❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❈â♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦
❖ ✉s♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ❈r✐♣t♦❣r❛✜❛ ❝♦♠♦
❡stí♠✉❧♦ ♠❛t❡♠át✐❝♦ ♥❛ s❛❧❛ ❞❡ ❛✉❧❛✳
▲❡❛♥❞r♦ ❘♦❞r✐❣✉❡s ❞❡ ❈❛r✈❛❧❤♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r✲ ❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡
❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ❊r✐❦❛ ❈❛♣❡❧❛t♦
Carvalho, Leandro Rodrigues de
O uso de elementos da criptografia como estímulo matemático na sala de aula / Leandro Rodrigues de Carvalho. - Rio Claro, 2016
78 f. : il., figs., tabs., fots.
Dissertação (mestrado) - Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas
Orientador: Erika Capelato
1. Teoria dos números. 2. Números primos. 3. Cifra de César. 4. Atividade para sala de aula. I. Título.
512.7 C331u
Ficha Catalográfica elaborada pela STATI - Biblioteca da UNESP Campus de Rio Claro/SP
❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖
▲❡❛♥❞r♦ ❘♦❞r✐❣✉❡s ❞❡ ❈❛r✈❛❧❤♦
❖ ✉s♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ❈r✐♣t♦❣r❛❢✐❛ ❝♦♠♦ ❡stí♠✉❧♦
♠❛t❡♠át✐❝♦ ♥❛ s❛❧❛ ❞❡ ❛✉❧❛✳
❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿
Pr♦❢❛✳ ❉r❛✳ ❊r✐❦❛ ❈❛♣❡❧❛t♦ ❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ❘❡♥❛t❛ ❩♦t✐♥ ●♦♠❡s ❞❡ ❖❧✐✈❡✐r❛ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ■●❈❊ ✲ ❯◆❊❙P
Pr♦❢❛✳ ❉r❛✳ ❈❛♠✐❧❛ ❋❡r♥❛♥❞❛ ❇❛ss❡tt♦ ❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ❡ ▲❡tr❛s ✲ ❯◆❊❙P
❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡ ❛❣r❛❞❡ç♦ ❛ ❉❊❯❙✱ ❣r❛♥❞❡ ❝r✐❛❞♦r ❞♦ ✉♥✐✈❡rs♦✱ s❡♠ ❡❧❡ ♥❡♠ ♠❡s♠♦ ❡st❛rí❛♠♦s ❛q✉✐✳ ❆♦s ♠❡✉s ♣❛✐s ♣♦r t❡r❡♠ ♠❡ ❝♦♥❝❡❜✐❞♦ ❡ ❝r✐❛❞♦ ❝♦♠ t♦❞♦ ❛♠♦r ❡ ❝❛r✐♥❤♦✳ ❆❣r❛❞❡ç♦ ✐♠❡♥s❛♠❡♥t❡ ❛ ♠✐♥❤❛ ❝♦♠♣❛♥❤❡✐r❛ ❡ ❡s♣♦s❛ ❈r✐st✐❛♥❡ ♣❡❧♦ ❛♣♦✐♦ ❡ ♠♦t✐✈❛çã♦ ❡ ❛♦s ♥♦ss♦s ❣❛t♦s q✉❡ s❡♠♣r❡ ♠❡ ❛❝♦♠♣❛♥❤❛r❛♠ ♥❛s ♠❛❞r✉❣❛❞❛s ❞❡ ❡s✲ t✉❞♦✳
❊♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ➱r✐❦❛ ❈❛♣❡❧❛t♦✱ ♣❡❧❛s ♦r✐❡♥t❛çõ❡s ❡ ❝♦rr❡çõ❡s✱ s❡♠♣r❡ s♦❧í❝✐t❛ ❡ ❡❞✉❝❛❞❛✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ❝✉rs♦✱ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❡ ♣❛❝✐ê♥❝✐❛✱ ♦ ♠❡✉ ❛♣r❡♥❞✐③❛❞♦ ❢♦✐ ❛❧é♠ ❞♦s ❝♦♥t❡ú❞♦s✳ ❊♠ ❡s♣❡❝✐❛❧ ❛❣r❛❞❡ç♦ ❛ ♣r♦❢❡ss♦r❛ ❘❡♥❛t❛ ❩♦t✐♥ ●♦♠❡s ❞❡ ❖❧✐✲ ✈❡✐r❛✱ ♣❡❧❛s ♦r✐❡♥t❛çõ❡s r❡❢❡r❡♥t❡s ❛♦ ❝❛♣ít✉❧♦ s♦❜r❡ ❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ❡ ❛ ♣r♦❢❡ss♦r❛ ❙✉③✐♥❡✐ ▼❛r❝♦♥❛t♦✱ ❝♦♦r❞❡♥❛❞♦r❛ ❞❡st❡ ♣r♦❣r❛♠❛ ❞✉r❛♥t❡ ♠✐♥❤❛ tr❛❥❡tór✐❛✱ q✉❡ s❡♠✲ ♣r❡ ♥♦s ❛✉①✐❧✐♦✉ ❞❡ ♠❛♥❡✐r❛ ♣r❡st✐♠♦s❛✳
◆ã♦ ♣♦❞❡r✐❛ ♠❡ ❡sq✉❡❝❡r ❞♦s ❛♠✐❣♦s ❡ ❛♠✐❣❛s ❞❡ t✉r♠❛✱ ♥♦ ❝♦♠❡ç♦ ♣❛r❡❝✐❛ ✉♠❛ ❝♦♠♣❡t✐çã♦ ✭s❛✉❞á✈❡❧✮ ❡ ❛♦ ❧♦♥❣♦ ❞♦ ❝✉rs♦ s❡ tr❛♥s❢♦r♠♦✉ ❡♠ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ❞❡✲ ❞✐❝❛çã♦✱ ♣❡r❝❡❜❡♠♦s q✉❡ ❞✐✈✐❞✐r ❝♦♥❤❡❝✐♠❡♥t♦ só ♥♦s ❡♥❣r❛♥❞❡❝❡✳ ❋♦r❛♠ t❛♥t❛s ❛s ❛♠✐③❛❞❡s q✉❡ s❡r✐❛ ✐♥❥✉st♦ ❡sq✉❡❝❡r ❛❧❣✉♥s ♥♦♠❡s✱ ♣♦r ✐ss♦ ❛❝❤❡✐ ♠❡❧❤♦r ♦♠✐t✐✲❧♦s✱ ♠❛s t♦❞♦s ❡stã♦ ♥♦ ♠❡✉ ❝♦r❛çã♦✳
❆ ♠❛t❡♠át✐❝❛ é ♦ ❛❧❢❛❜❡t♦ ❝♦♠ ♦ q✉❛❧ ❉❊❯❙ ❡s❝r❡✈❡✉ ♦ ✉♥✐✈❡rs♦✳
❘❡s✉♠♦
❖ ❣r❛♥❞❡ ❞❡s❛✜♦ ♥♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛✱ ♣❡❧♦ ♠❡♥♦s ♥♦ ♠❡✉ ♣♦♥t♦ ❞❡ ✈✐st❛ ❝♦♠♦ ♣r♦❢❡ss♦r ♥♦s ú❧t✐♠♦s ❞❡③ ❛♥♦s✱ é ❢❛③❡r ❝♦♠ q✉❡ ♦s ❛❧✉♥♦s ♣❡r❝❡❜❛♠ ❛ ✐♠♣♦rtâ♥❝✐❛ ❡ ❛ ♣r❛t✐❝✐❞❛❞❡ ❞❛ ♠❛t❡♠át✐❝❛ ❡♠ s✉❛s ✈✐❞❛s✳ ■ss♦ ✈❛✐ ❛❧é♠ ❞❛s t❡♦r✐❛s ❞❛ ❆r✐t♠ét✐❝❛✱ ➪❧❣❡❜r❛ ♦✉ ●❡♦♠❡tr✐❛ ❡♥s✐♥❛❞❛s ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✳ ❖s ❛❧✉♥♦s ♣r❡❝✐s❛♠ ♣❡r❝❡❜❡r q✉❡ ♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s sã♦ ❢❡rr❛♠❡♥t❛s q✉❡ ♦s ❛❥✉❞❛♠ ❛ ❝♦♠♣r❡❡♥❞❡r ♦ ♠✉♥❞♦ ❛ s✉❛ ✈♦❧t❛✳ ❉✐❛♥t❡ ❞✐st♦✱ ❡st❛ ❞✐ss❡rt❛çã♦ ❜✉s❝❛ ❛♣r❡s❡♥t❛r ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s q✉❡ ❧❡✈❛♠ à ❝♦♠♣r❡❡♥sã♦ ❞❛ ❈r✐♣t♦❣r❛✜❛✿ ❝♦♥❝❡✐t♦s ❞❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ❡ ❞❛ ➪❧❣❡❜r❛✳ ❋❛③❡♠♦s ❛✐♥❞❛✱ ✉♠ ❜r❡✈❡ ❤✐stór✐❝♦ s♦❜r❡ ❛ ❈r✐♣t♦❣r❛✜❛ ❞❡s❝r❡✈❡♥❞♦ ❛ ❝✐❢r❛ ❞❡ ❈és❛r ❡ ❛s ❝✐❢r❛s ❛✜♥s✱ ♦ ❙✐st❡♠❛ ❘❙❆ ❡ ❛❧❣✉♥s ♠ét♦❞♦s ❞❡ tr♦❝❛ ❞❡ ❝❤❛✈❡s✳ ❘❡❧❛t❛♠♦s ❛❧❣✉♥s tr❛❜❛❧❤♦s ❞❡s❡♥✈♦❧✈✐❞♦s ♣❡❧♦s ❡st✉❞❛♥t❡s ❞♦ P❘❖❋▼❆❚ ♥❡st❡ t❡♠❛ ❡ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ♣r♦♣♦st❛ ❞❡ ❛t✐✈✐❞❛❞❡ ♣❛r❛ ♦s ❡st✉❞❛♥t❡s ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦✳ ❊st❛ ❛t✐✈✐❞❛❞❡ ❝♦♥s✐st❡ ♥❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ❦✐t ❞❡ ❡♥❝r✐♣t❛çã♦ ❡ ❞❡❝r✐♣t❛çã♦ ✉t✐❧✐③❛♥❞♦ ❝♦♣♦s ❞❡s❝❛rtá✈❡✐s✳ ❈♦♠ ❞✐♥â♠✐❝❛s ✉♥✐♥❞♦ ❡❧❡♠❡♥t♦s ❞❛ ❈r✐♣t♦❣r❛✜❛ ❡ ♦ ❛♣❧✐❝❛t✐✈♦ ❲❤❛ts❛♣♣✱ ❝♦♠♦ ♠❡✐♦ ❞❡ tr♦❝❛ ❞❛s ♠❡♥s❛❣❡♥s ❝r✐♣t♦❣r❛❢❛❞❛s✱ ♠♦t✐✈❛♠♦s ❛ s❛❧❛ ❞❡ ❛✉❧❛ ♣❛r❛ ♦ ❛♣r❡♥❞✐③❛❞♦ ❞❛ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛ ❡ ❞❛ P❡r♠✉t❛çã♦✳ ❆❧é♠ ❞✐ss♦✱ ♣r❡t❡♥❞❡♠♦s ❞❡s♣❡rt❛r ♥♦s ❛❧✉♥♦s ♦ ✐♥t❡r❡ss❡ ❡♠ ❛♣r♦❢✉♥❞❛r✲s❡ ♥♦s ❡st✉❞♦s ❞❛ ▼❛t❡♠át✐❝❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✱ ❥á q✉❡ ❡st❛ é ✉♠❛ ❞❛s ❢❡rr❛♠❡♥t❛s ❢✉♥❞❛♠❡♥t❛✐s ♥♦ ❝♦♥t❡①t♦ ❞❛ ❈r✐♣t♦❣r❛✜❛✱ ✉♠❛ ❝✐ê♥❝✐❛ ❝♦♠ ❣r❛♥❞❡ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ♥❛ ❛t✉❛❧✐❞❛❞❡✳
