NÚMEROS P-ÁDICOS

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

PP●▼ ✲ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚

◆ú♠❡r♦s ♣✲á❞✐❝♦s

♣♦r

❮t❛❧♦ ▼♦r❛❡s ❞❡ ▼❡❧♦ ●✉s♠ã♦

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ ❉❡✉s✱ ❡♠ s✉❛ ❜♦♥❞❛❞❡ ✐♥✜♥✐t❛✱ ♣♦r ♣❡r♠✐t✐r q✉❡ ❡st❡ ♠♦♠❡♥t♦ ✜③❡ss❡ ♣❛rt❡ ❞❡ ♠✐♥❤❛ ❥♦r♥❛❞❛ t❡rr❡str❡✳

❆♦s ♠❡✉s ♣❛✐s✱ q✉❡ ❢♦r❛♠ r❡s♣♦♥sá✈❡✐s ♣❡❧♦ ♠❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♠♦r❛❧ ❡ ✐♥t❡✲ ❧❡❝t✉❛❧ ❛tr❛✈és ❞❡ t♦❞♦ ❛♣♦✐♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ❝❤❡❣❛r ♦♥❞❡ ❝❤❡❣✉❡✐✱ s❡♠ ❡❧❡s ♥❛❞❛ ❞✐st♦ ❛❝♦♥t❡❝❡r✐❛✳

❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❉♦✉t♦r ❇r✉♥♦ ❘✐❜❡✐r♦✱ ♣❛rt✐❝✐♣♦✉ ❝♦♠✐❣♦ ❛♦ ❧♦♥❣♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❞❡❞✐❝❛♥❞♦ s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ t❡♠♣♦ ♣❛r❛ ♠❡ ❣✉✐❛r à ❝♦♥❝❧✉sã♦ ❞♦ ♠❡s♠♦✳

❆ ❙✉❡❧❡♥✱ ♣♦r s✉❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❛♣♦✐♦ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❡str❡ss❛♥t❡s✱ ❝♦♠✲ ♣❧✐❝❛❞♦s ❡ ❞✐st❛♥t❡s q✉❡ ♦ ❛♥❞❛♠❡♥t♦ ❞♦ ❝✉rs♦ ♣r♦✈♦❝❛✳

❆ ❘❛❢❛❡❧✱ ♣♦r t♦❞♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❡st✉❞♦✱ ♣r❡s❡♥ç❛ ❝♦♥st❛♥t❡ ♥❡st❛ ❛♥❞❛❞❛ ❡♠ ❜✉s❝❛ ❞❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ ♠❡str❛❞♦✳

➚ t♦❞♦s q✉❡ ❡st✐✈❡r❛♠ ♣r❡s❡♥t❡s ♥❡st❛ ❧♦♥❣❛ ❝❛♠✐♥❤❛❞❛✳ ❆♦s ❝♦❧❡❣❛s ❞❡ P❘❖❋✲ ▼❆❚✱ ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s✱ q✉❡ ♣r♦♣♦r❝✐♦♥❛r❛♠ ❡st❛ ❡①♣❡r✐ê♥❝✐❛ ú♥✐❝❛✱ ♣♦❞❡r ❝♦♠✲ ♣❛rt✐❧❤❛r ♠♦♠❡♥t♦s ❞❡ ❢❡❧✐❝✐❞❛❞❡s ❡ ❞✐✜❝✉❧❞❛❞❡s✳

❆ ♠✐♥❤❛ ❢❛♠í❧✐❛✳ ➚ ❙✉❡❧❡♥✳ ➚q✉❡❧❡s q✉❡ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ❡st✐✈❡r❛♠ ♣r❡s❡♥t❡s ♥❡st❡ ❝✐❝❧♦✳ ❆ t♦❞♦s ♦s q✉❡ s❡ ❛❧❡❣r❛♠ ❝♦♠ ♦ s✉❝❡ss♦✳

❘❡s✉♠♦

❆♣r❡s❡♥t❛♠♦s ❡ ❞❡✜♥✐♠♦s ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s ❝♦♠♦ ♦ r❡s✉❧t❛❞♦ ❞❡ ✉♠❛ ❜✉s❝❛ ♣♦r s♦❧✉çõ❡s✱ ♣❛r❛ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s✱ q✉❡ ♣❛rt❡ ❞❡ ✉♠❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧✱ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❛❝✐♦♥❛✐s✳ ❈♦♥st❛t❛♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s é ❡str✐t❛♠❡♥t❡ ♠❛✐♦r q✉❡ ♦s ✐♥t❡✐r♦s✳ ▼♦str❛♠♦s ✉♠ ❝r✐tér✐♦ ♣❛r❛ q✉❡ ✉♠ r❛❝✐♦♥❛❧ ♣♦ss✉❛ ✉♠ ❝♦rr❡s♣♦♥❞❡♥t❡ ♥✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳ ❇✉s❝❛♠♦s ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ r❡♣r❡s❡♥t❛r♠♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ❡ ♥ú♠❡r♦s ❝♦♠♣❧❡✲ ①♦s ❝♦♠♦ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳ ❆❧❣❡❜r✐❝❛♠❡♥t❡✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s s❡rá ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡ ❡✱ ♣❛rt✐♥❞♦ ❞✐st♦✱ ❜✉s❝❛♠♦s ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ❝♦r♣♦ ❞❡ ❢r❛çõ❡s ❞♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✱ q✉❡ ❢♦r♠❛rã♦✱ ❛ss✐♠✱ ♦ ❝♦r♣♦ ❞♦s r❛❝✐♦♥❛✐s ♣✲á❞✐❝♦s✱ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ♣✉r❛♠❡♥t❡ ❛❧❣é❜r✐❝♦✳ ◆❛ s❡❣✉♥❞❛ ♣❛rt❡✱ ✈❛♠♦s ❡①♣♦r ♦s ❢✉♥❞❛✲ ♠❡♥t♦s ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠❛ ♥♦r♠❛ ❞✐❢❡r❡♥t❡ ❞❛ ❤❛❜✐t✉❛❧✱ ❡st❛❜❡❧❡❝❡♥❞♦ ❛ss✐♠ ✉♠❛ ♥♦✈❛ ♠étr✐❝❛✱ ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ ❡ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ❝♦r♣♦ ♥ã♦✲❛rq✉✐♠❡❞✐❛♥♦✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s❀ ❝♦r♣♦ ♥ã♦✲❛rq✉✐♠❡❞✐❛♥♦❀ ♥ú♠❡r♦s ♣✲ á❞✐❝♦s✳

❲❡ ✐♥tr♦❞✉❝❡ ❛♥❞ ❞❡✜♥❡ t❤❡ ♣✲❛❞✐❝s ✐♥t❡❣❡r ♥✉♠❜❡rs ❛s ❛ s❡❛r❝❤ ❢♦r s♦❧✉t✐♦♥s r❡s✉❧t✱ ❢♦r ❛ ❝♦♥❣r✉❡♥❝❡s s②st❡♠ t❤❛t ❞❡r✐✈❡s ❢r♦♠ ❛ ✈❛r✐❛❜❧❡ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥ ✇✐t❤ r❛t✐♦♥❛❧ ❝♦❡✣❝✐❡♥ts✳ ❲❡ ❡✈✐❞❡♥❝❡ t❤❛t t❤❡ ♣✲❛❞✐❝ ✐♥t❡❣❡rs s❡t ✐s str✐❝t❧② ❧❛r❣❡r t❤❛♥ t❤❡ ✐♥t❡❣❡rs✳ ❲❡ ♣r❡s❡♥t ❛ ❝r✐t❡r✐♦♥ s♦ t❤❛t ❛ r❛t✐♦♥❛❧ t❤❛t ❤♦❧❞s ❛ ❝♦rr❡s♣♦♥✲ ❞❡♥t ✐♥ ❛ ♣✲❛❞✐❝ ✐♥t❡❣❡rs s❡t✳ ❲❡ s❡❛r❝❤ ❢♦r t❤❡ ♣♦ss✐❜✐❧✐t② t♦ r❡♣r❡s❡♥t ✐rr❛t✐♦♥❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛s ♣✲❛❞✐❝s ✐♥t❡❣❡rs✳ ❆❧❣❡❜r❛✐❝❛❧❧②✱ t❤❡ ♣✲❛❞✐❝ ✐♥t❡❣❡rs s❡t ✇✐❧❧ ❜❡ ❛♥ ✐♥t❡❣r❛❧ ❞♦♠❛✐♥ ❛♥❞✱ ❢r♦♠ t❤✐s✱ ✇❡ s❡❛r❝❤ ❢♦r t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ♣✲❛❞✐❝ ✐♥t❡❣❡rs q✉♦t✐❡♥t ✜❡❧❞ s♦ t❤❛t s❤❛❧❧ ❢♦r♠ t❤❡ ♣✲❛❞✐❝ r❛t✐♦♥❛❧s ✜❡❧❞✱ ❢r♦♠ ❛ ♣✉r❡❧② ❛❧❣❡❜r❛✐❝❛❧❧② ♣♦✐♥t ♦❢ ✈✐❡✇✳ ■♥ t❤❡ s❡❝♦♥❞ ♣❛rt✱ ✇❡ ✇✐❧❧ ❡①♣♦s❡ t❤❡ ❜❛s❡s ❢♦r t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ♥♦r♠ t❤❛t✬s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ✉s✉❛❧✱ ❡st❛❜❧✐s❤✐♥❣ s♦ ❛ ♥❡✇ ♠❡tr✐❝ ✐♥ t❤❡ r❛t✐♦♥❛❧ ♥✉♠❜❡rs s❡t ❛♥❞ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ♥♦♥✲❛r❝❤✐♠❡❞✐❛♥ ✜❡❧❞✳

❑❡② ✇♦r❞s✿ ♥✉♠❜❡rs t❤❡♦r②❀ ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ✜❡❧❞❀ ♣✲❛❞✐❝ ♥✉♠❜❡rs✳

❙✉♠ár✐♦

❆❣r❛❞❡❝✐♠❡♥t♦s ✐✐✐

✶ ◆ú♠❡r♦s ✐♥t❡✐r♦s ❡ r❛❝✐♦♥❛✐s ❝♦♠♦ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s ✶ ✶✳✶ ▼♦t✐✈❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❖s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷✳✶ ❘❡s♦❧✈❡♥❞♦ ❝♦♥❣r✉ê♥❝✐❛s ♠ó❞✉❧♦ pn_{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹}

✶✳✷✳✷ ❆❞✐çã♦ ❡♠ Zp✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✶✳✷✳✸ ▼✉❧t✐♣❧✐❝❛çã♦ ❡♠ Zp✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✶✳✷✳✹ ❘❡♣r❡s❡♥t❛♥t❡s ♣✲á❞✐❝♦s ❞❡ ✐♥t❡✐r♦s ♥❡❣❛t✐✈♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷✳✺ (Zp,+) é ✉♠ ●r✉♣♦ ❆❜❡❧✐❛♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✶✳✷✳✻ (Zp,+,·)é ✉♠ ❆♥❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✶✳✷✳✼ (Zp,+,·)é ✉♠ ❉♦♠í♥✐♦ ❞❡ ■♥t❡❣r✐❞❛❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✶✳✷✳✽ ❘❛❝✐♦♥❛✐s ❝♦♠♦ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷✳✾ ■rr❛❝✐♦♥❛✐s ❛❧❣é❜r✐❝♦s ❝♦♠♦ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷✳✶✵ ❈♦♠♣❧❡①♦s ❝♦♠♦ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✷✳✶✶ ❖s r❛❝✐♦♥❛✐s ♣✲á❞✐❝♦s ❝♦♠♦ ❝♦r♣♦ ❞❡ ❢r❛çã♦ ❞❡ Zp✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✷ ❯♠❛ ✈✐sã♦ t♦♣♦❧ó❣✐❝❛ ❞♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s ✸✵

