♦❜r❡ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ♦s ❞❡ ♥ú♠❡r♦ ❱■■✱ ❱■■■ ❡ ■❳✳
✶✳✸ ◆ú♠❡r♦s ◆❛t✉r❛✐s ❡ ◆ú♠❡r♦s ■♥t❡✐r♦s
❙❡❣✉♥❞♦ ❇♦②❡r ✭❬✺❪✱ ♣✳ ✶✻✵✮✱ ✐♥❞í❝✐♦s ❞❛ ♣r❡s❡♥ç❛ ❞♦s ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s ❢♦r❛♠ ❡♥❝♦♥tr❛❞♦s ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❡♠ ✉♠❛ ♦❜r❛ ❞❡ ❇r❛❤♠❛❣✉♣t❛ ✭✈✐✈❡✉ ❡♠ ✻✷✽✮ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❛r✐t♠ét✐❝❛ s✐st❡♠❛t✐③❛❞❛ t❛♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♠♦ ❞♦ ③❡r♦✳
❆ ♥♦çã♦ ❞❡ ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s ❢♦✐ ❡♥❝♦♥tr❛❞❛✱ s❡❣✉♥❞♦ ❇♦②❡r ✭❬✺❪✱ ♣✳ ✶✹✺✲✶✹✼✮ ♥♦ á❜❛❝♦ ❝❤✐♥ês✱ q✉❡ ❞✐❢❡r❡♥t❡ ❞♦ ár❛❜❡✱ ♣♦ss✉✐❛ ✏❞✉❛s ❝♦❧❡çõ❡s ❞❡ ❜❛rr❛s ✖ ✉♠❛ ✈❡r♠❡❧❤❛ ♣❛r❛ ♦s ❝♦❡✜❝✐❡♥t❡s ♣♦s✐t✐✈♦s ♦✉ ♥ú♠❡r♦s ❡ ✉♠ ♣r❡t❛ ♣❛r❛ ♦s ♥❡❣❛t✐✈♦s✳✑ ✭♦s á❜❛❝♦s ♠❛✐s ♠♦❞❡r♥♦s ❝♦♥❤❡❝✐❞♦s ♥❛ ❈❤✐♥❛ sã♦ ❞♦ sé❝✉❧♦ ❞❡③❡ss❡✐s✱ ♠❛✐s ♦s ♦r✐❣✐♥ár✐♦s ♣❛r❡❝❡♠ t❡r s✐❞♦ ✉s❛❞♦s ❛ ♠✐❧ ❛♥♦s ❛trás✮✳
❙❡❣✉♥❞♦ ❇♦②❡r ✭❬✺❪✱ ♣✳ ✹✸✻✲✹✸✼✮✱ ●✐✉s❡♣♣❡ P❡❛♥♦ ✭✶✽✺✽ ✲ ✶✾✸✷✮ ❢♦✐ ✉♠ ❞♦s ♠❛t❡✲ ♠át✐❝♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❧ó❣✐❝❛ ♠❛t❡♠át✐❝❛ ♠❛✐s ❝♦♥❤❡❝✐❞♦s ❞♦ sé❝✉❧♦ ❞❡③❡♥♦✈❡✳ ❙✉❛ ❡❧❛❜♦r❛çã♦ ❞♦s ❆①✐♦♠❛s ❞❡ P❡❛♥♦✱ ❡♠ ✶✽✽✾✱ ❢♦✐ ❜❛s❡❛❞❛ ❡♠ ✉♠ s✐♠❜♦❧✐s♠♦ ♣✉r♦ ❢♦r✲ ♠❛❧✱ ♦✉ s❡❥❛✱ ❞❡s❝♦♥s✐❞❡r♦✉ ❛ ❞❡s❝r✐çã♦ ❡♠ ♣❛❧❛✈r❛s q✉❡ ❝❛✉s❛✈❛♠ t❛♥t❛s ❛♠❜✐❣✉✐❞❛❞❡s ❡ ❤✐♣ót❡s❡s ♦❝✉❧t❛s ✭♠ét♦❞♦ ♣♦st✉❧❛❝✐♦♥❛❧✮✳ ❆ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❡❧❛❜♦r❛çã♦ ❞❡ss❛ ♦❜r❛ ♥ã♦ é s♦♠❡♥t❡ ♣❡❧❛ ❢♦r♠❛❧✐③❛çã♦ ❞❛ ❝♦♥str✉çã♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ♠❛s t❛♠❜é♠ ♣♦r ❡♠❜❛s❛r ❝♦♥str✉çõ❡s ❢♦r♠❛✐s ❞❛ á❧❣❡❜r❛ ❡ ❞❛ ❛♥á❧✐s❡✱ ❝♦♠♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦✳
❈❛♣ít✉❧♦ ✷
❈♦♥str✉çã♦ ❞♦s ◆ú♠❡r♦s ◆❛t✉r❛✐s
◆❡ss❡ ❝❛♣ít✉❧♦ ❝♦♥str✉✐r❡♠♦s ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ■♥✐❝✐❛❧♠❡♥t❡ ❡♥✉♥✲ ❝✐❛r❡♠♦s ♦s ❆①✐♦♠❛s ❞❡ P❡❛♥♦✳ ❊♠ s❡❣✉✐❞❛✱ ❛s ❞❡♠♦♥str❛çõ❡s s❡rã♦ ❜❛s❡❛❞❛s ♥♦ ❆①✐♦♠❛ ✸✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦✳ ❉❡✜♥✐r❡♠♦s ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❝♦♠ s✉❛s r❡s♣❡❝t✐✈❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❈♦♥t✐♥✉❛♥❞♦ ♦ ❝❛♣ít✉❧♦✱ ❡st❛❜❡❧❡✲ ❝❡r❡♠♦s ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ ♥❡ss❡ ❝♦♥❥✉♥t♦✳ ❆❧é♠ ❞✐ss♦✱ ❢❛❧❛r❡♠♦s s♦❜r❡ ♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♠✿ q✉❛❧q✉❡r s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡Nt❡♠ ✉♠ ♠❡♥♦r ♥ú♠❡r♦✳ ❊ ✜♥❛❧✐✲