❆❜str❛❝t
❚❤❡ ❣r❡❛t ❝❤❛❧❧❡♥❣❡ ✐♥ t❡❛❝❤✐♥❣ ♠❛t❤❡♠❛t✐❝s✱ ❛t ❧❡❛st ✐♥ ♠② ♣♦✐♥t ♦❢ ✈✐❡✇ ❛s ❛ t❡❛❝❤❡r ✐♥ t❤❡ ♣❛st t❡♥ ②❡❛rs ✐s t♦ ♠❛❦❡ st✉❞❡♥ts ✉♥❞❡rst❛♥❞ t❤❡ ✐♠♣♦rt❛♥❝❡ ❛♥❞ ♣r❛❝✲ t✐❝❛❧✐t② ♦❢ ♠❛t❤❡♠❛t✐❝s ✐♥ t❤❡✐r ❧✐✈❡s✳ ❚❤✐s ❣♦❡s ❜❡②♦♥❞ t❤❡ t❤❡♦r✐❡s ♦❢ ❛r✐t❤♠❡t✐❝✱ ❛❧❣❡❜r❛ ♦r ❣❡♦♠❡tr② t❛✉❣❤t ✐♥ ❜❛s✐❝ ❡❞✉❝❛t✐♦♥✳ ❙t✉❞❡♥ts ♥❡❡❞ t♦ r❡❛❧✐③❡ t❤❛t ♠❛t❤❡✲ ♠❛t✐❝❛❧ ❝♦♥❝❡♣ts ❛r❡ t♦♦❧s t❤❛t ❤❡❧♣ t❤❡♠ ✉♥❞❡rst❛♥❞ t❤❡ ✇♦r❧❞ ❛r♦✉♥❞ t❤❡♠✳ ■♥ ✈✐❡✇ ♦❢ t❤✐s✱ t❤✐s ❞✐ss❡rt❛t✐♦♥ ❛✐♠s t♦ ♣r❡s❡♥t ♠❛t❤❡♠❛t✐❝❛❧ ❝♦♥❝❡♣ts t❤❛t ❧❡❛❞ t♦ ✉♥❞❡rs✲ t❛♥❞✐♥❣ ♦❢ ❝r②♣t♦❣r❛♣❤②✿ ❝♦♥❝❡♣ts ♦❢ ♥✉♠❜❡r t❤❡♦r② ❛♥❞ ❛❧❣❡❜r❛✳ ❲❡ ❛❧s♦ ❛ ❜r✐❡❢ ❤✐st♦r② ♦♥ t❤❡ ❊♥❝r②♣t✐♦♥ ❞❡s❝r✐❜✐♥❣ t❤❡ ❈❛❡s❛r ❝✐♣❤❡r ❛♥❞ r❡❧❛t❡❞ ✜❣✉r❡s✱ t❤❡ ❘❙❆ s②st❡♠ ❛♥❞ s♦♠❡ ♠❡t❤♦❞s ♦❢ ❦❡② ❡①❝❤❛♥❣❡✳ ❲❡ r❡♣♦rt s♦♠❡ ✇♦r❦ ❞♦♥❡ ❜② st✉❞❡♥ts P❘❖❋▼❆❚ t❤✐s t❤❡♠❡ ❛♥❞ ♣r❡s❡♥t ❛ ♣r♦♣♦s❛❧ ❛❝t✐✈✐t② ❢♦r st✉❞❡♥ts ♦❢ ❜❛s✐❝ ❡❞✉❝❛t✐♦♥✳ ❚❤✐s ❛❝t✐✈✐t② ❝♦♥s✐sts ✐♥ ❜✉✐❧❞✐♥❣ ❛ ❦✐t ♦❢ ❡♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ✉s✐♥❣ ❞✐s♣♦s❛✲ ❜❧❡ ❝✉♣s✳ ❲✐t❤ ❞②♥❛♠✐❝ ❧✐♥❦✐♥❣ ❡❧❡♠❡♥ts ❊♥❝r②♣t✐♦♥ ❛♥❞ ❲❤❛ts❛♣♣ ❛♣♣❧✐❝❛t✐♦♥ ❛s ❛ ♠❡❛♥s ♦❢ ❡①❝❤❛♥❣❡ ♦❢ ❡♥❝r②♣t❡❞ ♠❡ss❛❣❡s✱ ✇❡ ♠♦t✐✈❛t❡ t❤❡ ❝❧❛ssr♦♦♠ ❢♦r ❧❡❛r♥✐♥❣ ❊✉❝❧✐❞❡❛♥ ❞✐✈✐s✐♦♥ ❛♥❞ ♣❡r♠✉t❛t✐♦♥✳ ■♥ ❛❞❞✐t✐♦♥✱ ✇❡ ✐♥t❡♥❞ t♦ ❛r♦✉s❡ st✉❞❡♥ts✬ ✐♥t❡r❡st ✐♥ ❞❡❡♣❡♥✐♥❣ t❤❡ st✉❞② ♦❢ ♠❛t❤❡♠❛t✐❝s✱ ❡s♣❡❝✐❛❧❧② ✐♥ ◆✉♠❜❡r ❚❤❡♦r②✱ ❛s t❤✐s ✐s ♦♥❡ ♦❢ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ t♦♦❧s ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❝r②♣t♦❣r❛♣❤②✱ ❛ s❝✐❡♥❝❡ ✇✐t❤ ❣r❡❛t ❛♣♣❧✐❝❛❜✐❧✐t② t♦❞❛②✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✸✳✶ ❊①❡♠♣❧♦ ❞❡ ❝✉r✈❛ ❡❧í♣t✐❝❛ y2 = x3 + 1x+ 1✱ ✐♠❛❣❡♠ ❡❧❛❜♦r❛❞❛ ♣❡❧♦ ♣ró♣r✐♦ ❛✉t♦r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✷ ❊①❡♠♣❧♦ ❞❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ y2 = x3 + 1x + 1 ✉t✐❧✐③❛♥❞♦ Z
11✳ ❋♦♥t❡✿
✇✇✇✳❝r✐♣t♦r❡❞✳✉♣♠✳❡s✴❝r②♣t✹②♦✉✴t❡♠❛s✴❊❈❈✴❧❡❝❝✐♦♥✶✴❧❡❝❝✐♦♥✶✳❤t♠❧✳ ❆❝❡ss♦ ❡♠ ✵✶✴✵✷✴✷✵✶✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✶ ❊①❡♠♣❧♦ ❞❛ ❡st❡❣❛♥♦❣r❛✜❛ ❛♣❧✐❝❛❞❛ ♥♦ ◗❘✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✷ ❇❛stã♦ ❞❡ ▲✐❝✉r❣♦✳ ❋♦♥t❡✿❤tt♣s✿✴✴s✐r✐❛r❛❤✳✇♦r❞♣r❡ss✳❝♦♠✴✷✵✶✸✴✵✺✴✶✸✴❝r✐♣t♦❣r❛✜❛✲
❜❛st❛♦✲❞❡✲❧✐❝✉r❣♦✲s❝②t❛❧❡✲❡♠✲♣②t❤♦♥✴✳ ❆❝❡ss♦ ❡♠ ✶✷✴✵✷✴✷✵✶✻✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✸ ▼áq✉✐♥❛ ❡♥✐❣♠❛✳ ❋♦♥t❡✿ ❤tt♣✿✴✴✇✇✇✳❡❧❞✐❛r✐♦✳❡s✴t✉r✐♥❣✴❝r✐♣t♦❣r❛✜❛✴❛❧❛♥✲
t✉r✐♥❣✲❡♥✐❣♠❛✲❝♦❞✐❣♦✲✵✲✷✷✻✵✼✽✵✹✷✳❤t♠❧✱ ❛❝❡ss❛❞♦ ❡♠ ✶✷✴✵✷✴✷✵✶✻✳ ✳ ✳ ✳ ✺✶ ✹✳✹ ❈r✐♣t♦❣r❛✜❛ ❞❡ ❝❤❛✈❡ s✐♠étr✐❝❛✳ ❋♦♥t❡✿ ❤tt♣✿✴✴❜✐❜❧✐♦♦✳✐♥❢♦✴❝❡rt✐✜❝❛❝❛♦✲
❞✐❣✐t❛❧✴✳ ❆❝❡ss♦ ❡♠ ✶✷✴✵✷✴✷✵✶✻✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✺ ❈r✐♣t♦❣r❛✜❛ ❞❡ ❝❤❛✈❡ ❛ss✐♠étr✐❝❛✳❋♦♥t❡✿ ❤tt♣✿✴✴❜✐❜❧✐♦♦✳✐♥❢♦✴❝❡rt✐✜❝❛❝❛♦✲
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✾
✷ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ✲ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ✷✸
✷✳✶ ❈♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦sZ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✷✳✷ ▼ó❞✉❧♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✸ ❈♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❡ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✹ ❆❧❣♦r✐t♠♦ ❊✉❝❧✐❞✐❛♥♦ ❞❛ ❞✐✈✐sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✺ ◆ú♠❡r♦s Pr✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✻ ▼á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✼ ❆r✐t♠ét✐❝❛ ▼♦❞✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✽ ❊q✉❛çõ❡s ❉✐♦❢❛♥t✐♥❛s ❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✾ ❈♦♥❣r✉ê♥❝✐❛ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✶✵ ❈❧❛ss❡s ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✶✶ ◆ú♠❡r♦s ❞❡ ▼❡rs❡♥♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✶✷ ◆ú♠❡r♦s ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✶✸ ❋✉♥çã♦ φ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✷✳✶✹ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✶✺ ❚❡st❡s ❞❡ ♣r✐♠❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✶✺✳✶ ❚❡st❡ ❞❡ ❢♦rç❛ ❜r✉t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✶✺✳✷ ❆♣❧✐❝❛çã♦ ❞♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✶✺✳✸ ◆ú♠❡r♦s ❞❡ ❈❛r♠✐❝❤❛❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✶✺✳✹ ❆❧❣♦r✐t♠♦ ❞❡ ▲✉❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✶✺✳✺ ▼ét♦❞♦ ❞❡ ▲✉❝❛s ▲❡❤♠❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✶✺✳✻ ❚❡st❡ ❞❡ ♣r✐♠❛❧✐❞❛❞❡ ❆❑❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✶✻ ❘❛í③❡s ♣r✐♠✐t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✸ ❊❧❡♠❡♥t♦s ❞❛ ➪❧❣❡❜r❛ ✹✸