✷✳✶ ◆♦t❛s ❤✐stór✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷ ◆♦r♠❛ ♦✉ ✈❛❧♦r ❛❜s♦❧✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷✳✶ ◆♦r♠❛ ❡♠ ✉♠ ❝♦r♣♦ q✉❛❧q✉❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷✳✷ ❱❛❧♦r✐③❛çã♦ ♣✲á❞✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✸ ▼étr✐❝❛ ❡ ❚♦♣♦❧♦❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✷

◆♦t❛çõ❡s ●❡r❛✐s

• ≡ (mod n) r❡♣r❡s❡♥t❛rá ❛ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ n✳

• (a, n) s❡rá ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡ a ❡n✳

• Z/nZé ♦ ❛♥❡❧ ❞❡ ✐♥t❡✐r♦s ♠ó❞✉❧♦ n✳ ❯s❛r❡♠♦sZ/pZq✉❛♥❞♦ n ❢♦r ♣r✐♠♦✳

• Zp s❡rá ♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳

• Qp s❡rá ♦ ❝♦♥❥✉♥t♦ ❞♦s r❛❝✐♦♥❛✐s ♣✲á❞✐❝♦s✳

• vp(n) s❡rá ❛ ✈❛❧♦r✐③❛çã♦ ♣✲á❞✐❝❛ ❞♦ r❛❝✐♦♥❛❧ n✳

• | · |p é ❛ ♥♦r♠❛ ♣✲á❞✐❝❛✳

• δ(x, y)é ❛ ♠étr✐❝❛ s♦❜r❡ Q ❝♦♠ ❛ ♥♦r♠❛ ♣✲á❞✐❝❛✳

■♥tr♦❞✉çã♦

❆ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ♣❛rt✐♥❞♦ ❞❡s❞❡ ♦s ♠❛✐s s✐♠♣❧❡s ❝♦♥❝❡✐t♦s ❝♦♠♦ ❞✐✈✐s✐✲ ❜✐❧✐❞❛❞❡ ❡ ❢❛t♦r❛çã♦ ❡♠ ♣r✐♠♦s✱ ❛♦s ♠❛✐s s♦✜st✐❝❛❞♦s ❝♦♠♦ ❛ r❡s♦❧✉çã♦ ❞♦ ú❧t✐♠♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ ❝❤❛♠❛ ❛t❡♥çã♦ ♣❡❧❛s ❜❡❧❛s ❛♣❧✐❝❛çõ❡s ❡ ✐♠♣❧✐❝❛çõ❡s ❧ó❣✐❝❛s✱ ❛❧é♠ ❞♦s r❡s✉❧t❛❞♦s q✉❡ ❞❡❧❛s ♦r✐❣✐♥❛♠✳ ❖ ❢❛s❝í♥✐♦ ❛♣r❡s❡♥t❛❞♦ ♥♦s ♣r✐♠❡✐r♦s ❝♦♥t❛t♦s ❝♦♠ ♦ t❡♠❛ ✐♠♣✉❧s✐♦♥❛r❛♠ ❛ ❜✉s❝❛ ♣♦r ❝♦♥t❡ú❞♦s r❡❧❛❝✐♦♥❛❞♦s✳

❇✉s❝❛♠♦s tr❛③❡r ✉♠❛ ❛♣r❡s❡♥t❛çã♦ s✉❝✐♥t❛✱ à ❣r❛❞✉❛♥❞♦s✱ ❣r❛❞✉❛❞♦s ❡ ♠❡str❛♥✲ ❞♦s ❡♠ ♠❛t❡♠át✐❝❛✱ q✉❡ ❥á t✐✈❡r❛♠ ❝♦♥t❛t♦ ❝♦♠ ❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❡♠ ❛❧❣✉♠ ♥í✈❡❧✱ ✉♠ t❡♠❛ q✉❡ ♥ã♦ é tã♦ ❝♦♠✉♠❡♥t❡ ❛❜♦r❞❛❞♦ ❡♠ ❝✉rs♦s ❞❡ t❡♦r✐❛ ❞♦s ♥ú♠❡✲ r♦s✱ ❝♦♠♦ ♦s ❛♣r❡s❡♥t❛❞♦s ♥♦s ❝✉rs♦s ❞❡ ❣r❛❞✉❛çã♦✳ ❊st❡ ❢♦✐ ♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠❛ t❡♦r✐❛ ❝♦♥❝❡❜✐❞❛ ❛ ♣♦✉❝♦ ♠❛✐s ❞❡ ✶✵✵ ❛♥♦s ♣♦r ❑✉rt ❍❡♥s❡❧✱ ❡ q✉❡ ❛♣r❡s❡♥t♦✉✲s❡ ❝♦♠♦ ✉♠❛ ❜❡❧❛ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ♦ ❛✉①í❧✐♦ ♥❛ r❡s♦❧✉çã♦ ❞❡ ✈ár✐♦s ♣r♦❜❧❡♠❛s✱ ❡♥tr❡ ❡❧❡s ✉♠ ❝é❧❡❜r❡✱ ♦ ú❧t✐♠♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ q✉❡ ♣❛ss♦✉ ♠❛✐s ❞❡ ✸✺✵ ❛♥♦s s❡♠ s♦❧✉çã♦✳

❆♣ós ✐♥ú♠❡r❛s ❜✉s❝❛s ♣♦r r❡❢❡rê♥❝✐❛s q✉❡ ❢✉♥❞❛♠❡♥t❛ss❡♠ ♦ ❝♦♥t❡ú❞♦ ♣r❡t❡♥✲ ❞✐❞♦✱ ♣❡r❝❡❜❡♠♦s q✉❡ ❛❧❣✉♥s ♣♦♥t♦s ♥❡❝❡ss✐t❛✈❛♠ ❞❡ ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ♠❛✐s ❞❡t❛✲ ❧❤❛❞❛✱ t❛❧✈❡③ ❝♦♠ ✉♠❛ ✈✐sã♦ ❞✐❢❡r❡♥❝✐❛❞❛✱ s❡r✈✐♥❞♦ ❛ss✐♠ ❝♦♠♦ ✉♠ ✐♠♣✉❧s♦ ✐♥✐❝✐❛❧ ♣❛r❛ ❛ ❛❜♦r❞❛❣❡♠ ❞♦ ❝♦♥t❡ú❞♦✳

❆ ✐❞❡✐❛ ❞❡ ❛❜♦r❞❛r ♦s ♥ú♠❡r♦s ♣✲á❞✐❝♦s s✉r❣✐✉ ❛♦ s❡ ❜✉s❝❛r ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ q✉❡ t♦r♥❛ ♣♦ssí✈❡❧ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ t♦❞❛ ✉♠❛ ❡str✉t✉r❛ ❞❡ ✉♠ ❝♦r♣♦ ❞❡ ❢r❛çõ❡s ❞✐❢❡r❡♥t❡ ❞♦ ❤❛❜✐t✉❛❧✳ P❛rt✐♥❞♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s r❛❝✐♦♥❛✐s s❛❜❡♠♦s q✉❡ ♣♦❞❡♠♦s ❝♦♥str✉✐r✱ ❛tr❛✈és ❞❡ s❡q✉ê♥❝✐❛s ❞❡ ❈❛✉❝❤②✱ s✉❛ ❡①t❡♥sã♦✱ ♦s ♥ú♠❡r♦s r❡❛✐s✳ ❙❛❜❡♠♦s q✉❡ ♥♦ ✈❛st♦ ❝❛♠♣♦ ❞❛ ♠❛t❡♠át✐❝❛✱ ♦s ♥ú✲ ♠❡r♦s r❡❛✐s ❡ s❡✉ ❢❡❝❤♦ ❛❧❣é❜r✐❝♦✱ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❞❡s❡♠♣❡♥❤❛♠ ✉♠ ♣❛♣❡❧ s✐♥❣✉❧❛r✳ ❱❡r❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ♦✉tr❛ ♠étr✐❝❛✱ ❞✐❢❡r❡♥t❡ ❞❛ ❤❛❜✐t✉❛❧✱ q✉❡ ♥♦s ❞á ✉♠ r❡s✉❧t❛❞♦ s✐♠✐❧❛r✳

❖ ❚r❛❜❛❧❤♦ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ ❞✉❛s ♣❛rt❡s✳ ❆ ♣r✐♠❡✐r❛ ♣❛rt❡ ❜✉s❝❛ ❛♣r❡s❡♥t❛r ♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s ❞❡ ♠❛♥❡✐r❛ ❛❧❣é❜r✐❝❛✱ ♣❛rt✐♥❞♦ ❞❡ ✉♠❛ ❛❜♦r❞❛❣❡♠ ♥ã♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦

x= 1 + 2x.

❚❛❧ r❡s♦❧✉çã♦✱ q✉❡ é ❛❜♦r❞❛❞❛ ❞❡ ❢♦r♠❛ ❝❧❛r❛ ♠❛✐s ❛ ❢r❡♥t❡✱ ❛♣r❡s❡♥t❛ ♣r♦❝❡❞✐♠❡♥t♦s ❝♦❡r❡♥t❡s✱ s❛❧✈♦ ♦ ❢❛t♦ ❞❡ ✉♠❛ sér✐❡ ❣❡♦♠étr✐❝❛ só ❝♦♥✈❡r❣✐r s❡ ❛ r❛③ã♦ ❡st✐✈❡r ❡♥tr❡ ③❡r♦ ❡ ✉♠✳

◆❛ s❡çã♦ ✶✳✷ ✐♥✐❝✐❛♠♦s ♥♦ss❛ ❝❛♠✐♥❤❛❞❛ ❝♦♠ ♦s ♥ú♠❡r♦s ♣✲á❞✐❝♦s✱ ❛✐♥❞❛ ❝♦♠ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s q✉❡ ♥♦s s❡r✈✐rã♦ ❞❡ ❜❛s❡s✳ ❱❡r❡♠♦s t❛♠❜é♠ ❛❧❣✉♠❛s ♣❛rt✐❝✉❧❛✲ r✐❞❛❞❡s ❞❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s ✉t✐❧✐③❛♥❞♦ ♦ q✉❡ ✐r❡♠♦s ❞❡✜♥✐r ❝♦♠♦

❱❡r❡♠♦s q✉❡ ❞❛❞♦ ✉♠ k _∈ Z+_{✱ ❛ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ❞❡ −k s❡rá ✐♥✜♥✐t❛✱}

❡ q✉❡ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r♠♦s ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ✐♥❞✐❝❛r ♦ ✐♥✈❡rs♦ ❛❞✐t✐✈♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♣✲á❞✐❝♦✳ ▼♦str❛♠♦s q✉❡ ♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s sã♦ ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡✳ ❆tr❛✈és ❞❡ ❡①❡♠♣❧♦s✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❝❛♠✐♥❤♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ♦s r❡♣r❡s❡♥t❛♥t❡s ♣✲á❞✐❝♦s ❞❡ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ ❡✈✐❞❡♥❝✐❛♥❞♦ ✉♠ ❝r✐tér✐♦ ♣❛r❛ q✉❡ ✉♠ r❛❝✐♦♥❛❧ ♣♦ss✉❛ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ♣❛r❛ ✉♠ ❞❛❞♦ p ✜①♦✳ ❚❛♠❜é♠ s❡rã♦

❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♥s ✐rr❛❝✐♦♥❛✐s ❛❧❣é❜r✐❝♦s ❝♦♠♦ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳ ❱❡r✐✜❝❛r❡♠♦s q✉❡ é ♣♦ssí✈❡❧ r❡♣r❡s❡♥t❛r♠♦s ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ❝♦♠♦ ✉♠ ♥ú♠❡r♦ ♣✲á❞✐❝♦✳