③❛♥❞♦ ♦ ❝❛♣ít✉❧♦ ♠♦str❛r❡♠♦s ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ❡ ♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♠✳ ❚♦❞❛ ❝♦♥str✉çã♦ s❡rá ❜❛s❡❛❞❛ ❡♠ ❞❡✜♥✐çõ❡s ❡ ❛①✐♦♠❛s ❛❧é♠ ❞❡ t❡♦r❡✲ ♠❛s ❡ ♣r♦♣♦s✐çõ❡s ❝♦♠ s✉❛s r❡s♣❡❝t✐✈❛s ❞❡♠♦♥str❛çõ❡s✳ ❊ss❡ ❝❛♣ít✉❧♦ ❢♦✐ ❜❛s❡❛❞♦ ❡♠ ❋❡rr❡✐r❛ ✭❬✶✵❪✮✱ ❡♠ ▼✐❧✐❡s ✭❬✶✺❪✮ ❡ ❡♠ ❙❤♦❦r❛♥✐❛♥ ✭❬✷✵❪✮✳ ❆ ❝♦♥str✉çã♦ ❝♦♥s✐❞❡r❛rá ♦ ③❡r♦ ❝♦♠♦ ♥ú♠❡r♦ ♥❛t✉r❛❧✱ ♠❛s ♣❛r❛ ❡st✉❞❛r ✉♠❛ ❝♦♥str✉çã♦ ❡♠ q✉❡ ♦ ③❡r♦ ♥ã♦ ❡stá ✐♥❝❧✉s♦ ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❜❛st❛ ❝♦♥s✉❧t❛r ▲✐♠❛ ✭❬✶✹❪✮✳
✷✳✶ ❆①✐♦♠❛s ❞❡ P❡❛♥♦
❯t✐❧✐③❛♥❞♦ ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ❝♦♥❤❡❝✐❞♦s✱ ❝♦♠♦ ❝♦♥❥✉♥t♦ ❡ ❢✉♥çã♦✱ ✈❛♠♦s ❡①♣♦r ♦s ❛①✐♦♠❛s ❞❡ P❡❛♥♦ ❛ s❡❣✉✐r✿
❊①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ N ❡ ✉♠❛ ❢✉♥çã♦s :N→N q✉❡ ❛❞♠✐t❡ ♦s s❡❣✉✐♥t❡s ❛①✐♦♠❛s✿
❆①✐♦♠❛ ✶✳ sé ✐♥❥❡t♦r❛✱ ♦✉ s❡❥❛✱ ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s ❞♦ ❞♦♠í♥✐♦ tê♠ ✐♠❛❣❡♥s ❞✐❢❡r❡♥t❡s✳
❆①✐♦♠❛ ✷✳ ❊①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ❡♠ N✱ ❞❡♥♦t❛❞♦ ♣♦r 0 ✭❝❤❛♠❛❞♦ ❞❡ ③❡r♦✮ t❛❧ q✉❡ 0∈/ Im(s)✳
❆①✐♦♠❛ ✸✳ ❙❡❥❛A ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ N t❛❧ q✉❡✿
✐✮0∈A❀
✐✐✮n ∈A⇒s(n)∈A✳
❊♥tã♦A=N✳
❖ ❝♦♥❥✉♥t♦Né ❝❤❛♠❛❞♦ ❞❡ ❈♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ◆❛t✉r❛✐s✳ ❖ ❆①✐♦♠❛ ✸ ❞❡✜♥✐❞♦
❛❝✐♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✱ ♦✉ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❈♦♠♣❧❡t❛✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦✳
❉❡✜♥✐r❡♠♦s ❛❣♦r❛ ✉♠❛ ❢✉♥çã♦ s✉❝❡ss♦r ❡ ❡♠ s❡❣✉✐❞❛ ♠♦str❛r❡♠♦s ❛❧❣✉♠❛s ❞❡ s✉❛s ❝❛r❛❝t❡ríst✐❝❛s✱ ❡①♣♦st❛s ♥♦ ❚❡♦r❡♠❛ ❛ s❡❣✉✐r✳ ❊ss❡ ♣❛ss♦ é ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❞❡✜♥✐r♠♦s ❛s ♦♣❡r❛çõ❡s ❡♠N✳
❉❡✜♥✐çã♦ ✷✳✶✳✶✳ ❉✐③❡♠♦s q✉❡ s : N → N é ✉♠❛ ❢✉♥çã♦ s✉❝❡ss♦r q✉❛♥❞♦ é ✐♥❥❡t♦r❛
❡ ❝❛❞❛ ❡❧❡♠❡♥t♦x ❞♦ ❞♦♠í♥✐♦ é ❛ss♦❝✐❛❞♦ ❛♦ ❡❧❡♠❡♥t♦x+ 1 ❞♦ ❝♦♥tr❛❞♦♠í♥✐♦✱ s❡♥❞♦
s(0) = 1❡ 0∈/ Im(s)✳
❚❡♦r❡♠❛ ✷✳✶✳✶✳ ❙❡ s:N→N é ❛ ❢✉♥çã♦ s✉❝❡ss♦r✱ ❡♥tã♦✱ t❡♠✲s❡✿
✐✮ s(n) 6= n✱ ♣❛r❛ t♦❞♦ n ∈ N✱ ♦✉ s❡❥❛✱ ♥❡♥❤✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ é s✉❝❡ss♦r ❞❡ s✐
♠❡s♠♦❀
✐✐✮Im(s) =N\ {0}✱ ♦✉ s❡❥❛✱ ♦ ③❡r♦ é ♦ ú♥✐❝♦ ♥❛t✉r❛❧ q✉❡ ♥ã♦ é s✉❝❡ss♦r ❞❡ ♥❡♥❤✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳
❉❡♠♦♥str❛çã♦✳ ✐✮ ❙❡❥❛ A ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ N t❛❧ q✉❡
A ={n ∈N:s(n)6=n}.
❚❡♠♦s q✉❡ 0∈A✱ ♣♦✐s s(0) = 0✱ ❥á q✉❡6 0∈/ Im(s) ♣❡❧♦ ❆①✐♦♠❛ ✷✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞♦ ❝♦♥❥✉♥t♦ A t❡♠♦s q✉❡
k∈A ⇔s(k)6=k.