✸✳✸✳✷ Pr♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✹ ❈r✐♣t♦❣r❛✜❛ ✹✾
✹✳✶ ❯♠ ❜r❡✈❡ ❤✐stór✐❝♦ s♦❜r❡ ❝r✐♣t♦❣r❛✜❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✷ ❈✐❢r❛s ❞❡ ❈és❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✸ ❈✐❢r❛s ❆✜♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✹ ❙✐st❡♠❛ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✺ ▼ét♦❞♦ ♣❛r❛ tr♦❝❛ ❞❡ ❝❤❛✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✳✺✳✶ Pr♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✳✺✳✷ ▼ét♦❞♦ ❞❡ ❉✐✣❡✲❍❡❧❧♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✺✳✸ ▼ét♦❞♦ ❞❡ ❊❧●❛♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✻ ❈r✐♣t♦❣r❛✜❛ ❜❛s❡❛❞❛ ❡♠ ❝✉r✈❛s ❡❧í♣t✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✻✳✶ ▼ét♦❞♦ ❞❡ ❉✐✣❡✲❍❡❧❧♠❛♥ ❝♦♠ ❝✉r✈❛s ❡❧í♣t✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✻✳✷ ❆❧❣♦r✐t♠♦ ❝r✐♣t♦❣rá✜❝♦ ▼❡♥❡③❡s✲❱❛♥st♦♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
✺ Pr♦♣♦st❛ ❞❡ ❆t✐✈✐❞❛❞❡ ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ✻✸
✺✳✶ P❛rt❡ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✺✳✷ P❛rt❡ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✺✳✸ P❛rt❡ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾
✻ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✼✸
✶ ■♥tr♦❞✉çã♦
❆ ✐♥✐❝✐❛t✐✈❛ ♣❛r❛ ♦ t❡♠❛ ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ s✉r❣✐✉ ❞❛ ♥❡❝❡ss✐❞❛❞❡ ❡♠ ♠♦t✐✈❛r♠♦s ♦s ❡st✉❞❛♥t❡s ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦ ♥❛s ❛✉❧❛s ❞❡ ♠❛t❡♠át✐❝❛✳ ❉✉r❛♥t❡ ❞❡③ ❛♥♦s ❧❡❝✐♦♥❛♥❞♦ ♠❛t❡♠át✐❝❛ ♥❛s ❡s❝♦❧❛s ❡st❛❞✉❛✐s ❡ ♣❛rt✐❝✉❧❛r❡s ❞❛ r❡❣✐ã♦ ❞❡ ❆♠❡r✐❝❛♥❛✲❙P ♣❛r❛ ❡st✉✲ ❞❛♥t❡s ❞♦ q✉✐♥t♦ ❛♦ ♥♦♥♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ❞♦ ♣r✐♠❡✐r♦ ❛♦ t❡r❝❡✐r♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ s❡♠♣r❡ ♠❡ ❞❡♣❛r❡✐ ❝♦♠ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ❛❧✉♥♦s q✉❡ ❢❛③✐❛♠ ❛s ♠❡s✲ ♠❛s ♣❡r❣✉♥t❛s✿ ✧✳✳✳♣❛r❛ q✉❡ s❡r✈❡ ❡ss❡ ❝♦♥t❡ú❞♦❄✧♦✉ ✧✳✳✳♦♥❞❡ ✉s❛r❡✐ ✐ss♦❄✧ ❊♥❝♦♥tr❛r r❛③õ❡s ♣❛r❛ s❡ ❛♣r❡♥❞❡r ♠❛t❡♠át✐❝❛ ♣❛r❡❝❡♠ ó❜✈✐❛s ♣❛r❛ q✉❡♠✱ ♣❡❧❛ ♣ró♣r✐❛ ♥❛t✉r❡③❛ ❡ ❜❡❧❡③❛ ❞❛ ♠❛t❡♠át✐❝❛✱ s❡ ✐♥t❡r❡ss❛ ♣♦r ❡st❛ ❝✐ê♥❝✐❛✳ P♦ré♠✱ ♣❛r❛ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ❛❧✉♥♦s q✉❡ t❡♠ ❢♦❜✐❛ ♦✉ ❛✈❡rsã♦ ❛♦ s❡✉ ❡st✉❞♦✱ ✐st♦ ♥ã♦ é ✉♠❛ t❛r❡❢❛ ❢á❝✐❧✳ ❆t✉❛❧♠❡♥t❡✱ ❛s ❢❛❝✐❧✐❞❛❞❡s ❞♦ ❝♦t✐❞✐❛♥♦ ❞❡ ✉♠ ❡st✉❞❛♥t❡✱ ♦♥❞❡ q✉❛s❡ t✉❞♦ ❥á ✈❡♠ ✧❝❛❧❝✉❧❛❞♦✧✱ ❝❛✉s❛ ❛ ❢❛❧s❛ ✐❧✉sã♦ ❞❡ q✉❡ ♥ã♦ ❤á ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❛♣r❡♥❞❡r ♠❛t❡♠át✐❝❛ ❛❧é♠ ❞❛s ♦♣❡r❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s✳ ◆❡ss❡ s❡♥t✐❞♦ ❛ss♦❝✐❛r ✉♠ ❝♦♥❝❡✐t♦ ♠❛t❡♠át✐❝♦✱ q✉❡ ✈❛✐ ❛❧é♠ ❞❛s s✐t✉❛✲ çõ❡s ❝♦♠✉♥s ❞♦ ❞✐❛ ❛ ❞✐❛✱ ❛ ❛❧❣♦ ❝♦♥❝r❡t♦ ♦✉ q✉❡ ❢❛ç❛ ♣❛rt❡ ❞❛ r❡❛❧✐❞❛❞❡ ❞♦s ❡st✉❞❛♥t❡s ♣♦❞❡ ♠❡❧❤♦r❛r ♦ s❡✉ ✐♥t❡r❡ss❡ ♣❡❧❛ ♠❛t❡♠át✐❝❛✳
❈♦♠ ❜❛s❡ ♥❛ ♣r❡♠✐ss❛ ❛❝✐♠❛✱ s✉r❣✐✉ ❛ ✐❞❡✐❛ ❞❡ ✉t✐❧✐③❛r♠♦s ❛ ❝r✐♣t♦❣r❛✜❛✱ ♦✉ s❡✉s ❝♦♥❝❡✐t♦s ♠❛✐s ❡❧❡♠❡♥t❛r❡s✱ ♣❛r❛ ❛✉①✐❧✐❛r ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❛♣r❡♥❞✐③❛❞♦ ♠❛t❡♠á✲ t✐❝♦ ❝♦♠ ❛ ✉t✐❧✐③❛çã♦ ❞♦ s♠❛rt♣❤♦♥❡ ❛tr❛✈és ❞♦ ✉s♦ ❞❡ ✉♠ s♦❢t✇❛r❡ ❞❡ ♠❡♥s❛❣❡♠ ✐♥st❛♥tâ♥❡❛✳
P❡sq✉✐s❛♥❞♦ s♦❜r❡ tr❛❜❛❧❤♦s r❡❧❛❝✐♦♥❛❞♦s à ❝r✐♣t♦❣r❛✜❛ ♥❛ ♣❧❛t❛❢♦r♠❛ ❞♦ P❘❖❋✲ ▼❆❚ ❡♥❝♦♥tr❛♠♦s ♦♥③❡ ❞✐ss❡rt❛çõ❡s s♦❜r❡ ❡st❡ t❡♠❛ ❞❡❢❡♥❞✐❞❛s ♥♦ ❛♥♦ ❞❡ ✷✵✶✸✱ tr❡③❡ ♥♦ ❛♥♦ ❞❡ ✷✵✶✹ ❡ ❞❡③ ♥♦ ❛♥♦ ❞❡ ✷✵✶✺✳ ❆ s❡❣✉✐r ❢❛③❡♠♦s ✉♠❛ sí♥t❡s❡ s♦❜r❡ ❛❧❣✉♥s ❞♦s tr❛❜❛❧❤♦s q✉❡ ♥♦s ❝❤❛♠❛r❛♠ ♠❛✐s ❛ ❛t❡♥çã♦✳
❖ ❛✉t♦r ❡♠ ❬✶✽❪✱ ❢❛③ ✉♠ ❧❡✈❛♥t❛♠❡♥t♦ ❤✐stór✐❝♦ ❞❛ ❝r✐♣t♦❣r❛✜❛✱ ❛❜♦r❞❛♥❞♦ ❝♦♠♦ ❛ ♠❛t❡♠át✐❝❛ ❛✉①✐❧✐❛ ❛ ❝r✐♣t♦❣r❛✜❛✱ ❢❛③❡♥❞♦ ✉s♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✳ ❆♣r❡s❡♥t❛ ❛s ♣❡rs♣❡❝t✐✈❛s ♣❛r❛ ♦ ❢✉t✉r♦ ❞❛ ❝r✐♣t♦❣r❛✜❛✱ ❛tr❛✈és ❞❛ ❝r✐♣t♦❣r❛✜❛ q✉â♥✲ t✐❝❛ ❡ ♣ós✲q✉â♥t✐❝❛ ❡ s✉❛s ✐♠♣❧✐❝❛çõ❡s ♣❛r❛ ♦ ❢✉t✉r♦✱ ♣♦ré♠ ♥ã♦ ❛♣r❡s❡♥t❛ ♣r♦♣♦st❛ ❞❡ ❛t✐✈✐❞❛❞❡s ♣❛r❛ s❛❧❛ ❞❡ ❛✉❧❛✳
❊♠ ❬✶✾❪ ♦ ❛✉t♦r t❡♠ ❝♦♠♦ ❢♦❝♦ ♣r✐♥❝✐♣❛❧ ♦ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ❘❙❆✳ ■♥✐❝✐❛♥❞♦ ❝♦♠ ✉♠❛ sí♥t❡s❡ ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ♦ ❛✉t♦r t❛♠❜é♠ ❛❜♦r❞❛ ♦s ❛s♣❡❝t♦s ❤✐stór✐❝♦s ❞❛ ❝r✐♣t♦❣r❛✜❛✱ s✉❛s ♦r✐❣❡♥s ❡ ♠♦t✐✈❛çõ❡s✳ ❋✐♥❛❧✐③❛ ♦ tr❛❜❛❧❤♦ ❝♦♠ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡ ✉♠ ❛❧❣♦rít✐♠♦ ❝r✐♣t♦❣rá✜❝♦ ❜❛s❡❛❞♦ ❡♠ ❘❙❆✱ ✉t✐❧✐③❛♥❞♦ ❡①❡♠♣❧♦s ♣❛r❛ ❛✉①✐❧✐❛r ❛
✷✷ ■♥tr♦❞✉çã♦