P❛rt✐♥❞♦ ❞♦s ❡①❡♠♣❧♦s ❥á ✈✐st♦s ✜❝❛ ❢á❝✐❧ ❞❡ ♣❡r❝❡❜❡r q✉❡ ♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s é ❡str✐t❛♠❡♥t❡ ♠❛✐♦r q✉❡ ♦s ✐♥t❡✐r♦s✱ ♣❛r❛ t♦❞♦p ♣r✐♠♦ ❡✱ ❛❧é♠ ❞✐ss♦✱ ✈❡r❡♠♦s q✉❡ ♦

❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s ♥ã♦ é ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦✱ ♦ q✉❡ ♣♦❞❡♠♦s ♥♦t❛r q✉❛♥❞♦ ❡st❛♠♦s t❡♥t❛♥❞♦ ❞❡t❡r♠✐♥❛r ♦ ♥ú♠❡r♦ á✉r❡♦ ♥♦s ♣✲á❞✐❝♦s ✭❡①❡♠♣❧♦ ✶✳✶✵✮✱ ♣♦✐s ♣❛r❛ p= 2,3,5 ❡ 7 ♥ã♦ é ♣♦ssí✈❡❧✳ ❆ ❝♦♥❝❧✉sã♦ ❞❛ ♣r✐♠❡✐r❛ ♣❛rt❡ s❡ ❞á ❝♦♠ ❛

❝♦♥str✉çã♦ ❞♦ ❝♦r♣♦ ❞❡ ❢r❛çõ❡s ❞❡Zp✳ ❖♥❞❡ ❝♦♥❝❧✉✐r❡♠♦s ❛♣r❡s❡♥t❛♥❞♦ ✉♠❛ r❡❧❛çã♦

❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡F rac(Zp) ❡ s✉❛s r❡s♣❡❝t✐✈❛s ❡①♣❛♥sõ❡s ❡♠ sér✐❡s✳

❆ s❡❣✉♥❞❛ ♣❛rt❡ ♥♦s trás ✉♠❛ ✈✐sã♦ t♦♣♦❧ó❣✐❝❛ ❞♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✱ ✐♥✐❝✐❛♥❞♦ ❝♦♠ ❛❧❣✉♠❛s ♥♦t❛s ❤✐stór✐❝❛s ♥❛ s❡çã♦ ✷✳✶✳ ◆❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③✐r❡♠♦s ❛ ✐❞❡✐❛ ❞❡ ♥♦r♠❛ ♦✉ ✈❛❧♦r ❛❜s♦❧✉t♦ ♣✲á❞✐❝♦✱ ✐♥✐❝✐❛❧♠❡♥t❡ ♦❜s❡r✈❛♥❞♦ ♠✐♥✉❝✐♦s❛♠❡♥t❡ ❛s ❞❡✜♥✐✲ çõ❡s ❞❡ ✈❛❧♦r✐③❛çã♦ ♣✲á❞✐❝❛✱ ♣❛rt✐♥❞♦ ❞❛ ✈❛❧♦r✐③❛çã♦ ❞❡ ✉♠ ✐♥t❡✐r♦✱ ❛té ❝❤❡❣❛r♠♦s ❛♦ ❢❛t♦ ❞❡ q✉❡ ❛ ✈❛❧♦r✐③❛çã♦ ♣✲á❞✐❝❛ ❞❡ ✉♠ r❛❝✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡ ❞♦ q✉♦❝✐❡♥t❡ ❞❡ ✐♥t❡✐r♦s ❝♦♥s✐❞❡r❛❞♦✱ s❡❥❛ s✉❛ ❢♦r♠❛ ✐rr❡❞✉tí✈❡❧ ♦✉ ❝♦♠ r❡♣r❡s❡♥t❛çõ❡s ❡q✉✐✈❛❧❡♥t❡s✳ ❚❡r❡✲ ♠♦s ✐♥tr♦❞✉③✐❞❛ ❛ ✐❞❡✐❛ ❞❡ ✈❛❧♦r ❛❜s♦❧✉t♦ ♥ã♦✲❛rq✉✐♠❡❞✐❛♥♦ ❡ ✈❡r❡♠♦s q✉❡ ❛ ♥♦r♠❛ ♣✲á❞✐❝❛ é ✉♠ ❡①❡♠♣❧♦ ❞❡st❛✳ ❆♣r❡s❡♥t❛r❡♠♦s ❛s ✐❞❡✐❛s ✐♥✐❝✐❛✐s s♦❜r❡ ❛ ♠étr✐❝❛ ♥♦ ❝♦r♣♦ ❞♦s r❛❝✐♦♥❛✐s ♣✲á❞✐❝♦s✱ ❛❧é♠ ❞✐ss♦✱ ❛ ♣❛rt✐r ❞♦ ♠♦♠❡♥t♦ q✉❡ t❡♠♦s ✉♠ ✈❛❧♦r ❛❜s♦❧✉t♦ ❜❡♠ ❞❡✜♥✐❞♦✱ ♣♦❞❡♠♦s ❞❡s❝r❡✈❡r ✉♠❛ ♠étr✐❝❛ s♦❜r❡ ❡st❡ ❝♦r♣♦✳

❙❡rá ❛♣r❡s❡♥t❛❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡ ✉❧tr❛♠étr✐❝❛✳ ❊st❛ ♠étr✐❝❛ ✐♥❞✉③ ✉♠❛ t♦♣♦❧♦❣✐❛ ❝♦♠ ❝❛r❛❝t❡ríst✐❝❛s ♥ã♦ ❝♦♠✉♥s✱ ❡♥tr❡ ❡❧❛s ❢❛③ ❝♦♠ q✉❡ t♦❞♦s ♦s tr✐â♥❣✉❧♦s s❡❥❛♠ ✐sós❝❡❧❡s ❡ q✉❡✱ ❞❡✜♥✐❞❛s ❛s ❜♦❧❛s ✧❛❜❡rt❛s✧ ❡ ✧❢❡❝❤❛❞❛s✧✱ r❡s♣❡❝t✐✲ ✈❛♠❡♥t❡

B(c, t) ={x∈K: |x−c|< t}

B(c, t) = {x∈K: |x−c| ≤t},

✈❡r❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ B(c, t) é ❛❜❡rt♦ ❡ ❢❡❝❤❛❞♦ ❡ ♦ ❝♦♥❥✉♥t♦ B(c, t) t❛♠❜é♠ é

❛❜❡rt♦ ❡ ❢❡❝❤❛❞♦✳

P❛r❛ ✉♠ ♠❡❧❤♦r ❡♠❜❛s❛♠❡♥t♦ ❛s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ✉t✐❧✐③❛❞❛s ❢♦r❛♠ ❬✹❪✱ ❬✷❪ ❡ ❬✻❪ r❡❢❡r❡♥t❡s ❛♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✱ ❛❧é♠ ❞❡ ✉♠ ❡♠❜❛s❛♠❡♥t♦ t❡ór✐❝♦ s♦❜r❡ á❧❣❡❜r❛ ❡ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❝♦♠ ❬✸❪✱ ❬✶❪✱ ❬✺❪ ❡ ❬✽❪✳

❈❛♣ít✉❧♦ ✶

◆ú♠❡r♦s ✐♥t❡✐r♦s ❡ r❛❝✐♦♥❛✐s ❝♦♠♦

✐♥t❡✐r♦s ♣✲á❞✐❝♦s

✶✳✶ ▼♦t✐✈❛çã♦

❙❡❣✉♥❞♦ ❬✹❪✱ ❑✉rt ❍❡♥s❡❧ ✭1897✮ ✐♥tr♦❞✉③✐✉ ❛ ✐❞❡✐❛ ❞❡ ♥ú♠❡r♦ ♣✲á❞✐❝♦ ♣❛r❛ ♥♦s

♠♦str❛r q✉❡ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r♠♦s ✉♠❛ ❝♦♥str✉çã♦ ❞✐❢❡r❡♥t❡ ❞❛ ❤❛❜✐t✉❛❧ ♣❛r❛ ❛ ♠étr✐❝❛✱ ❡ ❛ss✐♠ ♦❜t❡r♠♦s ✉♠❛ t♦♣♦❧♦❣✐❛ s♦❜r❡ ♦s r❛❝✐♦♥❛✐s ❞✐st✐♥t❛ ❞❛ ✉s✉❛❧✳ ❆♥t❡s ❞❡ ❛♣r❡s❡♥t❛r♠♦s ❛ ✐❞❡✐❛ ❞❡ ❍❡♥s❡❧✱ ✈❛♠♦s ♦❜s❡r✈❛r ❛ s❡❣✉✐♥t❡ s✐t✉❛çã♦✿

❘❡s♦❧✈❛ ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦✿✶

x= 1 + 2x

❙✐♠♣❧❡s♠❡♥t❡ ♣♦❞❡rí❛♠♦s ♣❡♥s❛r ♥✉♠❛ r❡s♦❧✉çã♦ ❞✐r❡t❛✱ ❡♥❝♦♥tr❛♥❞♦ x = −1

❝♦♠♦ s♦❧✉çã♦✳ ▼❛s ✉♠ ♦❜s❡r✈❛❞♦r ❛t❡♥t♦ ❡ ✉♠ ♣♦✉❝♦ ♠❛✐s ❝r✐❛t✐✈♦ ♣♦❞❡r✐❛ ♣❡♥s❛r ❡♠ t♦♠❛r ✉♠ ♠❡✐♦ ✐t❡r❛t✐✈♦ ❡✱ ❝♦♠♦x= 1 + 2x✱ s✉❜st✐t✉✐r✐❛ ♦x ♣❡❧♦ s❡✉ ✈❛❧♦r✱ q✉❡

♥✉♠ ♣r✐♠❡✐r♦ ♣❛ss♦ ✜❝❛r✐❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

x= 1 + 2(1 + 2x)

❞❡♣♦✐s t❡rí❛♠♦s

x= 1 + 2(1 + 2(1 + 2x))

❡

x= 1 + 2(1 + 2(1 + 2(1 + 2x))),

❝♦♠ ❡st❡ ♣r♦❝❡ss♦ ♣♦❞❡♥❞♦ s❡ r❡♣❡t✐r ✈ár✐❛s ✈❡③❡s ✭❞❡ ♠❛♥❡✐r❛ ✐♥✜♥✐t❛ ✐♥❝❧✉s✐✈❡✮✳

❖❜s❡r✈❛♥❞♦ t❛❧ ❢❛t♦✱ ♣♦❞❡♠♦s ♣❡♥s❛r ♥✉♠❛ ❢♦r♠❛ ❞❡ ❝♦♥str✉✐r ✉♠❛ s❡q✉ê♥❝✐❛ t❛❧ q✉❡

xn+1 = 1 + 2xn.