❆♣❧✐❝❛♥❞♦ s ❡♠ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❡ s(k) 6= k✱ ♦❜t❡♠♦s s(s(k)) 6= s(k)✱ ❥á q✉❡ s é
✐♥❥❡t♦r❛ ♣❡❧♦ ❆①✐♦♠❛ ✶✳ ❆ss✐♠✱ s(k)∈A✳ ▲♦❣♦✱ ❝♦♠♦ t❡♠♦s
0∈A
❡
k ∈A⇒s(k)∈A,
♣❡❧♦ ❆①✐♦♠❛ ✸✱ ❝♦♥❝❧✉í♠♦s q✉❡A=N✳
✐✐✮ ❚♦♠❡♠♦s ♦ ❝♦♥❥✉♥t♦ A = N∪Im(s)✳ ◆♦t❡ q✉❡ A ⊂N✳ ❚❡♠♦s q✉❡ 0 ∈A ❡ s❡ k∈A ❡♥tã♦ s(k)∈Im(s)✱ ❛ss✐♠s(k)∈A✳ P❡❧♦ ❆①✐♦♠❛ ✸ t❡♠♦s q✉❡A=N✱ ❡ ❝♦♠♦
0∈/ Im(s)✱ ♣❡❧♦ ❆①✐♦♠❛ ✷✱ ❡♥tã♦ Im(s) =N\ {0}✳
◆♦t❛çã♦ ✷✳✶✳✶✳ N\ {0}=N∗.
❉❡✜♥✐çã♦ ✷✳✶✳✷✳ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧n 6= 0✱ ♦ ♥ú♠❡r♦ ♥❛t✉r❛❧mt❛❧ q✉❡s(m) =
n ❝❤❛♠❛✲s❡ ❛♥t❡❝❡ss♦r ❞❡n✱ ❡n ❝❤❛♠❛✲s❡ s✉❝❡ss♦r ❞❡m✳
✷✳✷ ❆❞✐çã♦ ❞❡ ◆ú♠❡r♦s ◆❛t✉r❛✐s
❉❡✜♥✐çã♦ ✷✳✷✳✶✳ ❆ ❛❞✐çã♦ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ m ❡ n✱ é r❡♣r❡s❡♥t❛❞❛ ♣♦r m+n ❡
❞❡✜♥✐❞❛ r❡❝✉rs✐✈❛♠❡♥t❡ ❝♦♠♦✿ ✐✮m+ 0 =m❀
✐✐✮m+s(n) =s(m+n)✳
Pr♦♣♦s✐çã♦ ✷✳✷✳✶✳ ❆ ❛❞✐çã♦ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡♠ N✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ A ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s n ❝✉❥❛s s♦♠❛s m+n ❡stã♦
❜❡♠ ❞❡✜♥✐❞❛s ❡♠ N✳ P❡❧❛ ❝♦♥❞✐çã♦ ✐✮ ❞❛ ❉❡✜♥✐çã♦ ✷✳✷✳✶ t❡♠♦s q✉❡ 0 ∈ A ♣♦✐s m =
m+ 0∈A✳ P❡❧❛ ❝♦♥❞✐çã♦ ✐✐✮ t❛♠❜é♠ ❞❛ ❉❡✜♥✐çã♦ ✷✳✷✳✶ t❡♠♦s q✉❡✱ s❡m+n ❡stá ❜❡♠
❞❡✜♥✐❞♦✱ ♣❛r❛ t♦❞♦n ∈N✱ ❡♥tã♦ m+s(n) = s(m+n) t❛♠❜é♠ ❡stá ❞❡✜♥✐❞♦✱ ♦✉ s❡❥❛✱
n ∈A⇒s(n)∈A.
▲♦❣♦✱ ♣❡❧♦ ❆①✐♦♠❛ ✸ ❛ s♦♠❛ ❡stá ❞❡✜♥✐❞❛ ❡♠N✳
❉❡✜♥✐çã♦ ✷✳✷✳✷✳ ■♥❞✐❝❛r❡♠♦s ♣♦r 1✭❧ê✲s❡ ✉♠✮ ♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ q✉❡ é s✉❝❡ss♦r ❞❡ 0✱ ♦✉ s❡❥❛✱s(0) = 1✳
❆❣♦r❛ q✉❡ ❥á ❞❡✜♥✐♠♦s ❛ ♦♣❡r❛çã♦ ❞❛ ❛❞✐çã♦ ❡♠ N ❡ ♠♦str❛♠♦s q✉❡ ❡stá ❜❡♠
❞❡✜♥✐❞❛✱ ✈❛♠♦s ❞❡♠♦♥str❛r ❛ ✈❛❧✐❞❛❞❡ ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳
Pr♦♣♦s✐çã♦ ✷✳✷✳✷✳ P❛r❛ t♦❞♦ ♥❛t✉r❛❧ m✱ t❡♠✲s❡ s(m) = m + 1 ❡ s(m) = 1 +m✳
P♦rt❛♥t♦m+ 1 = 1 +m✳
❉❡♠♦♥str❛çã♦✳ P❛r❛ ♠♦str❛r ❛ ✐❣✉❛❧❞❛❞❡s(m) =m+ 1✱ ❜❛st❛ ♦❜s❡r✈❛r q✉❡
m+ 1 =m+s(0) =s(m+ 0) =s(m).
❆❣♦r❛✱ ♣❛r❛ ♠♦str❛r ❛ ✐❣✉❛❧❞❛❞❡ s(m) = 1 +m✱ ❝♦♥s✐❞❡r❡♠♦s ✉♠ ❝♦♥❥✉♥t♦
A={m∈N:s(m) = 1 +m}.
◆♦t❡ q✉❡ 0∈A✱ ♣♦✐ss(0) = 1 = 1 + 0✳ ❚♦♠❛♥❞♦m ∈A✱ ♠♦str❛r❡♠♦s q✉❡ s(m)∈ A✳
❉❡ ❢❛t♦✱
s(m) = 1 +m⇒s(s(m)) =s(1 +m) = 1 +s(m).