❝♦♠♣r❡❡♥sã♦✳
❯♠❛ r❡✢❡①ã♦ s♦❜r❡ ❛ ❡❞✉❝❛çã♦ ♠❛t❡♠át✐❝❛ ❡ s✉❛s ✐♠♣❧✐❝❛çõ❡s ♥♦ ✉s♦ ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ♥♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ é ❢❡✐t❛ ❡♠ ❬✸❪✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❛✉t♦r❛ ❛♣r❡s❡♥t❛ ❛ ❤✐stór✐❛ ❞❛ ❝r✐♣t♦❣r❛✜❛ ❝♦♠ ê♥❢❛s❡ ♥♦ ❛❧❣♦r✐t♠♦ ❘❙❆✱ ❡♠ s❡❣✉✐❞❛ ❛♣r❡s❡♥t❛ ✉♠ ❡st✉❞♦ ❞❛ s✐t✉❛çã♦ ❞♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ ♥♦ ❇r❛s✐❧ ❡♥tr❡ ♦s ❛♥♦s ❞❡ ✶✾✾✺ ❡ ✷✵✵✺✱ ✐♥❝❧✉✐♥❞♦ ❛s ❞✐✜❝✉❧❞❛❞❡s ❡ ❞❡s❛✜♦s ❛♦ ❛♣r❡♥❞✐③❛❞♦ ♠❛t❡♠át✐❝♦✳ P❛r❛ ✜♥❛❧✐③❛r✱ ❛♣r❡s❡♥t❛ ✉♠❛ ♣r♦♣♦st❛ ❞❡ tr❛❜❛❧❤♦ ❝♦♠ ❢♦❝♦ ♥❛ ❝r✐♣t♦❣r❛✜❛ ✐♥t✉✐t✐✈❛✱ ❛tr❛✈és ❞❛ ❡s❝r✐t❛ ❡♠ ❇r❛✐❧❧❡ ❡ ❛ ✉t✐❧✐③❛çã♦ ❞❛ ❝✐❢r❛ ❞❡ ❈és❛r ♣❛r❛ ❛✉①í❧✐♦ ❞♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ ♥❛ s❛❧❛ ❞❡ ❛✉❧❛✳ ❊♠ ❬✾❪ ♦ ❛✉t♦r ❝♦♠❡ç❛ ❝♦♠ ❛ ❤✐stór✐❛ ❞❛ ❝r✐♣t♦❣r❛✜❛✱ ♦r✐❣❡♥s ❡ ♠ét♦❞♦s ❛♥t✐❣♦s ❞❡ ♦❝✉❧t❛çã♦ ❞❡ ♠❡♥s❛❣❡♥s ❛té ❛ ❝r✐♣t♦❣r❛✜❛ ✉t✐❧✐③❛❞❛ ❛t✉❛❧♠❡♥t❡ ♥♦s ❝♦♠♣✉t❛❞♦r❡s✳ ❆❜♦r❞❛ ❝♦♥❝❡✐t♦s ❞❛ ➪❧❣❡❜r❛ ❡ ❛r✐t♠ét✐❝❛ ♥❡❝❡ssár✐♦s ♣❛r❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ❜❛s❡❛❞♦ ♥♦ ♣r♦t♦❝♦❧♦ ❉✐✣❡✲❍❡❧❧♠❛♥✳ ❊st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ três ♣r♦✲ ♣♦st❛s ❞❡ ❛t✐✈✐❞❛❞❡s ♣❛r❛ ✉s♦ ❞❛ ❝r✐♣t♦❣r❛✜❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ ◆❛ ♣r✐♠❡✐r❛ ♣r♦♣♦st❛✱ ❛♣r❡s❡♥t❛ ♦ ✉s♦ ❞❡ ❢✉♥çõ❡s ♣❛r❛ ❝r✐❛çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❝r✐♣t❛çã♦ ❡ ❞❡❝r✐♣t❛çã♦ ❞❡ ♠❡♥s❛❣❡♥s ❝♦♠ ♦s ❛❧✉♥♦s✳ ❆ s❡❣✉♥❞❛ ♣r♦♣♦st❛ ✉t✐❧✐③❛ ♠❛tr✐③❡s ❝♦♠♦ ❡❧❡♠❡♥t♦ ❝❡♥tr❛❧ ❞♦ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦✳ P♦r ú❧t✐♠♦ ❛♣r❡s❡♥t❛ ✉♠❛ ❢♦r♠❛ ❞❡ s❡ ❝r✐❛r ✉♠ s✐st❡♠❛ ❝r✐♣✲ t♦❣rá✜❝♦ ❜❛s❡❛❞♦ ♥♦ ♣r♦t♦❝♦❧♦ ❉✐✣❡✲❍❡❧❧♠❛♥ ♣❛r❛ tr♦❝❛ ❞❡ ♠❡♥s❛❣❡♥s s❡❝r❡t❛s ❡♥tr❡ ❣r✉♣♦s ❞❡ ❛❧✉♥♦s✳ ❊♠ t♦❞❛s ❛s ♣r♦♣♦st❛s ♦ ❛✉t♦r ❡s♣❡❝✐✜❝❛ ❝❛❞❛ ♣❛rt❡ ❝♦♠♦ tr❛❜❛❧❤❛r ❝♦♠ ♦s ❛❧✉♥♦s✳
❖ tr❛❜❛❧❤♦ ❬✶✶❪ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛ ✉t✐❧✐③❛çã♦ ❞❛s ❡q✉❛çõ❡s ❞✐♦❢❛♥t✐♥❛s ❡ ❛ ❝r✐♣✲ t♦❣r❛✜❛✳ ◆❛ ♣❛rt❡ ❞❛s ❡q✉❛çõ❡s ❞✐♦❢❛♥t✐♥❛s ♦ ❛✉t♦r ✉t✐❧✐③❛ ✉♠❛ s✐t✉❛çã♦ ♣r♦❜❧❡♠❛ ❡♥✈♦❧✈❡♥❞♦ ❝♦♠♣r❛s ❞❡ ♠❡r❝❛❞♦r✐❛s✳ ❊♠ s❡❣✉✐❞❛ ❛♣r❡s❡♥t❛ ♦s s✐st❡♠❛s ❝r✐♣t♦❣rá✜❝♦s ❜❛s❡❛❞♦s ♥❛ ❝✐❢r❛ ❞❡ ❈és❛r ❡ ❘❙❆ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞♦ ❊①❝❡❧ ♣❛r❛ r❡❛❧✐③❛çã♦ ❞♦s ❝á❧❝✉❧♦s✱ ♣♦ré♠ ♥ã♦ ❛♣r❡s❡♥t❛ ♣r♦♣♦st❛ ♣❛r❛ ✉t✐❧✐③❛çã♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳
◆♦ tr❛❜❛❧❤♦ ❬✼❪ ♦ ❛✉t♦r ❢❛③ ✉♠❛ ✐♥tr♦❞✉çã♦ à ❝r✐♣t♦❣r❛✜❛ ❡ ❞❡♣♦✐s ❛♣r❡s❡♥t❛ ✉♠ r❡s✉♠♦ ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ♥❡❝❡ssár✐❛ ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ s✐st❡♠❛ ❘❙❆✳ P♦r ✜♠✱ ✉t✐❧✐③❛ ♦ s♦❢t✇❛r❡ ▼❛①✐♠❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❝á❧❝✉❧♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ❘❙❆✳ ❊st❡ tr❛❜❛❧❤♦ t❛♠❜é♠ ♥ã♦ ❛♣r❡s❡♥t❛ ♣r♦♣♦st❛ ♣❛r❛ s❡ tr❛❜❛❧❤❛r ❝♦♠ ❛❧✉♥♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳
➱ ♣♦ssí✈❡❧ ♦❜s❡r✈❛r♠♦s ❛✐♥❞❛ q✉❡ ❛ ❧✐t❡r❛t✉r❛ ❛♣r❡s❡♥t❛ ✉♠❛ ❣r❛♥❞❡ ❞✐✈❡rs✐❞❛❞❡ ❞❡ tr❛❜❛❧❤♦s q✉❡ ❛♣❧✐❝❛♠ ❛ ❝r✐♣t♦❣r❛✜❛ ♥♦ ❡♥s✐♥♦ ❞❡ ❝♦♥t❡ú❞♦s ♠❛t❡♠át✐❝♦s✳ ❆ss✐♠✱ ❛ ♠♦t✐✈❛çã♦ ♣r✐♥❝✐♣❛❧ ❞❡st❛ ❞✐ss❡rt❛çã♦ é ❝♦♥tr✐❜✉✐r ♣❛r❛ ❛ ❞✐s❝✉ssã♦ q✉❡ ✈❡♠ s❡♥❞♦ ❢❡✐t❛ ♣❡❧♦s ❡st✉❞❛♥t❡s ❞♦ P❘❖❋▼❆❚ ❡ ❞❡ ♦✉tr♦s ♣❡sq✉✐s❛❞♦r❡s✱ q✉❡ ✈❡❡♠ ♥❡st❡ t❡♠❛✱ ❡❧❡♠❡♥t♦s ♠♦t✐✈❛❞♦r❡s ❞❡♥tr♦ ❞❛ s❛❧❛ ❞❡ ❛✉❧❛ ❞❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ▼❛t❡♠át✐❝❛✳
✷✸
✷ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ✲ ❝♦♥❝❡✐t♦s
❜ás✐❝♦s
❆ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s é ✉♠ r❛♠♦ ❞❛ ▼❛t❡♠át✐❝❛ q✉❡ t❡✈❡ s✉❛ ♦r✐❣❡♠ ♥❛ ❛♥t✐❣❛ ●ré✲ ❝✐❛ ❡ ❤♦❥❡✱ ✐♥s♣✐r❛✱ ❞❡♥tr❡ ♦✉tr❛s ❛♣❧✐❝❛çõ❡s✱ ♦ ♣r♦❝❡ss♦ ❞❡ ❝r✐♣t♦❣r❛✜❛ ❡♠ tr❛♥s❛çõ❡s ✜♥❛♥❝❡✐r❛s✳ ❆❧❣✉♥s ♣r♦❜❧❡♠❛s ❡♠ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❞❡♠♦r❛r❛♠ sé❝✉❧♦s ♣❛r❛ s❡r❡♠ r❡s♦❧✈✐❞♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♦ ú❧t✐♠♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t q✉❡ ✐♥st✐❣♦✉ ♠✉✐t♦s ♣❡s✲ q✉✐s❛❞♦r❡s ❞✉r❛♥t❡ ♠❛✐s ❞❡ ✸✵✵ ❛♥♦s ❡ ❢♦✐ ✜♥❛❧♠❡♥t❡ ❞❡♠♦♥str❛❞♦ ♣♦r ❆♥❞r❡✇ ❲✐❧❡s ❡♠ ✶✾✾✺✳ ❊st❡ ❝❛♣ít✉❧♦ ❡stá ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡ ♦s r❡s✉❧t❛❞♦s ❢♦r❛♠ r❡t✐r❛❞♦s ❞❡ ❬✶✵❪ ❡ ❬✶✷❪✳
✷✳✶ ❈♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s
Z
❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❞❡♥♦t❛❞♦ ♣♦r Z✱ ❝✉❥♦ sí♠❜♦❧♦ ✈❡♠ ❞❛ ♣❛❧❛✈r❛
❩❛❤❧❡♥ ✭q✉❡ s✐❣♥✐✜❝❛ ♥ú♠❡r♦ ❡♠ ❛❧❡♠ã♦✮ é ❛ ✉♥✐ã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉✲ r❛✐s N = {0,1,2,3,4,5,6...}✱ ❥á ✐♥❝❧✉✐♥❞♦ ♦ ③❡r♦✱ ❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s
✧♥❡❣❛t✐✈♦s✧{−1,−2,−3,−4,−5,−6,· · · }♦✉ s❡❥❛✱
Z={...−3,−2,−1,0,1,2,3, ...}.