✶_{❆ r❡s♦❧✉çã♦ ❞❡ ❢❛t♦ s❡rá ❛❜♦r❞❛❞❛ ♥♦ ❡①❡♠♣❧♦ ✶✳✸✳}

◆❡st❡ ♣♦♥t♦ ♥♦s ❜❛st❛ ❡s❝♦❧❤❡r ✉♠ x0✱ ♣♦r ❡①❡♠♣❧♦ ♣❛rt✐♥❞♦ ❞❡ ✉♠ x0 = 1✱ ♣❛r❛

♣♦❞❡r♠♦s ❞❡t❡r♠✐♥❛r✱ ♥✉♠❛ ❝♦♥str✉çã♦ r❡❝♦rr❡♥t❡✱ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♠♦❞♦ q✉❡

x0 = 1

x1 = 1 + 2x0 = 1 + 2·1

x2 = 1 + 2x1 = 1 + 2·(1 + 2·1)

x3 = 1 + 2x2 = 1 + 2·(1 + 2·(1 + 2·1))

✳✳✳

❘❡❛❧✐③❛♥❞♦ ❛s ♦♣❡r❛çõ❡s ❡♠ ❝❛❞❛ t❡r♠♦ xi ❞❛ s❡q✉ê♥❝✐❛ t❡♠♦s q✉❡✱ ♥❡st❡ ❝❛s♦✱

❡st❛♠♦s s♦♠❛♥❞♦ ♣♦tê♥❝✐❛s ❞❡2✱ ♦❜s❡r✈❡✿

x0 = 1

x1 = 1 + 2

x2 = 1 + 2 + 4

x3 = 1 + 2 + 4 + 8

✳✳✳

◗✉❡ ♣❛r❛ n ♣❛ss♦s ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❝♦♠♦✿

xn= 1 + 2 + 22 + 23+. . .+ 2n

❱❛❧❡ r❡ss❛❧t❛r q✉❡✱ s❡ t♦♠❛r♠♦s ✐♥✜♥✐t♦s ♣❛ss♦s ✭nt❡♥❞❡♥❞♦ ❛♦ ✐♥✜♥✐t♦✮✱ t❡r❡♠♦s

✉♠❛ s♦♠❛ ✐♥✜♥✐t❛ ❞❡ ♣♦tê♥❝✐❛s✳ ❆ss✐♠ s❡♥❞♦✱ t❛♠❜é♠ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛✱ ♥❡ss❡ ❝❛s♦ ❝♦♠ r❛③ã♦ 2❡ a0 = 1✱ ✐♥✜♥✐t❛✳

S =

+∞

X

i=0

2i

◆❡st❡ ♣♦♥t♦ ✈❛❧❡ ❧❡♠❜r❛r q✉❡ ✉♠❛ s♦♠❛ ✐♥✜♥✐t❛ ❞❡ ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ t❡♠ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

1 +α+α2+. . .= 1 1₋α,

♦♥❞❡ s✉❜st✐t✉✐♥❞♦α♣♦r2✱ ♦❜t❡r❡♠♦s−1✳ ❆♣❡s❛r ❞❛ s✉❜st✐t✉✐çã♦ s❡r ✐♥❞❡✈✐❞❛✱ ♣♦✐s

❛ s♦♠❛ ❞❡ ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ ✐♥✜♥✐t❛ só ❝♦♥✈❡r❣❡ s❡ ❛ r❛③ã♦ ❡st✐✈❡r ❡♥tr❡ ③❡r♦ ❡ ✉♠✱ ✐st♦ ♥♦s ❛♣r❡s❡♥t❛ ✉♠ ❢❛t♦ ✐♥t❡r❡ss❛♥t❡✱ ♣♦✐s ✈♦❧t❛♥❞♦ ♣❛r❛ ❛ ❡q✉❛çã♦ ♦r✐❣✐♥❛❧✱ ❛♦ r❡s♦❧✈ê✲❧❛ ❞❡ ❢♦r♠❛ ❝♦♥✈❡♥❝✐♦♥❛❧✱ t❛♠❜é♠ ♦❜t❡♠♦s ♦ r❡s✉❧t❛❞♦−1✳ ❚❛❧

❢❛t♦ ❛♣❛r❡♥t❡♠❡♥t❡ ♣♦❞❡ ♥♦s ♣❛r❡❝❡r ✉♠ s✐♠♣❧❡s ❝❛s♦ ✐s♦❧❛❞♦✱ ♠❛s ♣♦r ♦✉tr♦ ❧❛❞♦ ♣♦❞❡♠♦s s✉s♣❡✐t❛r ❞❡st❡ ♣r♦❝❡ss♦ ❡ ❛♥❛❧✐s❛r ♠❛✐s ❛ ❢✉♥❞♦ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ✈❛❧✐❞❛r ❡st❡ ❝❛♠✐♥❤♦✳

◆♦t❡♠♦s q✉❡ s❡❣✉✐♠♦s ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ❝♦❡r❡♥t❡✱ ♦❜s❡r✈❛♥❞♦ q✉❡ ♣♦❞❡♠♦s tr❛✲ t❛r ❛ ❡q✉❛çã♦ s✉❣❡r✐❞❛ ❝♦♠♦ ✉♠❛ sér✐❡✱ ❛té ♦ ❢❛t♦ ❞❡ ♥❡❝❡ss✐t❛r♠♦s q✉❡ ❛ sér✐❡ s❡❥❛ ❝♦♥✈❡r❣❡♥t❡✳ ❙❡rá ♣♦ssí✈❡❧ q✉❡✱ ❛tr❛✈és ❞❡ ❛❧❣✉♠❛ ❝♦♥str✉çã♦ ♥ã♦ ❝♦♥✈❡♥❝✐♦♥❛❧✱ ♣♦❞❡♠♦s ❢❛③❡r ✉♠❛ sér✐❡ q✉❡ ❛♣❛r❡♥t❡♠❡♥t❡ ❝r❡s❝❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✱ ❝♦♥✈❡r❣✐r❄

❱❡r❡♠♦s q✉❡ s✐♠✱ é ♣♦ssí✈❡❧ q✉❡ ✐st♦ ❛❝♦♥t❡ç❛✳ ▼❛s ❛♥t❡s ♣r❡❝✐s❛r❡♠♦s ❝♦♥st❛t❛r t♦❞❛ ❛ ❝♦♥str✉çã♦ ❞❛ ❡str✉t✉r❛ ♠❛t❡♠át✐❝❛ q✉❡ ♥♦s ♣r♦♣♦r❝✐♦♥❛rá ❜❡❧♦s r❡s✉❧t❛❞♦s ❛❝❡r❝❛ ❞❡st❛s ✐♥❞❛❣❛çõ❡s✳

✶✳✷ ❖s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳

❆ ✐❞❡✐❛ ❞❡ ❍❡♥s❡❧ ❡stá ❧✐❣❛❞❛ ❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ♠ó❞✉❧♦

pn✳ ❊st❡ s❡rá ♦ ❝❛♠✐♥❤♦ q✉❡ ✉t✐❧✐③❛r❡♠♦s ♣❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳

❆♥t❡s ❞❡ ❝♦♥s✐❞❡r❛r♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s✱ ✈❛♠♦s r❡❧❡♠❜r❛r ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s q✉❡ ♥♦s s❡rã♦ út❡✐s✳

❉❡✜♥✐çã♦ ✶✳✶ ❙❡❥❛♠ a✱ b ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡ n ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ❉✐r❡♠♦s q✉❡ a é ❝♦♥❣r✉❡♥t❡ ❛ b ♠ó❞✉❧♦ n s❡ a ❡ b ❞❡✐①❛♠ ♦ ♠❡s♠♦ r❡st♦ ♥❛ ❞✐✈✐sã♦ ♣♦r n✱ ❡♠

♦✉tr❛s ♣❛❧❛✈r❛s✱ q✉❛♥❞♦ (b−a) ♥ã♦ ❞❡✐①❛ r❡st♦ ♥❛ ❞✐✈✐sã♦ ♣♦r n✳ ❙❡ ✐st♦ ❛❝♦♥t❡❝❡✱

❡s❝r❡✈❡✲s❡

a_≡b (mod n).

P♦r ❡①❡♠♣❧♦✱19_≡3 (mod 8)❡ 21_≡1 (mod 4)✱ ♣♦✐s 19₋3é ❞✐✈✐sí✈❡❧ ♣♦r 8✱ ❞❛

♠❡s♠❛ ❢♦r♠❛✱21−1é ❞✐✈✐sí✈❡❧ ♣♦r 4✳

◆♦t❡ q✉❡✱ s❡ t♦♠❛r♠♦s ✉♠ ♥ú♠❡r♦d♥❛t✉r❛❧✱ ❡❧❡ s❡♠♣r❡ s❡rá ❝♦♥❣r✉❡♥t❡ ♠ó❞✉❧♦ n ❛♦ s❡✉ r❡st♦ ♥❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ♣♦rn✱ s❡♥❞♦ ❛ss✐♠ s❡rá ❝♦♥❣r✉❡♥t❡ ♠ó❞✉❧♦n ❛

❛❧❣✉♠ ❞♦s ♥ú♠❡r♦s0,1,2,3, . . . , n−1✳ P♦r ❝♦♥s❡q✉ê♥❝✐❛✱ ❞♦✐s ❞❡ss❡s ♥ú♠❡r♦s ♥ã♦

s❡rã♦ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ n ❡♥tr❡ s✐✳

❆q✉✐ ❞❡✐①❛♠♦s ❛♦ ❧❡✐t♦r ❛ ❢á❝✐❧ ✈❡r✐✜❝❛çã♦ ❞❡ q✉❡ s❡ ♠❛♥té♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ r❡✢❡①✐✈✐❞❛❞❡✱ s✐♠❡tr✐❛✱ tr❛♥s✐t✐✈✐❞❛❞❡✱ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❝♦♠ ❛ s♦♠❛ ❡ ❛ ❞✐❢❡r❡♥ç❛✱ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❝♦♠ ♦ ♣r♦❞✉t♦ ❡ ❝♦♠ ❛ r❡❣r❛ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦✳ ❖ ❧❡✐t♦r ♣♦❞❡ ✈❡r✐✜❝❛r ❡st❡s r❡s✉❧t❛❞♦s ❡♠ ❬✽❪ ❡ ♦❜t❡r ✉♠ ❡st✉❞♦ ♠❛✐s ❞❡t❛❧❤❛❞♦ s♦❜r❡ ❝♦♥❣r✉ê♥❝✐❛s ❡♠ ❬✺❪✳

P♦rt❛♥t♦✱ t❡♠♦s q✉❡ ❛ r❡❧❛çã♦ ≡ (mod n) ✭s❡r ❡q✉✐✈❛❧❡♥t❡ ❛♦ r❡st♦ ❞❛ ❞✐✈✐sã♦

♣♦r n✮ t❡♠ ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ♠✉✐t♦ ♣❛r❡❝✐❞♦ ❝♦♠ ❛ r❡❧❛çã♦ ❞❡ ✐❣✉❛❧❞❛❞❡ ✉s✉❛❧✱

❝♦♠ ✉♠ ♣♦✉❝♦ ♠❛✐s ❞❡ ❛t❡♥çã♦✱ ✈❡♠♦s q✉❡✱ t❛♥t♦ ❛ r❡❧❛çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❝♦♠♦ ❛ ✐❣✉❛❧❞❛❞❡✱ sã♦ r❡❧❛çõ❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛s ❡♥✈♦❧✈❡♥❞♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳

❊♠ ❣❡r❛❧✱ t❡♠♦s ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ∼ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ ❣❡♥ér✐❝♦ X

q✉❛♥❞♦✿

✭✶✮ ❋♦r r❡✢❡①✐✈❛✿ x_∼x _∀ x_∈X✳

✭✷✮ ❋♦r s✐♠étr✐❝❛✿ x∼y=⇒y∼x✳

✭✸✮ ❋♦r tr❛♥s✐t✐✈❛✿ x_∼y ❡ y_∼z =_⇒x_∼z✳

◆ã♦ é ❢♦❝♦ ❞❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛r ❞❡ ❢♦r♠❛ ♠✐♥✉❝✐♦s❛ ❛s ♣❛rt✐❝✉❧❛r✐❞❛❞❡s ❞❛ ❝♦♥str✉çã♦ ❞❡st❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ r❡s✉❧t❛♥t❡ ❞❛ r❡❧❛çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛✳ ❋✐❝❛♥❞♦ ❛ ❝❛r❣♦ ❞♦ ❧❡✐t♦r✱ ❝❛s♦ q✉❡✐r❛ ❢❛♠✐❧✐❛r✐③❛r✲s❡ ♠❡❧❤♦r ♦✉ s✐♠♣❧❡s♠❡♥t❡ r❡❧❡♠✲ ❜r❛r✱ ✈❡r ❬✽❪✱ ♣❛r❛ ✉♠ ❡st✉❞♦ ♠❛✐s ❝♦♠♣❧❡t♦✱ ❝♦♥s✉❧t❛r ❬✶❪✳