❆ss✐♠✱ s(m)∈A✳ ❈♦♠ ✐ss♦✱ ♣❡❧♦ ❆①✐♦♠❛ ✸✱ t❡♠♦s A=N✳
❆ ♣❛rt✐r ❞❛q✉✐ ✉t✐❧✐③❛r❡♠♦s t❛♠❜é♠ ❛ ♥♦t❛çã♦ ✐♥❞♦✲❛rá❜✐❝❛ ♣❛r❛ ♦s ❡❧❡♠❡♥t♦s ❡♠
N✱ ♦✉ s❡❥❛✱ ♦ s✉❝❡ss♦r ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧m♣♦❞❡rá s❡r ✐♥❞✐❝❛❞♦ ❝♦♠♦m+1 =s(m)✳
❚❡♦r❡♠❛ ✷✳✷✳✶✳ ❙❡❥❛♠ m✱ n ❡ p ♥ú♠❡r♦s ♥❛t✉r❛✐s q✉❛✐sq✉❡r✳ ❙ã♦ ✈❡r❞❛❞❡✐r❛s ❛s
s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿
✐✮ Pr♦♣r✐❡❞❛❞❡ ❆ss♦❝✐❛t✐✈❛ ❞❛ ❆❞✐çã♦✿ m+ (n+p) = (m+n) +p❀
✐✐✮ Pr♦♣r✐❡❞❛❞❡ ❈♦♠✉t❛t✐✈❛ ❞❛ ❆❞✐çã♦✿ m+n =n+m❀
✐✐✐✮ ▲❡✐ ❞♦ ❈❛♥❝❡❧❛♠❡♥t♦ ❞❛ ❆❞✐çã♦✿ m+p=n+p⇒m=n✳
❉❡♠♦♥str❛çã♦✳ ✐✮ ❋✐①❛♥❞♦ ♦s ♥ú♠❡r♦s m ❡ n ✉t✐❧✐③❛r❡♠♦s ✐♥❞✉çã♦ ❡♠ p✳ ❚♦♠❡ ♦
❝♦♥❥✉♥t♦
A(m,n) ={p∈N:m+ (n+p) = (m+n) +p}.
❚❡♠♦s q✉❡0∈A(m,n)✱ ♣♦✐s
m+ (n+ 0) =m+n= (m+n) + 0.
❚♦♠❛♥❞♦ ✉♠k ∈A(m,n)✱ ♦✉ s❡❥❛✱ m+ (n+k) = (m+n) +k✱ ❡ ✉t✐❧✐③❛♥❞♦ ❛ ❞❡✜♥✐çã♦
❞❡ ❛❞✐çã♦ ❡♠ N t❡♠♦s q✉❡
m+ (n+s(k)) =m+s(n+k) = s(m+ (n+k)) = s((m+n) +k) = (m+n) +s(k).
▲♦❣♦✱ ♣❡❧♦ ❆①✐♦♠❛ ✸✱ A(m,n) = N✳ ❈♦♠♦ m ❡ n sã♦ q✉❛✐sq✉❡r✱ ❡♥tã♦ ❛ ♣r♦♣r✐❡❞❛❞❡
❛ss♦❝✐❛t✐✈❛ ❞❛ ❛❞✐çã♦ ❡♠N ❡stá ♣r♦✈❛❞❛✳
✐✐✮ ❋✐①❛♥❞♦ ♦ ♥ú♠❡r♦m q✉❛❧q✉❡r ♣r♦✈❛r❡♠♦s ♣♦r ✐♥❞✉çã♦ ❡♠n✳ ❚♦♠❡ ♦ ❝♦♥❥✉♥t♦
A(m)={n ∈N:m+n =n+m}.
❚❡♠♦s q✉❡0∈Am✱ ♣♦✐s
m+ 0 =m= 0 +m.
❚♦♠❛♥❞♦k ∈Am✱ ♦✉ s❡❥❛✱ m+k=k+m✱ ❡ ✉t✐❧✐③❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✷✳✷✳✷ t❡♠♦s q✉❡
m+s(k) =s(m+k) =s(k+m) = (k+m) + 1 = 1 + (k+m) = (1 +k) +m=s(k) +m.
▲♦❣♦✱ ♣❡❧♦ ❆①✐♦♠❛ ✸✱ A(m) =N✱ ❡ ❝♦♠♦ m é q✉❛❧q✉❡r✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ❝♦♠✉t❛t✐✈❛ ❞❛
❛❞✐çã♦ ❡♠N ❡stá ♣r♦✈❛❞❛✳
✐✐✐✮ ❋✐①❛♥❞♦ ♦s ♥ú♠❡r♦s m ❡ n q✉❛✐sq✉❡r ❡ t♦♠❛♥❞♦ ✉♠ ❝♦♥❥✉♥t♦
A(m,n)={p∈N:m+p=n+p⇒m=n},
♣r♦✈❛r❡♠♦s ♣♦r ✐♥❞✉çã♦ ❡♠ pq✉❡ A(m,n) =N✳ ❚❡♠♦s q✉❡0∈A(m,n)✱ ♣♦✐s
m+ 0 =n+ 0⇒m=n.
❚♦♠❛♥❞♦k ∈A(m,n)✱ ♦✉ s❡❥❛✱ m+k =n+k ⇒ m =n ❡ ❧❡♠❜r❛♥❞♦ q✉❡s é ✐♥❥❡t♦r❛✱
♣❡❧♦ ❆①✐♦♠❛ ✶✱ t❡♠♦s q✉❡
m+s(k) =n+s(k)⇒s(m+k) =s(n+k)⇒m+k=n+k ⇒m=n.
▲♦❣♦✱ ♣❡❧♦ ❆①✐♦♠❛ ✸✱A(m,n) =N✱ ❡ ❝♦♠♦ m❡ n sã♦ q✉❛✐sq✉❡r✱ ❛ ❧❡✐ ❞❡ ❝❛♥❝❡❧❛♠❡♥t♦
❞❛ ❛❞✐çã♦ ❡♠ N ❡stá ♣r♦✈❛❞❛✳
Pr♦♣♦s✐çã♦ ✷✳✷✳✸✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛u∈N t❛❧ q✉❡m+u=m✱ ♦✉ q✉❡ u+m =m✱
♣❛r❛ t♦❞♦ m ∈ N✳ ❊♥tã♦ u = 0✱ ❡ ❡st❡ é ♦ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ♣❛r❛ ❛ ♦♣❡r❛çã♦ ❞❡
❛❞✐çã♦ ❡♠N✳
❉❡♠♦♥str❛çã♦✳ P❛r❛ ✉♠ t❛❧u t❡♠♦s✿ 0 = 0 +u=u✳
Pr♦♣♦s✐çã♦ ✷✳✷✳✹✳ ❙❡❥❛♠m, n∈N t❛✐s q✉❡ m+n= 0✳ ❊♥tã♦ m=n= 0✳
❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡ n 6= 0✳ ▲♦❣♦ n é s✉❝❡ss♦r ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧n1✱
♦✉ s❡❥❛✱n =n1+ 1✱ ❛ss✐♠✿
0 = m+n=m+ (n1+ 1) = (m+n1) + 1 =s(m+n1).