❆ss✉♠✐r❡♠♦s ❛①✐♦♠❛t✐❝❛♠❡♥t❡ q✉❡ ❡①✐st❡♠ ❞✉❛s ♦♣❡r❛çõ❡s ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú✲ ♠❡r♦s ✐♥t❡✐r♦s✱ ❛ ❛❞✐çã♦ ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❞❡♥♦t❛❞❛s r❡s♣❡❝t✐✈❛♠❡♥t❡ ❝♦♠♦ a +b ❡ a·b ✭♦✉ s✐♠♣❧❡s♠❡♥t❡ ab✮✱ ♣❛r❛ q✉❛✐sq✉❡r ✐♥t❡✐r♦s a, b∈ Z✱ s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s
♣r♦♣r✐❡❞❛❞❡s✿
Pr♦♣r✐❡❞❛❞❡ ✷✳✶✳ ❋❡❝❤❛♠❡♥t♦✿ a+b❡a·bsã♦ ✐♥t❡✐r♦s s❡♠♣r❡ q✉❡a❡b❢♦r❡♠ ✐♥t❡✐r♦s✳
Pr♦♣r✐❡❞❛❞❡ ✷✳✷✳ ❈♦♠✉t❛t✐✈❛✿ a+b=b+a ❡ a·b=b·a✱ ♣❛r❛ q✉❛✐sq✉❡r ✐♥t❡✐r♦s a
❡ b✳
Pr♦♣r✐❡❞❛❞❡ ✷✳✸✳ ❆ss♦❝✐❛t✐✈❛✿ (a+b) +c=a+ (b+c) ❡ (a·b)·c=a·(b·c)✳ ♣❛r❛
q✉❛✐sq✉❡r ✐♥t❡✐r♦s a✱b ❡c✳
Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳ ❉✐str✐❜✉t✐✈❛✿ (a+b)·c=a·c+b·c✱ ♣❛r❛ q✉❛✐sq✉❡ra, b❡c✐♥t❡✐r♦s✳
✷✻ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ✲ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s
Pr♦♣r✐❡❞❛❞❡ ✷✳✺✳ ❊①✐stê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦ ♥❡✉tr♦✿ a+ 0 = 0 +a ❡ a·1 = 1·a✱ ♣❛r❛
t♦❞♦ ✐♥t❡✐r♦a✱ s❡♥❞♦0 ❡1 ❡❧❡♠❡♥t♦s ♥❡✉tr♦s ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦✱
r❡s♣❡❝t✐✈❛♠❡♥t❡✳
Pr♦♣r✐❡❞❛❞❡ ✷✳✻✳ ❊①✐stê♥❝✐❛ ❞❡ ✐♥✈❡rs♦ ❛❞✐t✐✈♦✿ P❛r❛ t♦❞♦ ✐♥t❡✐r♦ a ✱ ❝♦♠ a 6= 0✱
❡①✐st❡ ✉♠ ✐♥t❡✐r♦ x t❛❧ q✉❡ a +x = 0✳ ❊ss❡ ✐♥t❡✐r♦ x é ❞❡♥♦♠✐♥❛❞♦ ✐♥✈❡rs♦ ❛❞✐t✐✈♦
♦✉ s✐♠étr✐❝♦ ❞❡ a✱ ❞❡♥♦t❛❞♦ ♣♦r −a✳ ❉❡ss❛ ❢♦r♠❛ ❛ s✉❜tr❛çã♦ é ❞❡✜♥✐❞❛ ❝♦♠♦ ❝♦♠♦ a+ (−b) = a−b ♣❛r❛ q✉❛✐sq✉❡r ✐♥t❡✐r♦s a ❡ b✳
✷✳✷ ▼ó❞✉❧♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦
❖ ♠ó❞✉❧♦✱ ♦✉ ✈❛❧♦r ❛❜s♦❧✉t♦✱ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ a é ❞❡♥♦t❛❞♦ ❝♦♠♦ |a| q✉❡✱
❣❡♦♠❡tr✐❝❛♠❡♥t❡✱ é ✐♥t❡r♣r❡t❛❞♦ ❝♦♠♦ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦ ♥ú♠❡r♦ ✐♥t❡✐r♦a ❡ ❛ ♦r✐❣❡♠
❞❛ r❡t❛ ♥✉♠ér✐❝❛✳
❉❡✜♥✐çã♦ ✷✳✶✳ ❙❡❥❛ a ✉♠ ✐♥t❡✐r♦ q✉❛❧q✉❡r✳ ❉❡✜♥✐♠♦s✿
|a|=a s❡ a≥0✳
|a|=−a s❡ a <0✳
❖ ♠ó❞✉❧♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ a t❛♠❜é♠ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❝♦♠♦ |a| = √a2 ❡ ♣♦ss✉✐ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿
Pr♦♣r✐❡❞❛❞❡ ✷✳✼✳ ❙❡❥❛♠ a ❡b q✉❛✐sq✉❡r ♥ú♠❡r♦s r❡❛✐s✳ ❊♥tã♦
✶✳ |a| ≥0
✷✳ |a|= 0 ⇐⇒ a= 0
✸✳ |a|=| −a|✳
✹✳ |ab|=|a||b|✳
✺✳ |a+b| ≤ |a|+|b| ✭❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✮✳
✷✳✸ ❈♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❡ ❉✐✈✐s✐❜✐❧✐❞❛❞❡
❉❛❞♦s ❞♦✐s ✐♥t❡✐r♦s a ❡ b✱ ❝♦♠ a 6= 0✱ ❞✐③❡♠♦s q✉❡ a ❞✐✈✐❞❡ b, ❞❡♥♦t❛❞♦ ♣♦r a | b✱
s❡ ❡①✐st✐r ✉♠ ✐♥t❡✐r♦ c t❛❧ q✉❡ b = ac. ❙❡ a ♥ã♦ ❞✐✈✐❞❡ b ❞❡♥♦t❛♠♦s ♣♦r a ∤ b. ❆❧é♠
❞✐ss♦✱ b/a ✐♥❞✐❝❛ ♦ q✉♦❝✐❡♥t❡✿ b ❞✐✈✐❞✐❞♦ ♣♦r a✳ ❆ s❡❣✉✐r ❡♥✉♠❡r❛♠♦s ✉♠❛ sér✐❡ ❞❡
♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞✐✈✐sã♦✿
Pr♦♣r✐❡❞❛❞❡ ✷✳✽✳ ❙❡❥❛♠ a, b, c, m ❡ n ✐♥t❡✐r♦s q✉❛✐sq✉❡r✳ ❊♥tã♦ s❡❣✉❡♠ ❛s s❡❣✉✐♥t❡s
❆❧❣♦r✐t♠♦ ❊✉❝❧✐❞✐❛♥♦ ❞❛ ❞✐✈✐sã♦ ✷✼
✶✳ 1|a.
✷✳ a|a.
✸✳ a|0.
✹✳ a|b ❡b 6= 0⇒ |a| ≤ |b|.
✺✳ a|b ❡b |a⇒ |a|=|b|.
✻✳ a|b ❡b |c⇒a|c.
✼✳ c|a ❡c|b⇒c|(ma+nb),q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ m, n∈Z.
✽✳ a|b ❡a6= 0 ⇒(b/a)|b.
❉❡♠♦♥str❛çã♦✳ ❆ ❞❡♠♦str❛çã♦ ❞❡ ✶ ❡ ✷ ❡stá ❣❛r❛♥t✐❞❛ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✺ ❡ ❛ ❞❡♠♦♥s✲ tr❛çã♦ ❞❡ ✸ s❛✐ ❞♦ ✐♥✈❡rs♦ ❛❞✐t✐✈♦ ♥❛ Pr♦♣♦s✐çã♦ ✷✳✻✳
P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✹✿ s❡ a|b ❡♥tã♦ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ c t❛❧ q✉❡ b = ac✱ ♠❛s
|a| ≤ |a||c|=|ac|=|b|✱ ❞❛ ♠❡s♠❛ ❢♦r♠❛✱ ❞❡♠♦♥str❛♠♦s ✺✳
P❛r❛ ❞❡♠♦♥str❛r♠♦s ✻✿ s❡ a|b ❡ b|c✱ ♣♦r ❞❡✜♥✐çã♦ ❡①✐st❡♠ m ❡ n ✐♥t❡✐r♦s t❛✐s q✉❡ b =ma ❡c=nb⇒c=n(ma)♦✉ c= (nm)a, ♣♦rt❛♥t♦ a |c.
◆❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✼✿ ❡①✐st❡♠ p, q ∈ Z, t❛✐s q✉❡ a = pc ❡ b =qc✳ ▼✉❧t✐♣❧✐❝❛♥❞♦
❡st❛s ✐❣✉❛❧❞❛❞❡s ♣♦r m❡ n,r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❡r❡♠♦s ma=mpc ❡nb=nqc. ❙♦♠❛♥❞♦
♦s ♠❡♠❜r♦s ma+nb=mpc+nqc ♦✉ma+nb=c(mp+nq)⇒c|(ma+bn).
◆❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✽ t❡♠♦s✱ ♣♦r ❤✐♣ót❡s❡s q✉❡ b = ma✱ ♣❛r❛ ❛❧❣✉♠ ✐♥t❡✐r♦ m✱
❡♥tã♦ b/a é ✉♠ ✐♥t❡✐r♦✳ ❈♦♠♦a 6= 0 t❡♠♦s (b/a)·a=b q✉❡ ✐♠♣❧✐❝❛ (b/a)|b.
❊①❡♠♣❧♦ ✷✳✶✳ ❖s ♣ró①✐♠♦s ❡①❡♠♣❧♦s ✐❧✉str❛♠ ❛❧❣✉♥s ✐t❡♥s ❞♦ r❡s✉❧t❛❞♦ ❛❝✐♠❛✿ ✐✮ 3|6 ❡6|24❡♥tã♦ 3|24 ✭✐t❡♠ ✻✮✳
✐✐✮ 3|6 ❡3|9 ❡♥tã♦ 3|(5·6 + 7·9) ♦ q✉❡ ♥♦s ❞á 3|93 ✭✐t❡♠ ✼✮✳
✐✐✐✮ 3|6 ❡6/3 = 2 ✐♠♣❧✐❝❛2|6 ✭✐t❡♠ ✽✮✳
✷✳✹ ❆❧❣♦r✐t♠♦ ❊✉❝❧✐❞✐❛♥♦ ❞❛ ❞✐✈✐sã♦
❉❛❞♦s ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥ã♦ ♥❡❣❛t✐✈♦s a ❡ b, ❝♦♠ b 6= 0✱ ♥❛ ❞✐✈✐sã♦ ❞❡ a ♣♦r b✱ s❡♠♣r❡ ❡①✐st❡♠ ♥ú♠❡r♦s ✐♥t❡✐r♦sq ✭q✉♦❝✐❡♥t❡✮ ❡ r ✭r❡st♦✮ q✉❡ s❛t✐s❢❛③❡♠ ❛ s❡❣✉✐♥t❡
r❡❧❛çã♦✿
a=q.b+r, ❝♦♠ 0≤r < b.
❊ss❛ r❡❧❛çã♦ é ❝❤❛♠❛❞❛ ❞❡ ❛❧❣♦r✐t♠♦ ❊✉❝❧✐❞✐❛♥♦ ❞❛ ❞✐✈✐sã♦ ✭❡ss❡ ❛❧❣♦r✐t♠♦ ❛♣❛r❡❝❡✉ ♥♦ ❧✐✈r♦ ❱■■ ❞♦s ✧❊❧❡♠❡♥t♦s✧❞❡ ❊✉❝❧✐❞❡s✱ ♣♦r ✈♦❧t❛ ❞♦ ❛♥♦ ✸✵✵ ❛✳❈✮ ❡ t❛♠❜é♠ s❡ ❛♣❧✐❝❛ ❛ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥❡❣❛t✐✈♦s✳ ◗✉❛♥❞♦ r = 0, ♦✉ s❡❥❛✱ ♦ r❡st♦ é ③❡r♦✱ ❞✐③❡♠♦s q✉❡ ❛
✷✽ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ✲ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s
❊①❡♠♣❧♦ ✷✳✷✳ ❆ ❞✐✈✐sã♦ ❞❡ ✷✶ ♣♦r ✹✱ t❡♠ ❝♦♠♦ r❡s✉❧t❛❞♦ ♦ ✐♥t❡✐r♦ ✺ ❡ ♦ r❡st♦ ✶✱ ❛ss✐♠
21 = 5.4 + 1✳
❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ s❡rá ✉s❛❞♦ ♣❛r❛ ❞❡♠♦♥str❛r ♦ ♣r✐♥❝✐♣❛❧ ❚❡♦r❡♠❛ ❞❡st❛ s❡çã♦✳ ❚❡♦r❡♠❛ ✷✳✶✳ ✭❚❡♦r❡♠❛ ❞❡ ❊✉❞♦①✐✉s✮ ❉❛❞♦s ♦s ✐♥t❡✐r♦s a ❡ b✱ ❝♦♠ b 6= 0✱ ❡♥tã♦ a é
♠ú❧t✐♣❧♦ ❞❡ b ♦✉ ❡♥❝♦♥tr❛✲s❡ ❡♥tr❡ ❞♦✐s ♠ú❧t✐♣❧♦s ❝♦♥s❡❝✉t✐✈♦s ❞❡ b✳ ■st♦ é✱ ❡①✐st❡ ✉♠
✐♥t❡✐r♦ k t❛❧ q✉❡ kb≤a <(k+ 1)b✳
❚❡♦r❡♠❛ ✷✳✷✳ ✭❆❧❣♦r✐t♠♦ ❊✉❝❧✐❞✐❛♥♦✮ ❙❡❥❛♠a❡b✐♥t❡✐r♦s✱b 6= 0.❊①✐st❡ ✉♠ ú♥✐❝♦ ♣❛r
❞❡ ✐♥t❡✐r♦sq ❡r, ❝❤❛♠❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ q✉♦❝✐❡♥t❡ ❡ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛
❞❡ a ♣♦r b, t❛✐s q✉❡ a=bq+r ❝♦♠ 0≤r <|b|.
❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❞♦ b >0,♦ ❝❛s♦ b <0 é ❛♥á❧♦❣♦✳
❙❡ a= 0 ✱ ❜❛st❛ t♦♠❛r q=r= 0✳
❙❡ a 6= 0, ♣❡❧♦ ❚❡♦r❡♠❛ ❊✉❞♦①✐✉s✱ ❡①✐st❡ q s❛t✐s❢❛③❡♥❞♦ qb ≤ a < (q + 1)b✱ q✉❡
✐♠♣❧✐❝❛ 0 ≤ a − qb ❡ a − qb < b✳ P❛r❛ ❞❡♠♦♥str❛r ❛ ✉♥✐❝✐❞❛❞❡✱ s✉♣♦♠♦s q✉❡ ❛
❡①✐stê♥❝✐❛ ❞❡ ♦✉tr♦ ♣❛r q1 ❡r1✿
a=q1b+r1✱ ❝♦♠ 0≤r1 < b✳
❆ss✐♠ t❡♠♦sa−a= 0❡♥tã♦ (qb+r)−(q1b+r1) = 0q✉❡ ✐♠♣❧✐❝❛ ❡♠b(q−q1) = r1−r✱ ❝♦♥❝❧✉í♠♦s q✉❡b |(r1−r)✳ ▼❛s ❝♦♠♦r1 < b❡r < b✱ t❡♠♦s|r1−r|< b✱ s❡ b|(r1−r) ❞❡✈❡♠♦s t❡r r1 −r = 0 ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ r = r1✳ ❈♦♠♦ b 6= 0 t❡♠♦s q−q1 = 0 q✉❡ ✐♠♣❧✐❝❛q =q1✳
P♦❞❡♠♦s ❡♥❝♦♥tr❛r ♦✉tr❛ ❞❡♠♦♥str❛çã♦ ♣❛r❛ ♦ ❆❧❣♦r✐t♠♦ ❊✉❝❧✐❞✐❛♥♦ ✉s❛♥❞♦ ♦ Pr✐♥✲ ❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✶ ❡♠ ❬✶✷❪✳
✷✳✺ ◆ú♠❡r♦s Pr✐♠♦s
❯♠ ♥ú♠❡r♦ ✐♥t❡✐r♦n ❝♦♠ (n > 1)q✉❡ ♣♦ss✉❛ ❛♣❡♥❛s ❞♦✐s ❞✐✈✐s♦r❡s ♣♦s✐t✐✈♦s✿ ♦ ✶ ❡
♦ ♣ró♣r✐♦ n, é ❝❤❛♠❛♥❞♦ ❞❡ ♥ú♠❡r♦ ♣r✐♠♦✳ ❈♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ s❡❣✉✐♥t❡ s❡q✉ê♥❝✐❛✿ 2,3,5,7,11,13,17,19,23✳✳✳
❆♣❡s❛r ❞❛ ❛♣❛r❡♥t❡ s✐♠♣❧✐❝✐❞❛❞❡✱ ❛ s❡q✉ê♥❝✐❛ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s r❡♣r❡s❡♥t❛ ✉♠ ❞❡s❛✜♦ s✉♣r❡♠♦✱ q✉❡ ❛tr❛✈❡ss❛ ❣❡r❛çõ❡s ❞❡ ❣r❛♥❞❡s ♠❛t❡♠át✐❝♦s✱ ❞❡✈✐❞♦ ❛♦ s❡✉ ❝❛rát❡r ❝❛ót✐❝♦ ❡ ❛❧❡❛tór✐♦✳ ❙❡❣✉♥❞♦ ❬✶✺❪ ❡ss❡s ♥ú♠❡r♦s sã♦ ♦s ♣ró♣r✐♦s át♦♠♦s ❞❛ ♥❛t✉r❡③❛✱ ❞❡✈✐❞♦ ❛ s✉❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❣❡r❛r ♦s ❞❡♠❛✐s ♥ú♠❡r♦s✳
❚❡♦r❡♠❛ ✷✳✸✳ ✭❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✮ ❚♦❞♦ ✐♥t❡✐r♦ ♠❛✐♦r q✉❡ ✶ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❞❡ ♠❛♥❡✐r❛ ú♥✐❝❛✱ ❛ ♠❡♥♦s ❞❛ ♦r❞❡♠✱ ❝♦♠♦ ♦ ♣r♦❞✉t♦ ❞❡ ❢❛t♦r❡s ♣r✐♠♦s✳
▼á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ✷✾
❉❡♠♦♥str❛çã♦✳ ❙❡né ♣r✐♠♦ ♥ã♦ ❤á ♦ q✉❡ s❡ ♣r♦✈❛r✳ ❙❡❥❛n ❝♦♠♣♦st♦✱ ❡♥tã♦n=p1.n1 ♦♥❞❡ 1 < p1 é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❡ r❡♣r❡s❡♥t❛ ♦ ♠❡♥♦r ❞♦s ❞✐✈✐s♦r❡s ❞❡ n ❡ n1 ✉♠ ✐♥t❡✐r♦ s❛t✐s❢❛③❡♥❞♦ 1< n1 < n✳ ❙❡n1 ❢♦r ♣r✐♠♦ ❛ ♣r♦✈❛ ❡stá ❝♦♠♣❧❡t❛✱ ❝❛s♦ ❝♦♥trár✐♦✱ t♦♠❡♠♦s p2✱ ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✱ ❝♦♠♦ ♦ ♠❡♥♦r ❢❛t♦r ❞❡ n1 ❡ n1 = p2.n2✳ ❊♥tã♦
n =p1.p2.n2✳
❘❡♣❡t✐♥❞♦ ❡st❡ ♣r♦❝❡ss♦✱ ♦❜t❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡❝r❡s❝❡♥t❡ ❞❡ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s
n1, n2, n3, n4, n5, .., nr✳ ❈♦♠♦ t♦❞♦s ♦s ni sã♦ ✐♥t❡✐r♦s ♠❛✐♦r❡s ❞♦ q✉❡ ✶✱ ❡st❡ ♣r♦❝❡ss♦
❞❡✈❡ t❡r♠✐♥❛r✳ ❈♦♠♦ ♦s ♣r✐♠♦s ♥❛ s❡q✉ê♥❝✐❛ p1, p2, p3, ..., pk ♥ã♦ sã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡
❞✐st✐♥t♦s✱ n t❡rá ❛ s❡❣✉✐♥t❡ ❢♦r♠❛ ❣❡r❛❧✿
n =pa11 .pa22 .pa33 ...pak
k
P❛r❛ ❞❡♠♦♥str❛r♠♦s ❛ ✉♥✐❝✐❞❛❞❡✱ ✉s❛r❡♠♦s ✐♥❞✉çã♦ ❡♠ n✳ P❛r❛ n = 2 ❛ ❛✜r♠❛çã♦
é ✈❡r❞❛❞❡✐r❛✱ ♣❛r❛ n > 2 t❡♠♦s q✉❡ n ♣♦❞❡ s❡r ♣r✐♠♦ ✭♥ã♦ ❤á ♥❛❞❛ ❛ ♣r♦✈❛r✮ ♦✉
❝♦♠♣♦st♦✳ ❱❛♠♦s s✉♣♦r q✉❡ n s❡❥❛ ❝♦♠♣♦st♦ ❡ q✉❡ t❡♥❤❛ ❞✉❛s ❢❛t♦r❛çõ❡s✿ n =p1.p2.p3...ps =q1.q2.q3...qr
P❛r❛ ♣r♦✈❛r♠♦s q✉❡ s = r ❡ q✉❡ ❝❛❞❛ pi é ✐❣✉❛❧ ❛ qj t❡♠♦s✿ ❝♦♠♦ p1 ❞✐✈✐❞❡ ♦ ♣r♦✲ ❞✉t♦ q1.q1.q3...qr ❡❧❡ ❞✐✈✐❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ❢❛t♦r❡s qj✳ ❙❡♠ ♣❡r❞❛ ❞❛ ❣❡♥❡r❛❧✐❞❛❞❡
♣♦❞❡♠♦s s✉♣♦r q✉❡ p1 | q1✱ ♠❛s ❛♠❜♦s sã♦ ♣r✐♠♦s✱ ❧♦❣♦ p1 = q1✳ ❚❡♠♦s ❡♥tã♦ q✉❡
n p1
= p2.p3...ps = q2.q3...qr ❡ 1 <
n p1
< n✳ ❆ss✐♠ ❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ♥♦s ❞✐③ q✉❡
❛s ❞✉❛s ❢❛t♦r❛çõ❡s sã♦ ✐❞ê♥t✐❝❛s✱ ❛ ♠❡♥♦s ❞❛ ♦r❞❡♠✳ ❈♦♥❝❧✉í♠♦s✱ q✉❡ p1p2.p3...ps ❡
q1.q1.q3...qr sã♦ ✐❣✉❛✐s✳
❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ é ✉♠ ❞♦s ♠❛✐s ❝❧áss✐❝♦s ❞❛ ▼❛t❡♠át✐❝❛✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❬✶✷❪ ❛té ♦♥❞❡ s❡ ❝♦♥❤❡❝❡✱ ❛ ❞❡♠♦♥str❛çã♦ ❛ s❡❣✉✐r ❢♦✐ ❛ ♣r✐♠❡✐r❛ ❞❡♠♦♥str❛çã♦ ❡s❝r✐t❛ ✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦ ❡ é ❞❡✈✐❞❛ ❛ ❊✉❝❧✐❞❡s ❝❡r❝❛ ❞❡ ✸✵✵ ❆✳❈✳
❚❡♦r❡♠❛ ✷✳✹✳ ✭❚❡♦r❡♠❛ ❞❡ ❊✉❝❧✐❞❡s✮ ❆ q✉❛♥t✐❞❛❞❡ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s é ✐♥✜♥✐t❛✳ ❉❡♠♦♥str❛çã♦✳ ❋❛r❡♠♦s ❛ ♣r♦✈❛ ♣♦r r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦✳ ❱❛♠♦s s✉♣♦r q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s s❡❥❛ ✜♥✐t❛✳ ❈♦♥s✐❞❡r❡♠♦sp1, p2, p3, .., pn❛ ❧✐st❛ ❞❡ t♦❞♦s ♦s ♣r✐♠♦s✳
❈♦♥s✐❞❡r❡♠♦s t❛♠❜é♠ ♦ ♥ú♠❡r♦R=p1+p2+p3+...+pn+1✳ ➱ ❡✈✐❞❡♥t❡ q✉❡Ré ♠❛✐♦r
q✉❡ t♦❞♦s ♦s ♣r✐♠♦s pi✱ i = 1,· · · , n ❞❛ ❧✐st❛✱ ❛❧é♠ ❞❡ ♥ã♦ s❡r ❞✐✈✐sí✈❡❧ ♣♦r ♥❡♥❤✉♠
❞♦s pi✳ P❡❧♦ ❚❡♦r❡♠❛ ✷✳✸✱ R é ♣r✐♠♦ ♦✉ ♣♦ss✉✐ ❛❧❣✉♠ ❢❛t♦r ♣r✐♠♦✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ♥❛
❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♣r✐♠♦ q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ à ❧✐st❛ ❝♦♥s✐❞❡r❛❞❛✳ P♦rt❛♥t♦✱ ❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ♥ã♦ ♣♦❞❡ s❡r ✜♥✐t❛✳
✷✳✻ ▼á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠
❖ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡ ❞♦✐s ✐♥t❡✐r♦sa ❡ b, ❛♠❜♦s ❞✐❢❡r❡♥t❡s ❞❡ ③❡r♦✱ ❞❡♥♦✲
t❛❞♦ ♣♦r ♠❞❝(a, b) ♦✉ s✐♠♣❧❡s♠❡♥t❡ (a, b), é ♦ ♠❛✐♦r ✐♥t❡✐r♦ q✉❡ ❞✐✈✐❞❡ a ❡ b s✐♠✉❧t❛✲
✸✵ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ✲ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s
❊①❡♠♣❧♦ ✷✳✸✳ ❚❡♠♦s ♠❞❝✭✽✱✶✷✮❂✹✱ ♣♦✐s ♦s ❞✐✈✐s♦r❡s ❞❡ ✽ sã♦{1,2,4,8}❡ ♦s ❞✐✈✐s♦r❡s
❞❡ ✶✷ sã♦ {1,2,3,4,6,12}.
❆ s❡❣✉✐r ✈❛♠♦s ❞❡s❝r❡✈❡r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳
Pr♦♣r✐❡❞❛❞❡ ✷✳✾✳ P♦❞❡♠♦s ❞❡♠♦♥str❛r q✉❡✿ ✶✳ ❙❡ a 6= 0 ❡♥tã♦ ♠❞❝(a,0) =|a|.
✷✳ ❙❡ a 6= 0 ❡♥tã♦ ♠❞❝(a, a) = |a|.
✸✳ ❚❡♠♦s ♠❞❝(a, b) =♠❞❝(b, a).