❉❡✜♥✐çã♦ ✶✳✷ ❯♠ ✐♥t❡✐r♦ ♣✲á❞✐❝♦ é ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❛ ❢♦r♠❛

[. . . , an, an−1, . . . , a2, a1]p ✭✶✳✶✮

♦♥❞❡ 06ai 6p−1 ❡ p ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳

❉❡✜♥✐çã♦ ✶✳✸ ❉❛❞♦ ✉♠ p ♣r✐♠♦✱ ✜①♦✱ ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s ✐♥t❡✐r♦s

♣✲á❞✐❝♦s s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦r Zp✳

❉❡✜♥✐çã♦ ✶✳✹ ❈❤❛♠❛r❡♠♦s ❞❡ ❛♥❡❧ ❞❡ ✐♥t❡✐r♦s ♠ó❞✉❧♦ n ♦ q✉♦❝✐❡♥t❡ ❞❡ Z ♣❡❧❛

r❡❧❛çã♦ ≡ (mod n)✳ ❉❡♥♦t❛r❡♠♦s✷ t❛❧ ❛♥❡❧ ♣♦r Z/nZ✳

P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ n = 2✱ t❡r❡♠♦s ♦ ❛♥❡❧ Z/2Z✳ ❋♦r♠❛❞♦ ♣❡❧❛s ❝❧❛ss❡s ❞❡ ❡q✉✐✲

✈❛❧ê♥❝✐❛ ❝♦rr❡s♣♦♥❞❡♥t❡s ❛ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ 2✱ ✐st♦ é✱ ♦s ❡❧❡♠❡♥t♦s ❝♦♥❣r✉❡♥t❡s

❛♦ 0 ❡ ♦s ❡❧❡♠❡♥t♦s ❝♦♥❣r✉❡♥t❡s ❛♦ 1✱ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r¯0 ❡ ¯1 r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❊♠ ❬✸❪ ♦ ❧❡✐t♦r ♣♦❞❡ ❛♣r♦❢✉♥❞❛r✲s❡ ♦✉ ❛♣❡♥❛s r❡❧❡♠❜r❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❡ r❡s✉❧t❛❞♦s ❛❝❡r❝❛ ❞♦s ❛♥é✐s q✉♦❝✐❡♥t❡s✳

P❛r❛ r❡s♦❧✈❡r♠♦s ❛s ❝♦♥❣r✉ê♥❝✐❛s✱ ❝♦♥s✐❞❡r❛r❡♠♦s ♦s ❛♥é✐sZ/pn_Z✱ ❝♦♠_n♥ú♠❡r♦

♥❛t✉r❛❧ ❡p ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✱ ♦✉ s❡❥❛✱ s❡rã♦ ♦s ❛♥é✐s ❣❡r❛❞♦s ♣❡❧❛s ♣♦tê♥❝✐❛s ❞❡ ✉♠

♥ú♠❡r♦ ♣r✐♠♦✳

✶✳✷✳✶ ❘❡s♦❧✈❡♥❞♦ ❝♦♥❣r✉ê♥❝✐❛s ♠ó❞✉❧♦

p

✳

❖ ❝❛♠✐♥❤♦ q✉❡ ❜✉s❝❛r❡♠♦s ♣❛r❛ ❞❡t❡r♠✐♥❛r ✉♠ ♥ú♠❡r♦ ♣✲á❞✐❝♦ ♣❛ss❛ ♣❡❧❛ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s✳ P♦rt❛♥t♦✱ t♦r♥❛♠✲s❡ ♥❡✲ ❝❡ssár✐❛s ❛s ❞❡✜♥✐çõ❡s q✉❡ s❡❣✉❡♠✳

❉❡✜♥✐çã♦ ✶✳✺ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❡q✉❛çõ❡s ❞❛ ❢♦r♠❛

X₋a_≡0 (mod pi₎

❝♦♠ 1≤i≤n✱ s❡rá ❞❡♥♦♠✐♥❛❞♦ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ♠ó❞✉❧♦pn_{✱ ❡ ❞❡♥♦t❛❞♦ ❞❛}

❢♦r♠❛✿

X₋a_≡0 (mod pn_).

❯t✐❧✐③❛r❡♠♦s ♦s s✐st❡♠❛s ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ♠ó❞✉❧♦ pn _{♣❛r❛ ❡♥❝♦♥tr❛r♠♦s ✐♥t❡✐r♦s}

♣✲á❞✐❝♦s✳ ◆♦ss♦ ♦❜❥❡t✐✈♦ ♥ã♦ é ❣❡♥❡r❛❧✐③❛r ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❛ r❡s♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ❞❡ss❡ t✐♣♦✳ ❱❛♠♦s s❡❣✉✐r ❛tr❛✈és ❞❡ ❡①❡♠♣❧♦s q✉❡ ♥♦s ❛♣r❡s❡♥t❡ ❛s ♣❛rt✐❝✉❧❛r✐❞❛❞❡s ❞❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳

❆♦ r❡s♦❧✈❡r♠♦s ✉♠ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s✱ ❡♥❝♦♥tr❛r❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ r❡s✉❧t❛❞♦s q✉❡ ❞❡✜♥✐r❡♠♦s ❝♦♠♦ s❡❣✉❡✳

❉❡✜♥✐çã♦ ✶✳✻ ❆ s❡q✉ê♥❝✐❛(xi)1≤i≤n ❞❛s s♦❧✉çõ❡s ❞❡ X−a≡0 (mod pn) s❡rá ❞✐t❛

❝♦❡r❡♥t❡ s❡

xi+1 ≡xi (mod pi)

♣❛r❛ t♦❞♦i✱ 1_≤i < n✳

✷_{P♦r ❝♦♥✈❡♥çã♦ ♥❡st❡ tr❛❜❛❧❤♦✱ ❞❡✐①❛r❡♠♦s ❛ ♥♦t❛çã♦ Z}

p✱ q✉❡ ❛♣r❡s❡♥t❛✲s❡ ♠❛✐s ♥❛t✉r❛❧ ♣❛r❛

♦s ❛♥é✐s q✉♦❝✐❡♥t❡s ❡♠ ♦✉tr❛s s✐t✉❛çõ❡s✱ ❞❡st✐♥❛❞❛ ❛ r❡♣r❡s❡♥t❛r ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳

P♦rt❛♥t♦✱ q✉❛♥❞♦ ❡st❛♠♦s r❡s♦❧✈❡♥❞♦ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ♠ó❞✉❧♦ pn_✱

❡st❛♠♦s ❜✉s❝❛♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❝♦❡r❡♥t❡s✳

❖✉tr♦ ❢❛t♦ ✐♠♣♦rt❛♥t❡ ❞❡ s❡ ♣❡r❝❡❜❡r é q✉❡ ❝❛❞❛ xi ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠❛

sér✐❡ ❞❡ ♣♦tê♥❝✐❛s ❞❡ t❛❧ ❢♦r♠❛ q✉❡ t❡♠♦s

x1 =a1 ·p0

x2 =a1·p0 +a2·p

x3 =a1·p0+a2·p+a3·p2

x4 =a1·p0+a2·p+a3·p2+a4·p3

✳✳✳

xn=a1·p0+a2·p+a3 ·p2+a4·p3+· · ·+an·pn.

P❛r❛ ❡s❝r❡✈❡r♠♦s ♥♦ss♦ ✐♥t❡✐r♦ ♣✲á❞✐❝♦✱ ❝♦♥s✐❞❡r❛r❡♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s

a1, a2, a3, . . .

❞❛ sér✐❡ ❞❡ ♣♦tê♥❝✐❛s ❡♥❝♦♥tr❛❞❛✳

❊①❡♠♣❧♦ ✶✳✶ ❉❡t❡r♠✐♥❛r ✉♠ ✐♥t❡✐r♦ ✷✲á❞✐❝♦ s♦❧✉çã♦ ♣❛r❛ ❛ ❡q✉❛çã♦ x₋3 = 0✳

❙♦❧✉çã♦✿ ❘❡s♦❧✈❡♥❞♦ ❛ ❝♦♥❣r✉ê♥❝✐❛ x−3≡0 (mod2n_{)t❡♠♦s✱}

♣❛r❛ n= 1✿

x1−3≡0 (mod 2) =⇒

x1 ≡3 (mod2) =⇒

x1 ≡1 (mod 2).

❚❡♠♦s ♥♦ss❛ ♣r✐♠❡✐r❛ s♦❧✉çã♦✱ ♣❛r❛n = 1✱ x1 = 1✳

P❛r❛ n= 2✱ ✜❝❛♠♦s ❝♦♠✿

x2−3≡0 (mod 22)

x2 ≡3 (mod 22)

❖❜s❡r✈❡ q✉❡ ❛ ♣❛rt✐r ❞♦x2✱ ❛s s♦❧✉çõ❡s s❡ r❡♣❡t❡♠✱ ♣♦✐s 3<2n❛ ♣❛rt✐r ❞❡n= 2✱

❧♦❣♦✱ t❡r❡♠♦s ❝♦♠♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦

s= (1,3),

q✉❡ é ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦❡r❡♥t❡✳ ❖ r❡♣r❡s❡♥t❛♥t❡ ✷✲á❞✐❝♦ s♦❧✉çã♦ ❞❡st❛ ❡q✉❛çã♦ ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞♦ ♦❜s❡r✈❛♥❞♦✲s❡ ❛ s♦♠❛ ❞❡ ♣♦tê♥❝✐❛s ❡ t♦♠❛♥❞♦ ❛♣❡♥❛s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❝❛❞❛ t❡r♠♦✱ ♥❡st❡ ❝❛s♦ ✜❝❛♠♦s ❝♦♠✿

x2 = 1·20 + 1·21.

❆ss✐♠✱ ❡♥❝♦♥tr❛♠♦s

[1,1]2.

❱❡❥❛♠♦s q✉❡ x−3 = 0 ♣♦ss✉✐ s♦❧✉çã♦ ♥♦s ✐♥t❡✐r♦s✱ ❛ s❛❜❡r x = 3 ❡✱ ♥ã♦ ♣♦r ❝♦✐♥✲

❝✐❞ê♥❝✐❛✱ ♦ r❡♣r❡s❡♥t❛♥t❡ ✷✲á❞✐❝♦ ❞❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ é ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❜✐♥ár✐♦ ❞♦3✳

❊♥❝♦♥tr❛r r❡♣r❡s❡♥t❛♥t❡s ✷✲á❞✐❝♦s ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s é ❜✉s❝❛r ♦ r❡✲ ♣r❡s❡♥t❛♥t❡ ❜✐♥ár✐♦ ❞❡st❡ ♥ú♠❡r♦✳ ❆ ú♥✐❝❛ ❞✐❢❡r❡♥ç❛ s❡rá ❛ r❡♣r❡s❡♥t❛çã♦✳

❊①❡♠♣❧♦ ✶✳✷ ❉❡t❡r♠✐♥❛r ♦ r❡♣r❡s❡♥t❛♥t❡ ✺✲á❞✐❝♦ ❞♦ ✹✸✻✳

❙♦❧✉çã♦✿ P♦❞❡♠♦s ❜✉s❝❛r ✉♠❛ ❡q✉❛çã♦ t❛❧ q✉❡ s✉❛ s♦❧✉çã♦ s❡❥❛ ♦ ✹✸✻✳ ▼❛s ❝♦♠♦ ✈✐♠♦s ♥♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r q✉❡ ♣❛r❛ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ♦ ♣r♦❝❡ss♦ é ✐❞ê♥t✐❝♦ ❛♦ ♠ét♦❞♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ♠✉❞❛♥ç❛ ❞❡ ❜❛s❡✱ ✈❛♠♦s ❜✉s❝❛r ❡s❝r❡✈❡r ♦ ✹✸✻ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ ♣♦tê♥❝✐❛s✳ ❖ r❡♣r❡s❡♥t❛♥t❡ ❞♦ 436 ♥♦s ✺✲á❞✐❝♦s s❡rá [3,1,2,1]5✱

♣♦✐s t❡♠♦s q✉❡ 436 ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ ♣♦tê♥❝✐❛s ❞❡ ❜❛s❡ 5 ❞❛

s❡❣✉✐♥t❡ ❢♦r♠❛✿

436 = 1_·50+ 2_·51+ 1_·52+ 3_·53.