▼❛s ✐ss♦ é ✉♠ ❛❜s✉r❞♦✱ ♣♦✐s ♣❡❧♦ ❆①✐♦♠❛ ✷ ♦ ③❡r♦ ♥ã♦ é s✉❝❡ss♦r ❞❡ ♥ú♠❡r♦ ❛❧❣✉♠✳ ▲♦❣♦✱ n= 0✳ ❆ss✐♠✱
0 = m+n=m+ 0 =m.
✷✳✸ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ◆ú♠❡r♦s ◆❛t✉r❛✐s
❉❡✜♥✐çã♦ ✷✳✸✳✶✳ ❆ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❞♦✐s ♥ú♠❡r♦sm ❡n ♥❛t✉r❛✐s é r❡♣r❡s❡♥t❛❞❛ ♣♦r m·n ✭♦✉ mn✮ ❡ ❞❡✜♥✐❞❛ r❡❝✉rs✐✈❛♠❡♥t❡ ❝♦♠♦✿
✐✮m·0 = 0;
✐✐✮m·(n+ 1) =mn+m.
❆ ♣❛rt✐r ❞❛ ❉❡✜♥✐çã♦ ✷✳✸✳✶ ♠♦str❛r❡♠♦s ❛ ✈❛❧✐❞❛❞❡ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ♠✉❧t✐♣❧✐✲ ❝❛çã♦ ❡♠ N✳
❚❡♦r❡♠❛ ✷✳✸✳✶✳ ❙❡❥❛♠ m✱ n ❡ p ♥ú♠❡r♦s ♥❛t✉r❛✐s q✉❛✐sq✉❡r✳ ❙ã♦ ✈❡r❞❛❞❡✐r❛s ❛s
s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿
✐✮mn∈N✱ ♦✉ s❡❥❛✱ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡♠ N✳
✐✐✮ ❊❧❡♠❡♥t♦ ◆❡✉tr♦ ❞❛ ▼✉❧t✐♣❧✐❝❛çã♦✿ 1·n=n·1 = n✳
✐✐✐✮ Pr♦♣r✐❡❞❛❞❡ ❉✐str✐❜✉t✐✈❛ ❞❛ ▼✉❧t✐♣❧✐❝❛çã♦ ❡♠ r❡❧❛çã♦ ❛ ❆❞✐çã♦✿ m·(n+p) =
mn+mp ❡ (m+n)·p=mp+np✳
✐✈✮ Pr♦♣r✐❡❞❛❞❡ ❆ss♦❝✐❛t✐✈❛ ❞❛ ▼✉❧t✐♣❧✐❝❛çã♦✿ m·(np) = (mn)·p✳
✈✮ mn= 0⇒m= 0 ♦✉n= 0✳
✈✐✮ Pr♦♣r✐❡❞❛❞❡ ❈♦♠✉t❛t✐✈❛ ❞❛ ▼✉❧t✐♣❧✐❝❛çã♦✿ nm=mn✳
❉❡♠♦♥str❛çã♦✳ ✐✮ ❙❡❥❛ A ♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s t❛✐s q✉❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦
❡♥tr❡ q✉❛✐sq✉❡r ❞♦✐s ❡❧❡♠❡♥t♦s ❞❡ss❡ ❝♦♥❥✉♥t♦ é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ P❡❧❛ ❉❡✜♥✐çã♦ ✷✳✸✳✶ ♦❜t❡♠♦s q✉❡ 0 ∈ A✱ ♣♦✐s ❞❛❞♦ ✉♠ m ∈ N q✉❛❧q✉❡r✱ m·0 = 0 ∈ N✱ ❧♦❣♦ 0 ∈ A✳
❙✉♣♦♥❤❛♠♦s q✉❡ ✉♠ ♥❛t✉r❛❧nq✉❛❧q✉❡r ❡st❡❥❛ ❡♠A✱ ❧♦❣♦✱ ❞❛❞♦m∈Nt❡♠♦smn∈N✳
◗✉❡r❡♠♦s ♠♦str❛r q✉❡s(n) =n+ 1∈A✱ ♦✉ s❡❥❛m·(n+ 1)∈N✳ P❡❧❛ ❉❡✜♥✐çã♦ ✷✳✸✳✶
t❡♠♦s
m·(n+ 1) =mn+m.
P♦r ❤✐♣ót❡s❡✱mn∈N❡ t❡♠♦sm ∈N✱ ❧♦❣♦mn+m∈N✱ ♣♦✐s ❛ s♦♠❛ ❡stá ❜❡♠ ❞❡✜♥✐❞❛
❡♠ N✳ ❆ss✐♠✱ ♣❡❧♦ ❆①✐♦♠❛ ✸✱ A=N ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡♠N✳
✐✐✮ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡n·1 =n✳ ❚❡♠♦s q✉❡✿
n·1 =n·(0 + 1) =n·0 +n·1 = 0 +n =n.
❆ss✐♠✱ n·1 =n✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ n = 0 t❡♠♦s q✉❡ 1·0 = 0✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ ✐❣✉❛❧❞❛❞❡1·n=n ✈❛❧❡ ♣❛r❛ ✉♠ n ♥❛t✉r❛❧ q✉❛❧q✉❡r✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ ✈❛❧❡ ♣❛r❛ n+ 1✱ ♦✉ s❡❥❛✱ 1·(n+ 1) =n+ 1✳ ❉❡ ❢❛t♦✱ ✉t✐❧✐③❛♥❞♦ ❛ ❉❡✜♥✐çã♦ ✷✳✸✳✶✱
1·(n+ 1) = 1·n+ 1·1 =n+ 1.
❆ss✐♠✱ ♣❡❧♦ ❆①✐♦♠❛ ✸✱ ❛ ❛✜r♠❛çã♦
1·n=n·1 = n
é ✈á❧✐❞❛✱ ♣❛r❛ t♦❞♦ n∈N✳
✐✐✐✮ Pr✐♠❡✐r♦ ✈❛♠♦s ♠♦str❛r q✉❡ m ·(n +p) = mn+mp✳ ❙❡❥❛♠ m, n ♥ú♠❡r♦s
♥❛t✉r❛✐s q✉❛✐sq✉❡r ❡ ❝♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦
A ={p∈N:m·(n+p) =mn+mp}.
❚❡♠♦s q✉❡0∈A✱ ♣♦✐s
m·(n+ 0) =m·n=mn+ 0 =mn+m·0.