✹✳ ❙❡ a |b ❡♥tã♦ ♠❞❝(a, b) =a.
✺✳ ❙❡ t ∈Z✱ ❡♥tã♦ ♠❞❝(t·a, t·b) =t·♠❞❝(b, a)✳
✻✳ ❙❡❥❛♠ a1, a2, a3, a4, ..., an−1, an ✉♠❛ ❝♦❧❡çã♦ ✜♥✐t❛ ❞❡ ✐♥t❡✐r♦s✱ ♥ã♦ t♦❞♦s ♥✉❧♦s✳
❊♥tã♦ ♠❞❝(a1, a2, a3, a4, ..., an−1, an) =♠❞❝(a1, a2, a3, a4, ...,♠❞❝(an−1, an)).
❉❡♠♦♥str❛çã♦✳ ❆s ❞❡♠♦♥str❛çõ❡s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❛❝✐♠❛ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✶✵❪✳
▲❡♠❛ ✷✳✶✳ ✭▲❡♠❛ ❞❡ ❇é③♦✉t✷✮ ❉❛❞♦s ✐♥t❡✐r♦sa ❡ b, ♥ã♦ ❛♠❜♦s ♥✉❧♦s✱ ❡①✐st❡♠ ✐♥t❡✐r♦s
m ❡ n, t❛✐s q✉❡ am+bn=♠❞❝(a, b).
❉❡♠♦♥str❛çã♦✳ ❙❡a, b❡csã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡ s❡c|a ❡c|b ❡♥tã♦ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡
✷✳✽ ✐t❡♠ ✼✱ c|am′ +bn′ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ m′, n′ ✐♥t❡✐r♦s ♦✉ s❡❥❛✱ am′ +bn′ =r·c✱ ♣❛r❛ ❛❧❣✉♠ r ✐♥t❡✐r♦✳
✷✳✼ ❆r✐t♠ét✐❝❛ ▼♦❞✉❧❛r
❊♠ ❬✶✵❪ ♦s ❛✉t♦r❡s ❞❡s❝r❡✈❡♠ q✉❡ ✶✽✵✶✱ ♦ ♣r♦❡♠✐♥❡♥t❡ ❥♦✈❡♠ ♠❛t❡♠át✐❝♦ ❈❛r❧ ❋r✐❡❞r✐❝❤ ●❛✉ss ✭✶✼✼✼✲✶✽✺✺✮ ♣✉❜❧✐❝♦✉ s❡✉ ❧✐✈r♦ ✐♥t✐t✉❧❛❞♦ ❉✐sq✉✐s✐t✐♦♥❡s ❆r✐t❤♠❡t✐❝❛❡✱ ❝♦♥s✐❞❡r❛❞♦ ♦ ♠❛r❝♦ ♣❛r❛ ♦ ♥❛s❝✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❝♦♠♦ ❞✐s❝✐♣❧✐♥❛ ♣r♦♣r✐❛✲ ♠❡♥t❡ ❞✐t❛✳ ❯♠❛ ❞❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❡ss❡ tr❛❜❛❧❤♦✱ ❢♦✐ ❛ ❝r✐❛çã♦ ❞❛ ❝❛❧❝✉❧❛❞♦r❛ r❡❧ó❣✐♦✳ ❙❡ ❡♠ ✉♠ r❡❧ó❣✐♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❞❡ ✶✷ ❤♦r❛s q✉❡ ❡stá ♠❛r❝❛♥❞♦ ✶✵ ❤♦r❛s ♥ós ❛❞✐❝✐♦✲ ♥❛r♠♦s ✺ ❤♦r❛s✱ ♦ ♣♦♥t❡✐r♦ ❞❛s ❤♦r❛s ❛✈❛♥ç❛ ❛té ❛s ✸ ❤♦r❛s✳ ❆ss✐♠ ❛ ❝❛❧❝✉❧❛❞♦r❛ r❡❧ó❣✐♦ ❞❡ ●❛✉ss ❞❛r✐❛ ❝♦♠♦ r❡s♣♦st❛ ✸ ❡ ♥ã♦ ✶✺✳ ❆ss✐♠✱ ♦ ❝♦♥❝❡✐t♦ ❜ás✐❝♦ ❞❡st❛ ❝❛❧❝✉❧❛❞♦r❛ r❡❧ó❣✐♦ ❝♦♥s✐st❡ ❡♠ ❢♦r♥❡❝❡r ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ✉♠ r❡s✉❧t❛❞♦ ♣♦r ✶✷✳ ❉❡ss❛ ❢♦r♠❛ ❛ ❝❛❧❝✉❧❛❞♦r❛ ❞❡ ●❛✉ss t♦r♥♦✉✲s❡ ✉♠❛ ❢❡rr❛♠❡♥t❛ ♠✉✐t♦ út✐❧ ♣❛r❛ s❡ tr❛❜❛❧❤❛r ❝♦♠ ♥ú♠❡r♦s ❣r❛♥❞❡s✳ P♦r ❡①❡♠♣❧♦✱ ♠❡s♠♦ s❡♠ s❛❜❡r ♦ ✈❛❧♦r ❞❡799,❛ ❝❛❧❝✉❧❛❞♦r❛ r❡❧ó❣✐♦ ❞✐③ q✉❡ ❡ss❡ ♥ú♠❡r♦ ❞❡✐①❛ r❡st♦ ✼ q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r ✶✷✳
❆r✐t♠ét✐❝❛ ▼♦❞✉❧❛r ✸✶
●❛✉ss ❧♦❣♦ ♣❡r❝❡❜❡✉ q✉❡ ♦ r❡❧ó❣✐♦ ♣♦❞✐❛ ❝♦♥t❡r ✈❛❧♦r❡s ❞✐❢❡r❡♥t❡s ❞❡ ✶✷ ❤♦r❛s✱ ❛ss✐♠ ❞❡s❡♥✈♦❧✈❡✉ ❛ ✐❞❡✐❛ ❞❡ s❡ r❡❛❧✐③❛r ❛ ❛r✐t♠ét✐❝❛ r❡❧ó❣✐♦ ♦✉ ❛r✐t♠ét✐❝❛ ♠♦❞✉❧❛r✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ m.
❉❡✜♥✐çã♦ ✷✳✷✳ ❙❡❥❛♠ a ❡ b ✐♥t❡✐r♦s✳ ❉✐③❡♠♦s q✉❡a é ❝♦♥❣r✉❡♥t❡ ❛b ♠ó❞✉❧♦ m,(m > 0) ❡ ❞❡♥♦t❛♠♦s ♣♦r a ≡ b(♠♦❞m), s❡ m | (a −b). ❙❡ m ∤ (a−b)✱ ❞✐③❡♠♦s q✉❡ a é
✐♥❝♦♥❣r✉❡♥t❡ ❛ b ♠ó❞✉❧♦ m ❡ ❞❡♥♦t❛♠♦s ♣♦r a6≡b(♠♦❞m).
P♦❞❡♠♦s ✐❧✉str❛r ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ❝♦♠ ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✿ ❊①❡♠♣❧♦ ✷✳✹✳ 15≡3(♠♦❞4)✱ ♣♦✐s4|(15−3)✳
Pr♦♣♦s✐çã♦ ✷✳✶✳ ❙❡ a ❡ b sã♦ ✐♥t❡✐r♦s✱ t❡♠♦s a ≡b(♠♦❞m) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st✐r
✉♠ ✐♥t❡✐r♦ k t❛❧ q✉❡ a=b+km✳
❉❡♠♦♥str❛çã♦✳ ❙❡ a≡b ✭♠♦❞ ♠✮✱ ❡♥tã♦ m|(a−b) ♦ q✉❡ ✐♠♣❧✐❝❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠
♥ú♠❡r♦ ✐♥t❡✐r♦ k t❛❧ q✉❡ a−b=km✱ ✐st♦ é✱ a=b+km✱ ❛ r❡❝í♣r♦❝❛ é tr✐✈✐❛❧✳
❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ❢♦r♥❡❝❡ ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s✳ Pr♦♣♦s✐çã♦ ✷✳✷✳ ❙❡ a, b, c, m ❡ d sã♦ ✐♥t❡✐r♦s✱ m > 0 ❡ a ≡ b(♠♦❞m), ❛s s❡❣✉✐♥t❡s
♣r♦♣r✐❡❞❛❞❡s sã♦ ✈❡r❞❛❞❡✐r❛s✿ ✶✳ a≡a ✭♠♦❞ m✮✳
✷✳ b ≡a ✭♠♦❞ m✮✳
✸✳ ❙❡ a ≡b ✭♠♦❞ m✮ ❡ b ≡d ✭♠♦❞ m✮✱ ❡♥tã♦ a≡d ✭♠♦❞ m✮✳
✹✳ a+c≡b+c✭♠♦❞ m✮✱ ♦♥❞❡ cé ✉♠ ✐♥t❡✐r♦ q✉❛❧q✉❡r✳
✺✳ a−c≡b−c ✭♠♦❞ m✮✱ ♦♥❞❡ c é ✉♠ ✐♥t❡✐r♦ q✉❛❧q✉❡r✳
✻✳ ac≡bc ✭♠♦❞ m✮✱ ♦♥❞❡ c é ✉♠ ✐♥t❡✐r♦ q✉❛❧q✉❡r✳
❉❡♠♦♥str❛çã♦✳ ✭✶✮ ❈♦♠♦ m |0❡♥tã♦ m|(a−a)✱ ♦ q✉❡ ✐♠♣❧✐❝❛a ≡a ✭♠♦❞ m✮✳
✭✷✮ ❙❡ a≡b ✭♠♦❞ m✮✱ ❡♥tã♦a=b+k1m✱ ♣❛r❛ ❛❧❣✉♠ ✐♥t❡✐r♦ k1 ❧♦❣♦✱ b=a−k1m✳ ❙❡ k2 =−k1 ♦❜t❡♠♦sb=a+k2m✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ b ≡a ✭♠♦❞ m✮✳
✭✸✮ P♦r ❞❡✜♥✐çã♦✱ ❡①✐st❡♠ k1 ❡ k2 t❛✐s q✉❡✱ a−b = k1m ❡ b−d = k2m✱ s♦♠❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ♦❜t❡♠♦s a−d= (k1+k2)m✳ ▲♦❣♦✱a ≡d ✭♠♦❞ m✮✳
✭✹✮ ❙❡ a≡ b ✭♠♦❞ m✮✱ ❡①✐st❡ k ✐♥t❡✐r♦ t❛❧ q✉❡✱ a−b =km✳ ❚❡♠♦s✱ ♣❛r❛ q✉❛❧q✉❡r c ✐♥t❡✐r♦✱ q✉❡a−b = (a+c)−(b+c)✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ a+c≡b+c✭♠♦❞ m✮ ✳
✭✺✮ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ à ♣r♦♣r✐❡❞❛❞❡ ❛♥t❡r✐♦r✱ s❡ a≡ b ✭♠♦❞ m✮✱ ❡①✐st❡ k ✐♥t❡✐r♦
t❛❧ q✉❡✱ a−b = km✳ P❛r❛ q✉❛❧q✉❡r c✐♥t❡✐r♦ a−b = (a−c)−(b−c)✱ ♦ q✉❡ ✐♠♣❧✐❝❛
❡♠ a−c≡b−c✭♠♦❞ m✮✳
✭✻✮ ❙❡ a ≡ b ✭♠♦❞ m✮✱ ❡①✐st❡ k ✐♥t❡✐r♦ t❛❧ q✉❡✱ a−b = km✳ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❡st❛
✸✷ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ✲ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s
❆❧é♠ ❞❡st❛s✱ ♦✉tr❛s ♣r♦♣r✐❡❞❛❞❡s ♣♦❞❡♠ s❡r ❞❡♠♦♥str❛❞❛s✿
Pr♦♣♦s✐çã♦ ✷✳✸✳ ❙❡ a, b, c, d ❡ m sã♦ ✐♥t❡✐r♦s t❛✐s q✉❡ a ≡ b ✭♠♦❞ m✮❡ c ≡ d ✭♠♦❞ m✮✱ ❡♥tã♦ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s✿