❱❡r❡♠♦s ♠❛✐s ❛ ❢r❡♥t❡✱ q✉❡ ♣❛r❛ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♦ ♣r♦❝❡ss♦ s❡♠♣r❡ s❡rá ✜♥✐t♦✳

❊①❡♠♣❧♦ ✶✳✸ ❯t✐❧✐③❛♥❞♦ ❝♦♥❣r✉ê♥❝✐❛s✱ ❞❡t❡r♠✐♥❛r ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ❛ ❡q✉❛çã♦ ♣r♦♣♦st❛ ♥♦ ✐♥í❝✐♦ ❞❡st❡ ❝❛♣ít✉❧♦ ✭x= 1 + 2x✮✱ ♣❛r❛ p= 2✳

❙♦❧✉çã♦✿ ❚♦♠❡♠♦s ♦ s✐st❡♠❛ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ♣❛r❛ ❛ ❝♦♥❣r✉ê♥❝✐❛

x≡1 + 2x (mod 2n).

■♥✐❝✐❛❧♠❡♥t❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✱ ❛♣ós s✐♠♣❧❡s ♠❛♥✐♣✉❧❛çõ❡s✿

xn≡ −1 (mod 2n),

q✉❡✱ ♣❛r❛ ♠❡❧❤♦r ✈✐s✉❛❧✐③❛çã♦ ❞♦ r❡s✉❧t❛❞♦✱ ✈❛♠♦s ❛♥❛❧✐s❛r ❛❧❣✉♥s ♣❛ss♦s✳ P❛r❛

n= 1✱ t❡r❡♠♦s✿

x1 ≡ −1 (mod 2) =⇒x1 ≡1 (mod2),

♣❛r❛ n= 2✿

x2 ≡ −1 (mod 22) =⇒x2 ≡3 (mod 4),

n= 3✿

x3 ≡ −1 (mod 23) =⇒x3 ≡7 (mod 8).

❚❡r❡♠♦s ❝♦♠♦ s♦❧✉çã♦✱ ❛ s❡q✉ê♥❝✐❛✿

(1,3,7,15,31,63, . . .)

❖❜s❡r✈❡ q✉❡ ❝❛❞❛ t❡r♠♦✱ ❡①❝❡t♦ ♦ ♣r✐♠❡✐r♦✱ é ❝♦❡r❡♥t❡ ❝♦♠ s❡✉ ❛♥t❡❝❡ss♦r✳ ❆❧é♠ ❞✐ss♦✱ ❝❛❞❛ xi ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ ♣♦tê♥❝✐❛s ❞❡ 2✳

x1 = 1 = 1·20

x2 = 3 = 1·20+ 1·21

x3 = 7 = 1·20 + 1·21+ 1·22

❘❡❛❧✐③❛♥❞♦ ❡st❡ ♣r♦❝❡ss♦ n ✈❡③❡s✱ t❡r❡♠♦s

xn = 1·20+ 1·21+ 1·22+ 1·23 +· · ·+ 1·2n

P♦rt❛♥t♦✱ ♦ ✐♥t❡✐r♦ ✷✲á❞✐❝♦ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ x= 1 + 2xs❡rá✿

[. . . ,1,1,1,1,1]2

P♦❞❡♠♦s ✈❡r q✉❡ ❛ ❡q✉❛çã♦ r❡s♦❧✈✐❞❛ ❛♥t❡r✐♦r♠❡♥t❡ ♣♦ss✉✐ ♦ −1 ❝♦♠♦ s♦❧✉çã♦

❡ ♠❛✐s ❛ ❢r❡♥t❡ ✈❡r❡♠♦s q✉❡ ♦ r❡s✉❧t❛❞♦ ❡♥❝♦♥tr❛❞♦ é ♦ r❡♣r❡s❡♥t❛♥t❡ ♣✲á❞✐❝♦ ❞♦ −1✳ ❱❡r❡♠♦s t❛♠❜é♠✱ ❛♣ós ❛ ❞❡✜♥✐çã♦ ❞❛ ❛❞✐çã♦ ❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♥♦s ✐♥t❡✐r♦s

♣✲á❞✐❝♦s✱ q✉❡ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ ♥❡❣❛t✐✈♦✱ s✉❛ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ s❡rá ✐♥✜♥✐t❛✳

✶✳✷✳✷ ❆❞✐çã♦ ❡♠

Z

✳

❙❡❥❛♠ a ❡ b ❞♦✐s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s ♣❛r❛ ✉♠ ❞❛❞♦ p ♣r✐♠♦ ✜①♦✳ ❱✐♠♦s q✉❡ ♣❛r❛

❝♦♥str✉✐r♠♦s ✉♠ r❡♣r❡s❡♥t❛♥t❡ ♣✲á❞✐❝♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ t❡r❡♠♦s ✉♠❛ s♦♠❛ ❞❡ ♣♦✲ tê♥❝✐❛s ❞❡p✳ P♦rt❛♥t♦✱ ♣❛r❛ ❝❛❞❛ ♣✲á❞✐❝♦ ❞✐st✐♥t♦✱ t❡r❡♠♦s ✉♠❛ sér✐❡ ❞❡ ♣♦tê♥❝✐❛s

❛ss♦❝✐❛❞❛✳ ❈♦♥s✐❞❡r❡♠♦s ❡st❛s sér✐❡s ❛ss♦❝✐❛❞❛s✳ ❙❡❥❛♠✿

a=X

i>0

aipi−1 ❡ b = X

i>0

bipi−1

❆ ❛❞✐çã♦ a+b s❡rá ✉♠ ♥♦✈♦ ✐♥t❡✐r♦ ♣✲á❞✐❝♦c✱

c=X

i>0

cipi−1

t❛❧ q✉❡ ck = ak+bk s❡ ak+bk ≤ p−1✱ ❝❛s♦ ❝♦♥trár✐♦✱ ck = ak+bk −p ❡ ✧tr❛♥s✲

♣♦rt❛♠♦s✧ +1 ♣❛r❛ ♦ ♣ró①✐♠♦ ❝♦❡✜❝✐❡♥t❡✱ ♥♦ ❝❛s♦ ak+1✳ ❆ss✐♠ ♣r♦ss❡❣✉✐♠♦s ❛té

♦ an +bn✳ ❋❛③❡♥❞♦ ✉♠❛ ❛♥❛❧♦❣✐❛ ❝♦♠ ♦ ❛❧❣♦r✐t♠♦ ❞❛ s♦♠❛ q✉❡ ✉t✐❧✐③❛♠♦s ✉s✉❛❧✲

♠❡♥t❡ ❡ ❝♦♥s✐❞❡r❛♥❞♦ ❛ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ❞♦s ♥ú♠❡r♦s✱ ♦❜s❡r✈❡♠♦s ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✳

❊①❡♠♣❧♦ ✶✳✹ ❉❡t❡r♠✐♥❛r ❛ s♦♠❛ ❞❡ [2,0,1,1,2,0]3 + [1,0,1,0,0,2]3✳

❙♦❧✉çã♦✿ P♦❞❡♠♦s r❡s♦❧✈❡r ❛ s♦♠❛[2,0,1,1,2,0]3+ [1,0,1,0,0,2]3 ❞❛ s❡❣✉✐♥t❡

❢♦r♠❛✿

1

[2, 0, 1, 1, 2, 0]3

+ [1, 0, 1, 0, 0, 2]3

[1, 0, 0, 2, 1, 2, 2]3

❖❜t❡♥❞♦ ❝♦♠♦ r❡s✉❧t❛❞♦ [1,0,0,2,1,2,2]3✳

❖❜s❡r✈❡♠♦s q✉❡

528 = [2,0,1,1,2,0]3 ❡ 272 = [1,0,1,0,0,2]3

❡ q✉❡ [1,0,0,2,1,2,2]3 = 800✱ q✉❡ é ❛ s♦♠❛ 528 + 272✳

❱♦❧t❡♠♦s ❛♦ ❡①❡♠♣❧♦ ✶✳✸✳ ◆❡❧❡ ❞❡t❡r♠✐♥❛♠♦s [. . . ,1,1,1,1,1]2 ❝♦♠♦ s♦❧✉çã♦ ✷✲

á❞✐❝❛ ❞❡ x= 1 + 2x✳ ❙❛❜❡♠♦s t❛♠❜é♠ q✉❡ ♥♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✱ ♣❛r❛ q✉❛❧q✉❡r p✱

♦1 é[1]p✳ ❚♦♠❡♠♦s ♦[1]2✳ ❱❡❥❛ q✉❡ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ♦[1]2 ♣♦r[. . . ,0,0,0,1]2✳

1 1 1 1

[. . . , 1, 1, 1, 1, 1]2

+ [. . . , 0, 0, 0, 0, 1]2

[. . . , 0, 0, 0, 0, 0]2

❊♥❝♦♥tr❛♠♦s ♦[0]2✳ ❆ r❡♣r❡s❡♥t❛çã♦ ✷✲á❞✐❝❛ ♣❛r❛ ♦ ③❡r♦✳ ▼❛s ❡st❛♠♦s s♦♠❛♥❞♦

1 ❝♦♠ ❛❧❣✉♠ ✈❛❧♦r q✉❡ ♥♦s ❞á ♦ ③❡r♦ ❝♦♠♦ r❡s✉❧t❛❞♦✱ ❡♥tã♦ t❛❧ ♥ú♠❡r♦ t❡♠ q✉❡

s❡r ❛ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ❞♦ −1✳ ❈♦rr♦❜♦r❛♥❞♦ ❝♦♠ ♦ q✉❡ ❥á ♦❜s❡r✈❛♠♦s s♦❜r❡

r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ❞❡ ✐♥t❡✐r♦s ♥❡❣❛t✐✈♦s✳

✶✳✷✳✸ ▼✉❧t✐♣❧✐❝❛çã♦ ❡♠

Z

✳

❚❛❧ ❝♦♠♦ ❛ s♦♠❛✱ é ♣♦ssí✈❡❧ ❝♦♥str✉✐r ♣❛r❛ ♦ ♣r♦❞✉t♦ ✉♠ ♣r♦❝❡ss♦ ❛♥á❧♦❣♦ ❛♦ r❡❛❧✐③❛❞♦ ❝♦♠ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ◆❡st❡ ❢❛t♦✱ q✉❛♥❞♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❞♦✐s ❝♦❡✜❝✐❡♥t❡s é ♠❛✐♦r q✉❡ p−1✱ ♣r♦❝❡❞❡r❡♠♦s ❞❡ ❢♦r♠❛ s✐♠✐❧❛r ❝♦♠♦ ❝♦♠ ❛ ❛❞✐çã♦✳