❙✉♣♦♥❤❛♠♦s q✉❡ q ∈ A✱ ♦✉ s❡❥❛✱ m ·(n +q) = mn+mq✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ q+ 1∈A✱ ♦✉ s❡❥❛✱
m·(n+ (q+ 1)) =mn+m·(q+ 1) =mn+ (mq+m).
❚❡♠♦s
m·(n+ (q+ 1)) =m·(n+s(q)) =m·s(n+q) = m·((n+q) + 1).
❯s❛♥❞♦ ❛ ❉❡✜♥✐çã♦ ✷✳✸✳✶✱ ♦❜t❡♠♦s
m·((n+q) + 1) =m·(n+q) +m.
❆❣♦r❛✱ ✉s❛♥❞♦ ❛ ❤✐♣ót❡s❡ ♦❜t❡♠♦s
m·(n+q) +m = (mn+mq) +m.
❈♦♠♦ ✈❛❧❡ ❛ ❧❡✐ ❛ss♦❝✐❛t✐✈❛ ♣❛r❛ ❛❞✐çã♦✱ ♦❜t❡♠♦s
m·(n+ (q+ 1)) =mn+ (mq+m),
❝♦♠♦ q✉❡rí❛♠♦s✳ ❆ss✐♠ ✱ ♣❡❧♦ ❆①✐♦♠❛ ✸✱A=N✱ ♦✉ s❡❥❛✱m·(n+p) = mn+mp✱ ♣❛r❛
q✉❛✐q✉❡rm, n, p ∈N.❘❡st❛ ♠♦str❛r (m+n)·p=mp+np✳ ■ss♦ s❡❣✉❡ ❞♦ ♣r♦♣r✐❡❞❛❞❡
❝♦♠✉t❛t✐✈❛ q✉❡ ❞❡♠♦♥str❛r❡♠♦s ♥♦ ✐t❡♠ ✈✐✮✳
✐✈✮ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ m·(np) = (mn)· p✳ ❙❡❥❛♠ m ❡ n ♥ú♠❡r♦s ♥❛t✉r❛✐s
q✉❛✐sq✉❡r ❡ ❝♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦
A ={p∈N:m·(np) = (mn)·p}.
❚❡♠♦s q✉❡0∈A✱ ♣♦✐s m·(n·0) =m·0 = 0 ❡(mn)·0 = 0✱ ❧♦❣♦✱m·(n·0) = (mn)·0.
❙✉♣♦♥❤❛♠♦s q✉❡q∈A✱ ♦✉ s❡❥❛✱m·(nq) = (mn)·q✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡q+ 1 ∈A✱
♦✉ s❡❥❛✱ m·(n·(q+ 1)) = (mn)·(q+ 1)✳ ❯s❛♥❞♦ ❛ ❉❡✜♥✐çã♦ ✷✳✸✳✶ ❡ ❛ ♣r♦♣r✐❡❞❛❞❡
❞✐str✐❜✉t✐✈❛ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ t❡♠♦s
m·(n·(q+ 1)) =m·(nq+n) =m·(nq) +mn.
❯t✐❧✐③❛♥❞♦ ❛ ❤✐♣ót❡s❡ t❡♠♦s
m·(nq) +mn= (mn)·q+mn.
❈♦♠♦(mn)·q+mn= (mn)·(q+ 1)✱ ♦❜t❡♠♦sm·(n·(q+ 1)) = (mn)·(q+ 1)❡ ❛ss✐♠✱ ♣❡❧♦ ❆①✐♦♠❛ ✸✱ A=N✳ ▲♦❣♦✱m·(np) = (mn)·p✱ ♣❛r❛ q✉❛✐sq✉❡r m, n, p ∈N✳
✈✮ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ mn= 0⇒m= 0 ♦✉n= 0✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱
s✉♣♦♥❤❛♠♦s q✉❡n 6= 0✳ ▲♦❣♦né s✉❝❡ss♦r ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧n1✱ ♦✉ s❡❥❛✱n =n1+1✱
❛ss✐♠✱
0 =mn=m·(n1+ 1) =mn1+m.
P❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✳✹ t❡♠♦s q✉❡mn1 =m= 0✱ ❧♦❣♦m = 0✳
✈✐✮ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ nm = mn✳ ❙❡❥❛ m ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ q✉❛❧q✉❡r ❡
❝♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦
A={n∈N:nm=mn}.
Pr✐♠❡✐r♦ ✈❛♠♦s ♠♦str❛r q✉❡ 0 ∈ A✳ ❉❡ ❢❛t♦✱ t❡♠♦s m·0 = 0✱ ♣❡❧❛ ❉❡✜♥✐çã♦ ✷✳✸✳✶✳ P❛r❛ ♣r♦✈❛r q✉❡0·m = 0 ❝♦♥s✐❞❡r❡♠♦s ♦ ❝♦♥❥✉♥t♦
B ={m∈N: 0·m= 0}.
❚❡♠♦s q✉❡ 0 ∈ B✱ ♣♦✐s 0· 0 = 0✱ ♣❡❧❛ ❉❡✜♥✐çã♦ ✷✳✸✳✶✳ ❈♦♥s✐❞❡r❛♥❞♦ ✉♠ ♥ú♠❡r♦ q✉❛❧q✉❡r m ∈ B✱ ♦✉ s❡❥❛✱ 0·m = 0✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ 0·(m + 1) = 0✳ P❡❧❛
❉❡✜♥✐çã♦ ✷✳✸✳✶ t❡♠♦s q✉❡0·(m+ 1) = 0·m+ 0✳ ❯t✐❧✐③❛♥❞♦ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡m ∈B✱
♦✉ s❡❥❛✱ 0·m= 0 ♦❜t❡♠♦s0·(m+ 1) = 0✳ ▲♦❣♦✱ ♣❡❧♦ ❆①✐♦♠❛ ✸✱ t❡♠♦s q✉❡B =N✱
♦✉ s❡❥❛✱0·m = 0✳ P♦rt❛♥t♦✱ 0∈A✳
❆❣♦r❛ s✉♣♦♥❤❛♠♦s q✉❡ nm = mn ♣❛r❛ ❛❧❣✉♠ n ∈ A✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ n+ 1∈A✱ ♦✉ s❡❥❛✱ (n+ 1)·m =m·(n+ 1)✳ P❡❧❛ ❉❡✜♥✐çã♦ ✷✳✸✳✶ ❡ ✉t✐❧✐③❛♥❞♦ ❛ ❤✐♣ót❡s❡
t❡♠♦s✱
m·(n+ 1) =mn+m=nm+m.