✶✳ a+c≡b+d ✭♠♦❞ m✮✳
✷✳ a−c≡b−d ✭♠♦❞ m✮✳
✸✳ ac≡bd✭♠♦❞ m✮✳
✹✳ ac≡bc ✭♠♦❞ m✮✱ ❡♥tã♦ a≡b✭♠♦❞ m/d✮ ♦♥❞❡ d=♠❞❝(c, m)✳
✺✳ P❛r❛ q✉❛❧q✉❡r ✐♥t❡✐r♦ ❦ ❃✵ ❡ a≡b✭♠♦❞ m✮✱ ❡♥tã♦ ak
≡bk✭♠♦❞ m✮✳
❉❡♠♦♥str❛çã♦✳ ✭✶✮ ❙❡a ≡b✭♠♦❞ m✮❡ c≡d✭♠♦❞ m✮ ❡①✐st❡♠ ✐♥t❡✐r♦s k1 ❡k2✱ t❛✐s q✉❡
a−b=k1m❡c−d=k2m✳ ❙♦♠❛♥❞♦ ❛s ✐❣✉❛❧❞❛❞❡s ♦❜t❡♠♦s✱(a+c)−(b+d) = (k1+k2)m✳ ▲♦❣♦✱ a+c≡b+d✭♠♦❞ m✮✳
✭✷✮ ❙❡ a≡b✭♠♦❞ m✮ ❡ c≡d✭♠♦❞ m✮✱ ❡①✐st❡♠ ✐♥t❡✐r♦sk1 ❡k2 t❛✐s q✉❡ a−b =k1m ❡c−d=k2m✳ ❙✉❜tr❛✐♥❞♦ ❛s ✐❣✉❛❧❞❛❞❡s ♦❜t❡♠♦s (a−c)−(b−d) = (k1−k2)m✳ ▲♦❣♦✱
a−c≡b−d✭♠♦❞ m✮✳
✭✸✮ ❙❡ a≡b✭♠♦❞ m✮ ❡ c≡d✭♠♦❞ m✮ ❡①✐st❡♠ ✐♥t❡✐r♦sk1 ❡k2✱ t❛✐s q✉❡a−b =k1m ❡ c−d = k2m. ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ✐❣✉❛❧❞❛❞❡ ♣♦r c ❡ ❛ s❡❣✉♥❞❛ ♣♦r b ♦❜t❡♠♦s
ac−bc=ck1m ❡ bc−bd=bk2m. ❙♦♠❛♥❞♦ ❛s ú❧t✐♠❛s ✐❣✉❛❧❞❛❞❡s✱ ♦❜t❡♠♦s ac−bd=
(ck1+bk2)m✳ ▲♦❣♦✱ ac≡bd✭♠♦❞ m✮✳
✭✹✮ ❈♦♠♦ ac ≡ bc✭♠♦❞ m✮✱ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ k t❛❧ q✉❡ ac− bc = c(a − b) = km✳ ❙❡ ❞✐✈✐❞✐r♠♦s ♦s ❞♦✐s ♠❡♠❜r♦s ♣♦r d, t❡r❡♠♦s (c/d)(a −b) = k(m/d)✳ ❆ss✐♠✱ (m/d)|(c/d)(a−b)❡ ❝♦♠♦ ♠❞❝(m/d, c/d) = 1,✉s❛♥❞♦ ♦ ❢❛t♦ q✉❡a|bc❡(a, b) = 1t❡♠♦s a|c. ❆ss✐♠✱ a≡b✭♠♦❞ m/d✮✳
✭✺✮ ❚❡♠♦s ak−bk= (a−b)(ak−1+ak−2b+ak−3b2+...+abk−2+bk−1)❡ m|(a−b) ❡♥tã♦ m|ak−bk✳ ▲♦❣♦✱ak≡bk✭♠♦❞ m✮✳
❉❡✜♥✐çã♦ ✷✳✸✳ ❙❡ h ❡ k sã♦ ❞♦✐s ✐♥t❡✐r♦s ❝♦♠ h ≡ k✭♠♦❞ m✮✱ ❞✐③❡♠♦s q✉❡ k é ✉♠
r❡sí❞✉♦ ❞❡ h ♠ó❞✉❧♦ ♠✳
❉❡✜♥✐çã♦ ✷✳✹✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s {r1, r2, ..., rs} é ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí✲
❞✉♦s ♠ó❞✉❧♦ ♠ s❡✿
✭✶✮ ri 6≡rj✭♠♦❞ ♠✮ ♣❛r❛ i6=j✳
✭✷✮ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ n ❡①✐st❡ ✉♠ ri t❛❧ q✉❡ n≡ri ✭♠♦❞m✮✳
❊①❡♠♣❧♦ ✷✳✺✳ ❙❡❥❛ h = 25 ❡ k ✉♠ ✐♥t❡✐r♦ t❛❧ q✉❡ 25 ≡ k✭♠♦❞ 7✮ ❡ k ♣❡rt❡♥❝❡ ❛♦
❊q✉❛çõ❡s ❉✐♦❢❛♥t✐♥❛s ❧✐♥❡❛r❡s ✸✸
✷✳✽ ❊q✉❛çõ❡s ❉✐♦❢❛♥t✐♥❛s ❧✐♥❡❛r❡s
❊q✉❛çõ❡s ❉✐♦❢❛♥t✐♥❛s✸ ❧✐♥❡❛r❡s✱ sã♦ ❡q✉❛çõ❡s ♥❛ ❢♦r♠❛ ax+by = c✱ ❝♦♠ a, b ❡ c ✐♥t❡✐r♦s ♥ã♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♥✉❧♦s✳ ❆ s♦❧✉çã♦ ❞❡st❡ t✐♣♦ ❞❡ ❡q✉❛çã♦ é ❞❛❞❛ ♣❡❧♦ ♣❛r ❞❡ ✐♥t❡✐r♦s ✭x0, y0✮✱ t❛❧ q✉❡ ax0 +by0 = c s❡❥❛ ✈❡r❞❛❞❡✐r❛✳ ❖✉ s❡❥❛✱ ❛s s♦❧✉çõ❡s ❞❛ ❊q✉❛çã♦ ❉✐♦❢❛♥t✐♥❛ ❧✐♥❡❛r sã♦ ♦s ♣♦♥t♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s ✐♥t❡✐r❛s ❞♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ q✉❡ ❡stã♦ ❞✐s♣♦st♦s s♦❜r❡ ❛ r❡t❛ q✉❡ ❡st❛ r❡♣r❡s❡♥t❛✳
❆♣r❡s❡♥t❛r❡♠♦s ❛ s❡❣✉✐r ❛❧❣✉♥s r❡s✉❧t❛❞♦s r❡t✐r❛❞♦s ❞❡ ❬✶✹❪ ♣❛r❛ t❛✐s ❡q✉❛çõ❡s✳ Pr♦♣♦s✐çã♦ ✷✳✹✳ ❯♠❛ ❡q✉❛çã♦ ❞✐♦❢❛♥t✐♥❛ ❧✐♥❡❛r ❞❛ ❢♦r♠❛ ax+by =c✱ ❡♠ q✉❡ a6= 0
♦✉ b6= 0✱ ❛❞♠✐t❡ s♦❧✉çã♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ d=♠❞❝(a, b) ❞✐✈✐❞❡ c✳
❉❡♠♦♥str❛çã♦✳ ✭⇒✮❙✉♣♦♥❤❛♠♦s q✉❡ ✭x0, y0✮ é ✉♠ ♣❛r ❞❡ ✐♥t❡✐r♦s s❛t✐s❢❛③❡♥❞♦ ax0 +
by0 =c✳ ❙❡♥❞♦ d =♠❞❝(a, b)✱ t❡♠♦s q✉❡ d | a ❡ d| b✱ ❧♦❣♦ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✽ ✐t❡♠ ✭✼✮✱ d |(ax0+by0), ♦✉ s❡❥❛✱ d|c✳
✭⇐✮ ❙❡❥❛ d = ♠❞❝(a, b) s✉♣♦♥❞♦ q✉❡ d | c, t❡♠♦s c = pd, ♣❛r❛ ❛❧❣✉♠ p ✐♥t❡✐r♦✳ ❙❡ d = mdc(a, b) t❡♠♦s d | a ❡ d | b, ❞❛ Pr♦♣♦s✐çã♦ ✷✳✽ ✐t❡♠ ✼✱ d | (ar+bs), ❝♦♠ r ❡ s
✐♥t❡✐r♦s✳ ❆ss✉♠✐♥❞♦ q✉❡ ar+bs= 1·d ❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣❡❧♦ ✐♥t❡✐r♦ p t❡♠♦s arp+bsp=pd, ♦✉ s❡❥❛✱ a(rp) +b(sp) =c. ❆ss✐♠(x0, y0) = (rp, sp) é s♦❧✉çã♦ ❞❡ ax+by =c✳
❚❡♦r❡♠❛ ✷✳✺✳ ❙❡❥❛♠a❡b✐♥t❡✐r♦s ❡d=mdc(a, b)✳ ❙❡d|c❡♥tã♦ ❛ ❡q✉❛çã♦ ❞✐♦❢❛♥t✐♥❛
❧✐♥❡❛r ax+by = c ♣♦ss✉✐ ✐♥✜♥✐t❛s s♦❧✉çõ❡s✳ ❙❡ (x0, y0) é ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❞❛ ❡q✉❛çã♦✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ t t❛❧ q✉❡ t♦❞❛s ❛s s♦❧✉çõ❡s (x, y) sã♦ ❞❛❞❛s ♣♦r✿
x=x0+ (b/d)t
y=y0−(a/d)t
❉❡♠♦♥str❛çã♦✳ ❙❡ d = ♠❞❝(a, b) ❡♥tã♦ d | a ❡ d | b. P❡❧♦ ❧❡♠❛ ❞❡ ❇é③♦✉t ❡①✐st❡♠
✐♥t❡✐r♦s n ❡ m✱ t❛✐s q✉❡an+bm=d✳ ❈♦♠♦ d |c ✱ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ t t❛❧ q✉❡ c=td✳
❙❡ ♠✉❧t✐♣❧✐❝❛r♠♦s ❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r ♣♦r tt❡♠♦s ant+bmt=td =c✳ ■st♦ ♥♦s ❞✐③ q✉❡
♦ ♣❛r (x0, y0) = (nt, mt) é ✉♠❛ s♦❧✉çã♦ ❞❡ax+by =c✳
❱❛♠♦s s✉♣♦r q✉❡ (x, y) s❡❥❛ ✉♠❛ s♦❧✉çã♦✳ ❈♦♠♦ ax0 +by0 = c ♦❜t❡♠♦s ax+by =
ax0 +by0, ♦ q✉❡ ✐♠♣❧✐❝❛ a(x −x0) = b(y0 − y)✳ ❉✐✈✐❞✐♥❞♦ ♦s ❞♦✐s ♠❡♠❜r♦s ♣♦r d t❡r❡♠♦s (a/d)(x−x0) = (b/d)(y0 − y). ▲♦❣♦ (b/d) | (x−x0) ❡ ♣♦rt❛♥t♦ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ t s❛t✐s❢❛③❡♥❞♦ x−x0 = t(b/d), ♦✉ s❡❥❛✱ x = x0+ (b/d)t✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ❝♦♠♦ (a/d) | (y0 −y), ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ t t❛❧ q✉❡ y0 −y = (a/d)t ❡ ❛ss✐♠ ♦❜t❡♠♦s
y=y0−(a/d)t✳
❈♦r♦❧ár✐♦ ✷✳✶✳ ❙❡ d = ♠❞❝(a, b) = 1 ❡ (x0, y0) é ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❞❛ ❡q✉❛çã♦ ❞✐♦❢❛♥t✐♥❛ ❧✐♥❡❛r ax+by =c, ❡♥tã♦ t♦❞❛s ❛s ♦✉tr❛s s♦❧✉çõ❡s s❡rã♦ ❞❛❞❛s ♣♦r✿