❙❡❥❛♠a❡b❞♦✐s ♥ú♠❡r♦s ♣✲á❞✐❝♦s✱ ❝♦♠ai ❡bj ❞í❣✐t♦s ❞❡ss❡s ♥ú♠❡r♦s✳ ❈♦♥s✐❞❡r❡♠♦s

t❛♠❜é♠qij ❝♦♠♦ ♦ q✉♦❝✐❡♥t❡ ❞❡ai·bj ♣♦rp❡rij ❝♦♠♦ ♦ r❡st♦ ❞❡st❛ ♠❡s♠❛ ❞✐✈✐sã♦✳

◗✉❛♥❞♦ ♠✉❧t✐♣❧✐❝❛r♠♦s ❞♦✐s ❞í❣✐t♦s ❞❡ss❡s ♥ú♠❡r♦s✱ ❝❛s♦ ♦ ♣r♦❞✉t♦ s❡❥❛ ♠❡♥♦r q✉❡

p✱ ❝♦♥t✐♥✉❛r❡♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♥♦r♠❛❧♠❡♥t❡✳ ❈❛s♦ s❡❥❛ ✐❣✉❛❧ ♦✉ ♠❛✐♦r q✉❡ p✱

❞✐✈✐❞✐r❡♠♦s ♦ r❡s✉❧t❛❞♦ ♣♦rp✱ s♦♠❛♥❞♦✲s❡ ♦ q✉♦❝✐❡♥t❡ ❛♦ ♣ró①✐♠♦ ❞í❣✐t♦ ❛ ❡sq✉❡r❞❛✱

✧❞❡s❝❡♥❞♦✧ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦✳ ❖❜s❡r✈❡ ♦s ❡①❡♠♣❧♦s q✉❡ s❡❣✉❡♠✿

❊①❡♠♣❧♦ ✶✳✺ ❈♦♥s✐❞❡r❡♠♦s n1 = 27✱ n2 = 8 ❡ p= 5✳ ❈❛❧❝✉❧❛r n1·n2✳

❙♦❧✉çã♦✿ ❚❡♠♦s q✉❡ 27 = [1,0,2]5 ❡ 8 = [1,3]5✳ ❖ ❛❧❣♦r✐t♠♦ ♣❛r❛ ♦ ♣r♦❞✉t♦

[1,0,2]5·[1,3]5 s❡r✐❛ ♦ s❡❣✉✐♥t❡✿

1 1 0 2

× 1 3

3 1 1 + 1 0 2

1 3 3 1

❈♦♠ ♦ ❛❧❣♦r✐t♠♦ t❡♠♦s q✉❡ [1,0,2]5 ·[1,3]5 = [1,3,3,1]5 = 216 = 27·8✳

❊①❡♠♣❧♦ ✶✳✻ ❱❛♠♦s ❝♦♥s✐❞❡r❛r ♦s r❡♣r❡s❡♥t❛♥t❡s ✼✲á❞✐❝♦s ♣❛r❛ ♦s ♥ú♠❡r♦s 741 ❡ 369✳ ❈❛❧❝✉❧❛r ♦ ♣r♦❞✉t♦ ❞❡st❡s r❡♣r❡s❡♥t❛♥t❡s✱ ♥♦s ✼✲á❞✐❝♦s✳

❙♦❧✉çã♦✿ ■♥✐❝✐❛❧♠❡♥t❡ ♦❜s❡r✈❡♠♦s q✉❡ 369 = [1,0,3,5]7 ❡ 741 = [2,1,0,6]7✳

❈♦♥str✉✐♥❞♦✱ ♣❛ss♦ ❛ ♣❛ss♦✱ t❡r❡♠♦s ✉♠ ❛❧❣♦r✐t♠♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

3 4

1 0 3 5

× 2 1 0 6

6 3 1 2 0 0 0 0

❱❛❧❡ ♦❜s❡r✈❛r ♥❡st❡ ♣♦♥t♦ q✉❡✿ ❛♦ ♠✉❧t✐♣❧✐❝❛r♠♦s ♦ 6 ♣♦r 5✱ q✉❡ ♥♦s ❞á 30 ❝♦♠♦

r❡s✉❧t❛❞♦✱ ❞✐✈✐❞✐♠♦s ♣♦r 7✱ ♦ 2 q✉❡ é s❡✉ r❡st♦✱ ✧❞❡s❝❡✧ ❡ ♦ 4✱ q✉❡ é ♦ q✉♦❝✐❡♥t❡✱

✧s♦❜❡✧ s♦❜r❡ ♦ ♣ró①✐♠♦ t❡r♠♦ ❞❛ ❡sq✉❡r❞❛✳ ❈♦♥t✐♥✉❛♥❞♦✱ t❡♠♦s

1

✁3 ✁4

1 0 3 5

× 2 1 0 6

6 3 1 2 0 0 0 0 1 0 3 5 + 2 1 0 3

2 2 1 6 1 1 2

❊♥❝♦♥tr❛♠♦s ❛ss✐♠ ♦ ♥ú♠❡r♦ [2,2,1,6,1,1,2]7 q✉❡ é ♦ r❡♣r❡s❡♥t❛♥t❡ ♣✲á❞✐❝♦ ♣❛r❛

273429✭r❡s✉❧t❛❞♦ ❞♦ ♣r♦❞✉t♦ ❞❡ 369 ♣♦r 741✮✳

✶✳✷✳✹ ❘❡♣r❡s❡♥t❛♥t❡s ♣✲á❞✐❝♦s ❞❡ ✐♥t❡✐r♦s ♥❡❣❛t✐✈♦s✳

❈❤❡❣❛♠♦s ❛ ❝♦♥❝❧✉sã♦ ❛♥t❡r✐♦r♠❡♥t❡ q✉❡ ♦ r❡♣r❡s❡♥t❛♥t❡ ♣✲á❞✐❝♦ ❞♦−1♣♦ss✉í❛

r❡♣r❡s❡♥t❛çã♦ ✐♥✜♥✐t❛✳✸ _{❆❣♦r❛ q✉❡ t❡♠♦s ❞❡✜♥✐❞♦ ♦s ❝♦♥❝❡✐t♦s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐✲}

♣❧✐❝❛çã♦ ❞❡ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✱ ♣♦❞❡r❡♠♦s ❛✜r♠❛r q✉❡ ❛ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❡❣❛t✐✈♦ s❡rá s❡♠♣r❡ ✐♥✜♥✐t❛✳

Pr♦♣♦s✐çã♦ ✶✳✶ ❙❡❥❛ k_∈Z+_{✱ ❡♥tã♦ ❛ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ❞❡ −k é ✐♥✜♥✐t❛✳}

❉❡♠♦♥str❛çã♦✿ ❏á ✈✐♠♦s q✉❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡stá ❞❡✜♥✐❞❛ ♥♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s ❡ q✉❡ ♣❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ♦ ✐♥✈❡rs♦ ❛❞✐t✐✈♦ ❞❡ q✉❛❧q✉❡r ♥ú♠❡r♦ ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r♠♦s ❡❧❡ ♣♦r−1✳ ❙❡ ❛ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ❞♦ ₋1 ❢♦r ✐♥✜♥✐t❛ ♣❛r❛ t♦❞♦p ♣r✐♠♦✱ t❡♠♦s

q✉❡ ❛ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ❞❡ q✉❛❧q✉❡r ✐♥t❡✐r♦ ♥❡❣❛t✐✈♦ s❡rá ✐♥✜♥✐t❛✳

P♦rt❛♥t♦✱ q✉❡r❡♠♦s ♠♦str❛r q✉❡−1♣♦ss✉✐ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ✐♥✜♥✐t❛ ♣❛r❛ q✉❛❧✲

q✉❡r p ♣r✐♠♦✳ P❛r❛ ✐ss♦✱ ❜✉s❝❛r❡♠♦s ♦ r❡♣r❡s❡♥t❛♥t❡ ♣✲á❞✐❝♦ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❛

❡q✉❛çã♦x+ 1 = 0✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛sx+ 1 ≡0 (mod pn_)✳

✸_{❋❛t♦ ♦❜s❡r✈❛❞♦ ♥❛ s❡çã♦ ✶✳✷✳✷✳}

❚❡r❡♠♦s

x1 ≡ −1 (mod p)→x1 =p−1

x2 ≡ −1 (mod p2)→x2 =p2−1

x3 ≡ −1 (mod p3)→x3 =p3−1

x4 ≡ −1 (mod p4)→x4 =p4−1

✳✳✳ ❋♦r♠❛♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s

(p₋1, p2_{−1, p}3 _{−1, p}4_{−1, . . .).}

➱ s✐♠♣❧❡s ♣❡r❝❡❜❡r q✉❡ t❛❧ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s é ❝♦❡r❡♥t❡✱ ♣♦✐s ❝❛❞❛ t❡r♠♦ xi

é ❝♦♥❣r✉❡♥t❡ ❛ xi−1 ♠ó❞✉❧♦ pi−1✳ P♦❞❡♠♦s ❛ss✐♠✱ ❡s❝r❡✈❡r ❝♦♠♦ ✉♠❛ sér✐❡ ❞❡

♣♦tê♥❝✐❛s ❞❛ ❢♦r♠❛✿

α= (p−1)·p0+ (p−1)·p1+ (p−1)·p2+ (p−1)·p3+· · · .

❈❤❡❣❛♥❞♦ ❛ss✐♠ ❛♦ r❡♣r❡s❡♥t❛♥t❡ ♣✲á❞✐❝♦ ❞♦−1✱ q✉❡ t❡rá ❛ s❡❣✉✐♥t❡ ❡str✉t✉r❛✿

[. . . ,(p₋1),(p₋1),(p₋1),(p₋1)]p, ∀p ♣r✐♠♦✳ ✭✶✳✷✮

❙❡♠♣r❡ q✉❡ k ∈ Z+_{✱ ♣❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ❛ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ❞♦ −k✱ ❜❛st❛}

♠✉❧t✐♣❧✐❝❛r♠♦s ❛ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ❞♦k ♣❡❧❛ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ❞♦ ₋1✱ q✉❡

s❡rá s❡♠♣r❡ ❞❛ ❢♦r♠❛ ✶✳✷✳

❙❡❥❛k = [an, an−1, . . . , a2, a1]p✱ ❢❛r❡♠♦s

−k = [. . . ,(p₋1),(p₋1),(p₋1),(p₋1)]p·[an, an−1, . . . , a2, a1]p.

❆ r❡♣r❡s❡♥t❛çã♦ ❞♦−k s❡rá s❡♠♣r❡ ✐♥✜♥✐t❛✱ ♣♦✐s ❛ r❡♣r❡s❡♥t❛çã♦ ❞♦−1s❡♠♣r❡ s❡rá

✐♥✜♥✐t❛✳

P♦rt❛♥t♦ ♣♦❞❡♠♦s ♥♦s ❢❛③❡r ❛ s❡❣✉✐♥t❡ ♣❡r❣✉♥t❛✿ ◗✉❛❧ ❛ ❡str✉t✉r❛ ❛❧❣é❜r✐❝❛ ❞❡

Zp❄ ❆s ✐♠♣❧✐❝❛çõ❡s s❡❣✉✐♥t❡s ♥♦s ❞❛rã♦ ✉♠❛ ✈✐sã♦ ♠❛✐s ❛♠♣❧❛ ♣❛r❛ r❡s♣♦♥❞❡r♠♦s

❡st❛ ♣❡r❣✉♥t❛✳

✶✳✷✳✺

(

Z

,

+)