❉❡ss❛ ❢♦r♠❛ t❡♠♦s
(n+ 1)·m= (n+s(0))·m =s(n+ 0)·m=s(n)·m.
P❡❧❛ ❉❡✜♥✐çã♦ ✷✳✸✳✶ ♦❜s❡r✈❛♠♦s q✉❡
n·2 =n·(1 + 1) =n·1 +n=n+n
| {z }
2 vezes
,
n·3 =n·(2 + 1) =n·2 +n=n+n+n
| {z }
3 vezes
,
n·4 = n·(3 + 1) = n·3 +n =n+n+n+n
| {z }
4 vezes
.
❉❡ss❛ ❢♦r♠❛ t❡♠♦s
s(n)·m=s(n) +s(n) +...+s(n)
| {z }
m vezes
= (n+ 1) + (n+ 1) +...+ (n+ 1)
| {z }
m vezes
.
❈♦♠♦ ❛ ❛❞✐çã♦ é ❛ss♦❝✐❛t✐✈❛ ❡ ❝♦♠✉t❛t✐✈❛ ❡ ✉s❛♥❞♦ ❛ ❉❡✜♥✐çã♦ ✷✳✸✳✶ t❡♠♦s
(n+ 1) + (n+ 1) +...+ (n+ 1)
| {z }
m vezes
=n+n+...+n
| {z }
m vezes
+ 1 + 1 +...+ 1
| {z }
m vezes
=nm+1·m=nm+m.
❖✉ s❡❥❛✱
(n+ 1)·m =nm+m=m·(n+ 1).
❆ss✐♠✱ ♣❡❧♦ ❆①✐♦♠❛ ✸✱ t❡♠♦s q✉❡A =N✱ ♦✉ s❡❥❛✱ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦
é ✈á❧✐❞❛ ❡♠ N✳
✷✳✹ ❘❡❧❛çã♦ ❞❡ ♦r❞❡♠ ❡♠
N
❱❛♠♦s ❛❣♦r❛ ♠♦str❛r q✉❡ Né ✉♠ ❝♦♥❥✉♥t♦ ♦r❞❡♥❛❞♦✳ P❛r❛ ✐ss♦ ✈❛♠♦s ❡st❛❜❡❧❡❝❡r
✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ ❡♠N ✉t✐❧✐③❛♥❞♦ ❛ ✐❞❡✐❛ ❞❡ r❡❧❛çã♦ ❜✐♥ár✐❛✳
❉❡✜♥✐çã♦ ✷✳✹✳✶✳ ❯♠❛ r❡❧❛çã♦ ❜✐♥ár✐❛ R ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ A ❞✐③✲s❡ ✉♠❛
r❡❧❛çã♦ ❞❡ ♦r❞❡♠ ❡♠A q✉❛♥❞♦ s❛t✐s❢❛③❡r ❛s ❝♦♥❞✐çõ❡s ❛ s❡❣✉✐r✱ ♣❛r❛ t♦❞♦sx, y, z ∈A✿
✐✮ ❘❡✢❡①✐✈✐❞❛❞❡✿ xRx✳
✐✐✮ ❆♥t✐ss✐♠❡tr✐❛✿ s❡ xRy ❡yRx✱ ❡♥tã♦ x=y✳
✐✐✐✮ ❚r❛♥s✐t✐✈✐❞❛❞❡✿ s❡ xRy ❡ yRz✱ ❡♥tã♦ xRz✳
❯♠ ❝♦♥❥✉♥t♦ q✉❛❧q✉❡r ♥ã♦ ✈❛③✐♦ ♠✉♥✐❞♦ ❞❡ ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ é ❝❤❛♠❛❞♦ ❞❡ ❝♦♥❥✉♥t♦ ♦r❞❡♥❛❞♦✳
❉❡✜♥✐çã♦ ✷✳✹✳✷✳ ❉❛❞♦sm, n∈N✱ ❞✐③❡♠♦s q✉❡mRns❡ ❡①✐st✐rp∈Nt❛❧ q✉❡n=m+p✳
❉❡✜♥✐çã♦ ✷✳✹✳✸✳ P❛r❛ m, n ∈ N✱ s❡ mRn ❞✐③❡♠♦s q✉❡ m é ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ n✳ P❛ss❛r❡♠♦s ❛ ❡s❝r❡✈❡r ♦ sí♠❜♦❧♦ ≤ ♥♦ ❧✉❣❛r ❞❡R✱ ❡ ❞❡ss❡ ♠♦❞♦✱ m≤n s✐❣♥✐✜❝❛rá mRn✳
◆♦t❛çã♦ ✷✳✹✳✶✳ ✶✿ ❙❡ m ≤ n✱ ♣♦ré♠ m 6= n✱ ❡s❝r❡✈❡♠♦s m < n ❡ ❞✐③❡♠♦s q✉❡ m é
♠❡♥♦r ❞♦ q✉❡n✳
✷✿ ❊s❝r❡✈❡♠♦s n ≥m ❝♦♠♦ ♦♣çã♦ ♣❛r❛ m ≤n✳ ▲ê✲s❡n é ♠❛✐♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ m✳
✸✿ ❊s❝r❡✈❡♠♦sn > m ❝♦♠♦ ♦♣çã♦ ♣❛r❛ m < n✳ ▲ê✲s❡ n é ♠❛✐♦r ❞♦ q✉❡ m✳
Pr♦♣♦s✐çã♦ ✷✳✹✳✶✳ ▲❡✐ ❞❛ ❚r✐❝♦t♦♠✐❛✿ ♣❛r❛ q✉❛✐sq✉❡rm, n∈N✱ t❡♠♦s ✉♠❛✱ ❡ s♦♠❡♥t❡
✉♠❛✱ ❞❛s r❡❧❛çõ❡s s❡❣✉✐♥t❡s ♦❝♦rr❡♥❞♦✿ ✐✮m < n✳
✐✐✮m =n✳
✐✐✐✮ m > n✳
❉❡♠♦♥str❛çã♦✳ ◆♦t❡ q✉❡ s❡ ♦❝♦rr❡r ✐✮ ❡ ✐✐✐✮✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ m < n ❡ m > n✱
s✉❜❡♥t❡♥❞❡✲s❡ q✉❡ m 6= n✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ✐✐✮✱ q✉❡ ❛✜r♠❛ m = n✳ ▲♦❣♦ ♥ã♦ ♣♦❞❡
♦❝♦rr❡r ❛s três r❡❧❛çõ❡s ❛♦ ♠❡s♠♦ t❡♠♣♦✳ ◆♦t❡ t❛♠❜é♠ q✉❡ ✐✮ ❡ ✐✐✐✮ sã♦ ✐♥❝♦♠♣❛tí✈❡✐s✱ ♣♦✐s s❡m < n ❡♥tã♦ ❡①✐st❡ p∈N∗ t❛❧ q✉❡ m+p=n ❡ s❡n < m ❡♥tã♦ ❡①✐st❡ p
1 ∈ N∗
t❛❧ q✉❡ n+p1 =m✳ ▲♦❣♦✱
m+p=n⇒(n+p1) +p=n⇒n+ (p1+p) =n⇒(p1+p) = 0.
P❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✳✹ t❡♠♦s q✉❡p1 =p= 0✱ ❝♦♥tr❛❞✐③❡♥❞♦ ❛s ❤✐♣ót❡s❡s ❛❞♦t❛❞❛s✳
❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ❛♣❡♥❛s ✉♠❛ ❞❛s r❡❧❛çõ❡s ♦❝♦rr❡✳ ❙❡❥❛
m ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ q✉❛❧q✉❡r ❡ ❝♦♥s✐❞❡r❡ ✉♠ ❝♦♥❥✉♥t♦
M ={x∈N:x=m, x > m ♦✉ x < m}.
❚❡♠♦s q✉❡0∈M✱ ♣♦✐s0 = m ♦✉0< m✳ ❙✉♣♦♥❤❛♠♦s q✉❡k ∈M✳ ◗✉❡r❡♠♦s ♠♦str❛r
q✉❡k+ 1∈M✳ ❈♦♠♦ k∈M ♣♦❞❡♠♦s t❡r ♦s s❡❣✉✐♥t❡s ❝❛s♦s✿
✶♦ ❝❛s♦✿ k=m✳ ❚❡♠♦s q✉❡ k+ 1 =m+ 1✱ ♦✉ s❡❥❛✱ k+ 1> m✳ ❆ss✐♠ k+ 1∈M✳
✷♦ ❝❛s♦✿ k < m✳ ❚❡♠♦s q✉❡ ❡①✐st❡ p∈N∗ t❛❧ q✉❡ k+p=m✱ ♦✉ s❡❥❛✱
(k+p) + 1 =m+ 1⇒k+ (p+ 1) =m+ 1⇒k+ (1 +p) =m+ 1⇒(k+ 1) +p=m+ 1.
❈♦♠♦p∈N∗✱ s❡ p= 1 ❡♥tã♦ (k+ 1) =m✳ ❆ss✐♠ k+ 1 ∈M✳
✸♦ ❝❛s♦✿ k > m✳ ❚❡♠♦s q✉❡ ❡①✐st❡ p∈N∗✱ t❛❧ q✉❡ k=m+p✱ ♦✉ s❡❥❛✱
k+ 1 = (m+p) + 1⇒k+ 1 =m+ (p+ 1) ⇒k+ 1 > m.
❆ss✐♠✱ k+ 1∈ M✳ P❡❧♦ ❆①✐♦♠❛ ✸ t❡♠♦s q✉❡ M =N✳ ❆ss✐♠✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ▲❡✐
❞❛ ❚r✐❝♦t♦♠✐❛ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦sm, n∈N✳
❯♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ q✉❡ ♦❜❡❞❡❝❡ à ▲❡✐ ❞❛ ❚r✐❝♦t♦♠✐❛ é ❝❤❛♠❛❞❛ ❞❡ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ t♦t❛❧✳ ❆ss✐♠✱ t❡♠♦s ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ t♦t❛❧ ❡♠N.
❚❡♦r❡♠❛ ✷✳✹✳✶✳ ❈♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❡♠N✿ s❡❥❛♠
a✱b ❡c ♥ú♠❡r♦s ♥❛t✉r❛✐s q✉❛✐sq✉❡r✳ ❙ã♦ ✈❡r❞❛❞❡✐r❛s ❛s s❡❣✉✐♥t❡s ✐♠♣❧✐❝❛çõ❡s✿
✐✮a ≤b⇒a+c≤b+c✳
✐✐✮a≤b ⇒ac≤bc✳
❉❡♠♦♥str❛çã♦✳ ✐✮ ❙❡ a≤b ❡♥tã♦ ❡①✐st❡ m∈N t❛❧ q✉❡ a+m=b✳ ❉❡ss❛ ❢♦r♠❛✱
(a+m) +c=b+c⇒a+ (m+c) =b+c⇒a+ (c+m) = b+c⇒(a+c) +m =b+c.
❆ss✐♠✱ a+c≤b+c✳
✐✐✮ ❙❡ a≤b ❡♥tã♦ ❡①✐st❡ m∈N t❛❧ q✉❡ a+m=b✳ ❉❡ss❛ ❢♦r♠❛✱
(a+m)·c=bc⇒ac+mc=bc.
❆ss✐♠✱ ac≤bc✳
❚❡♦r❡♠❛ ✷✳✹✳✷✳ ▲❡✐ ❞♦ ❈❛♥❝❡❧❛♠❡♥t♦ ❞❛ ▼✉❧t✐♣❧✐❝❛çã♦✿ s❡❥❛♠a, b, c∈N✱ ❝♦♠c6= 0✳ ❙❡ac=bc ❡♥tã♦ a =b✳
❉❡♠♦♥str❛çã♦✳ ❚♦♠❛♥❞♦a6=b t❡♠♦s ❛s s❡❣✉✐♥t❡s ♣♦ss✐❜✐❧✐❞❛❞❡s✿
✐✮ ❙❡ a > b ❡♥tã♦ ❡①✐st❡ m1 ∈N∗ t❛❧ q✉❡ a=b+m1✳ ❉❡ss❛ ❢♦r♠❛
ac= (b+m1)·c⇒ac=bc+m1c.
❚❡♠♦s m1 6= 0 ❡ c6= 0✱ ❛ss✐♠✱ ac > bc✳
✐✐✮ ❙❡ a < b ❡♥tã♦ ❡①✐st❡m2 ∈N∗ t❛❧ q✉❡ a+m2 =b✳ ❉❡ss❛ ❢♦r♠❛
(a+m2)·c=bc⇒ac+m2c=bc.