é ✉♠ ●r✉♣♦ ❆❜❡❧✐❛♥♦✳

P♦r ❛♥❛❧♦❣✐❛ ❝♦♠ ♦s ♥ú♠❡r♦s r❡❛✐s✱ é ❡✈✐❞❡♥t❡ q✉❡ ❛ ❛❞✐çã♦ ❞❡ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s é ❢❡❝❤❛❞❛✱ ❛ss♦❝✐❛t✐✈❛ ❡ ❝♦♠✉t❛t✐✈❛✱ ❡ q✉❡ ♦ ❡❧❡♠❡♥t♦ ❛❞✐t✐✈♦ ♥❡✉tr♦ é ❛♣❡♥❛s ❛ sér✐❡ ❝♦♠ t♦❞♦s ♦s ❝♦❡✜❝✐❡♥t❡s ✐❣✉❛✐s ❛ ③❡r♦✳ ▼❛s ♦ q✉❡ s❛❜❡♠♦s s♦❜r❡ ♦s ✐♥✈❡rs♦s ❛❞✐t✐✈♦s❄ ❆q✉✐ q✉❡r❡♠♦s ❞❡❝✐❞✐r ♦ s❡❣✉✐♥t❡✿ ❞❛❞♦ ✉♠ α ✐♥t❡✐r♦ ♣✲á❞✐❝♦✱ q✉❡ ❢♦r♠❛

t❡rá ♦−α❄

◆♦t❡♠♦s q✉❡ q✉❛❧q✉❡r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ t❡♠ r❡♣r❡s❡♥t❛çã♦ ♣✲á❞✐❝❛ ✜♥✐t❛✱ ♣♦✐s ♣❛r❛ ❝❛❞❛x♥ú♠❡r♦ ✐♥t❡✐r♦ é ♣♦ssí✈❡❧ t♦♠❛r♠♦s ✉♠ns✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ♣❛r❛ t❡r♠♦s

pn _{> x✳ ❚♦r♥❛♥❞♦✲♦✱ ♣♦r ❝♦♥str✉çã♦✱ ✜♥✐t♦✳}

❙❡❥❛[1]p ❛ ✉♥✐❞❛❞❡✳ ❙❡❥❛♠ t❛♠❜é♠

α= [. . . ,0,0,0, an, an−1, . . . , a3, a2, a1]p

✉♠ ♣✲á❞✐❝♦ ✜♥✐t♦✱ ❡

−α = [. . . , bn+3, bn+2, bn+1, bn, bn−1, bn−2, . . . , b3, b2, b1]p

♦ s❡✉ ✐♥✈❡rs♦ ❛❞✐t✐✈♦✳

❖❜s❡r✈❡ q✉❡ t❡♠♦sα+ (−α) = 0✱ ❛ss✐♠ ❛♦ s♦♠❛r♠♦s t❡r♠♦ ❛ t❡r♠♦ ❡♥❝♦♥tr❛r❡✲

♠♦s ♦ ♥ú♠❡r♦ ♣✲á❞✐❝♦ β = [_{· · ·},0,0,0,_{· · ·} ,0,0,0]✱ ♦❜s❡r✈❡ q✉❡ ♣❛r❛ ✐ss♦ ❛❝♦♥t❡❝❡r

❜❛st❛ q✉❡ ❛ s♦♠❛ ❞♦s t❡r♠♦s s❡❥❛ ✐❣✉❛❧ ❛p✱ ♣♦✐sp= 0 ❡♠ Zp✳

❆ss✐♠✱ t❡♠♦s✿

b1+a1 =p⇒b1 =p−a1 ⇒b1 = (p−1)−a1 + [1]p

❖❜s❡r✈❡ q✉❡✱ ❛ss✐♠ ❝♦♠♦ ♥❛ ❜❛s❡ ✶✵✱ q✉❛♥❞♦ ❛ s♦♠❛ ❡♥tr❡ ❞♦✐s t❡r♠♦s ❢♦r ✐❣✉❛❧ ♦✉ ♠❛✐♦r✱ ♥♦ ❝❛s♦ ❛ p✱ ❞❡✈❡✲s❡ ❛❝r❡s❝❡♥t❛r ✉♠❛ ✭♦✉ ♠❛✐s✮ ✉♥✐❞❛❞❡✭s✮ ❛ s♦♠❛ ❞♦s

t❡r♠♦s s❡❣✉✐♥t❡s✱ ♦✉ s❡❥❛✱

b2+a2 + 1 =p⇒b2 = (p−1)−a2

b3+a3 + 1 =p⇒b3 = (p−1)−a3

●❡♥❡r❛❧✐③❛♥❞♦ ❡ss❡ ♣r♦❝❡ss♦ ❝❤❡❣❛♠♦s ❛♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

bn+an+ 1 =p⇒bn= (p−1)−an

❱❡❥❛ t❛♠❜é♠ q✉❡✿

bn+1+ 0 + 1 = p⇒bn+1 =p−1

bn+2+ 0 + 1 = p⇒bn+2 =p−1

▲♦❣♦✱ t❡♠♦s q✉❡

bn+k+ 0 + 1 = p⇒bn+k=p−1

♣❛r❛ t♦❞♦k ≥1✳

❆ss✐♠✱ ♣♦❞❡♠♦s ❝r✐❛r ✉♠ ♠ét♦❞♦ rá♣✐❞♦ ❡ ❞✐r❡t♦ ♣❛r❛ ❡♥❝♦♥tr❛r ♦ s✐♠étr✐❝♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛ ❢♦r♠❛ ♣✲á❞✐❝❛✳

−α= [_{· · ·} , p₋1, p₋1,_{· · ·} ,(p₋1)₋αn,(p−1)−αn−1,· · · ,(p−1)−α2,(p−1)−α1] + [1]p

❆❣♦r❛✱ s❡ t❡♠♦s q✉❛❧q✉❡r ✐♥t❡✐r♦ ♣✲á❞✐❝♦ α = P

i>0αipi−1 ♣♦❞❡♠♦s ❞❡✜♥✐r s❡✉

✐♥✈❡rs♦ ❛❞✐t✐✈♦✳ ❚♦♠❡♠♦s ✉♠γ t❛❧ q✉❡

γ =X

i>0

(p−1−αi)pi−1. ✭✶✳✸✮

❊st❡ ♥ú♠❡r♦ é ❜❡♠ ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠ ✐♥t❡✐r♦ ♣✲á❞✐❝♦ ❞❡s❞❡ q✉❡0≤αi ≤p−1 =⇒

0_≤p₋1₋αi ≤p−1✳ P❡❧♦ ❛r❣✉♠❡♥t♦ ❞❡ q✉❡α+γ+1 = 0t❡♠♦s q✉❡ ♣❛r❛ q✉❛❧q✉❡r

α_∈Zt❡♠♦s ✉♠ ✐♥✈❡rs♦ ❛❞✐t✐✈♦✱ ♥♦t❛❞❛♠❡♥t❡γ+1✳ ❯♠❛ ✈❡③ q✉❡ ♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s

♣♦ss✉❡♠ ✐♥✈❡rs♦ ❛❞✐t✐✈♦✱Zp ❥✉♥t♦ ❝♦♠ ❛ ❛❞✐çã♦ ❢♦r♠❛♠ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✳ ❈♦♠♦

❝♦♥s❡q✉ê♥❝✐❛✱ ❛❣♦r❛ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ♦s ✐♥t❡✐r♦s ♥❡❣❛t✐✈♦s ❝♦♠♦ ✉♠ ♥ú♠❡r♦ ♣✲á❞✐❝♦✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ t❡♠♦sZ_⊂Zp✳

❊①❡♠♣❧♦ ✶✳✼ ❉❡t❡r♠✐♥❛r ♦ ✐♥✈❡rs♦ ❛❞✐t✐✈♦ ❞♦ ✐♥t❡✐r♦ ✺✲á❞✐❝♦ α= [1,2,0,3]5✳

❙♦❧✉çã♦✿ ❯t✐❧✐③❛♥❞♦ ✶✳✸✱ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r γ✳ ❚❡r❡♠♦s

γ = [. . . ,4,4,4,3,2,4,1]5.

❆❣♦r❛ ♥♦s ❜❛st❛ s♦♠❛r ❛ ✉♥✐❞❛❞❡✳ ❊♥❝♦♥tr❛♥❞♦

−α= [. . . ,4,4,4,3,2,4,2]5.

✶✳✷✳✻

(

Z

,

+,

·

)

é ✉♠ ❆♥❡❧✳

❚❛❧ q✉❛❧ ❛❝♦♥t❡❝❡ ❝♦♠ ❛ ❛❞✐çã♦✱ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♥♦s ✐♥t❡✐r♦s ♣✲á❞✐❝♦s é ❢❡❝❤❛❞❛✱ ❛ss♦❝✐❛t✐✈❛ ❡ ❝♦♠✉t❛t✐✈❛✱ ♣♦r ❛♥❛❧♦❣✐❛ ❝♦♠ ♦s ♥ú♠❡r♦s r❡❛✐s✳ ❆ ✐❞❡♥t✐❞❛❞❡ ❞❛ ♠✉❧✲ t✐♣❧✐❝❛çã♦ é ❥✉st❛♠❡♥t❡ ♦1✱ ♣♦✐s 1 = 1 + 0_·p+ 0_·p2_{+ 0·p}3_{+. . . ♣❛r❛ t♦❞♦ ♣r✐♠♦}

p∈Z.

✶✳✷✳✼

(

Z

,

+,

·

)

é ✉♠ ❉♦♠í♥✐♦ ❞❡ ■♥t❡❣r✐❞❛❞❡✳

❈♦♠♦ ❥á t❡♠♦s ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦✱ ♥♦s ❜❛st❛ ♠♦str❛r q✉❡ Zp ♥ã♦ ♣♦ss✉✐ ❞✐✈✐✲

s♦r❡s ❞❡ ③❡r♦✳

Pr♦♣♦s✐çã♦ ✶✳✷ P❛r❛ t♦❞♦ a ❡ b ∈ Zp✱

a_·b= 0_⇐⇒a= 0 ♦✉ b= 0.

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ a =P

i>0aipi−1✱ b = P

i>0bipi−1 ∈ Z✳ ❱❛♠♦s s✉♣♦r q✉❡

a·b = c = P

i>0cipi−1✳ ❙❡ a ❡ b sã♦ ❛♠❜♦s ❞✐❢❡r❡♥t❡s ❞❡ ③❡r♦✱ ❝❛❞❛ ✉♠ ❞❡❧❡s t❡♠

✉♠ ❝♦❡✜❝✐❡♥t❡ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ ♣♦r ❡①❡♠♣❧♦ an ♣❛r❛ a ❡ bm ♣❛r❛ b✳ P❡❧❛ ❞❡✜♥✐çã♦

❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣✲á❞✐❝❛✱ t❡♠♦s

cn+m =anbm mod p.

❉❡s❞❡ q✉❡ p ♥ã♦ ❞✐✈✐❞❛ an ♦✉ bm✱ ❡st❡ ♥ã♦ ❞✐✈✐❞❡ an ·bm =⇒ cn+m 6= 0 ❡♥tã♦

c₆= 0 ❡ ❛ss✐♠

a_·b= 0_⇐⇒a= 0 ♦✉ b= 0.

✶✳✷✳✽ ❘❛❝✐♦♥❛✐s ❝♦♠♦ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳

◆❡st❛ s❡çã♦✱ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ❛♣r❡s❡♥t❛r ♦ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ r❡♣r❡✲ s❡♥t❛♥t❡ ♣✲á❞✐❝♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✳ ❆❧é♠ ❞✐ss♦✱ ✈❡r q✉❡ ♥❡♠ s❡♠♣r❡ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r♠♦s ✉♠ r❡♣r❡s❡♥t❛♥t❡ ♣✲á❞✐❝♦ ♣❛r❛ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ❞❛❞♦ ✉♠p ✜①♦✳

❊①❡♠♣❧♦ ✶✳✽ ❉❡t❡r♠✐♥❛r ✉♠❛ s♦❧✉çã♦ ♣❛r❛2x₋3 = 0 ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ✺✲á❞✐❝♦s✳

❙♦❧✉çã♦✿ ❆♥❛❧✐s❛r❡♠♦s ❛s s♦❧✉çõ❡s ♣❛r❛ ❛ ❝♦♥❣r✉ê♥❝✐❛ 2x₋3 _≡ 0 (mod pn₎

❝♦♠ n t❡♥❞❡♥❞♦ ❛♦ ✐♥✜♥✐t♦✳ ❋✐❝❛♠♦s ❝♦♠✿

2x_≡3 (mod5) =_⇒x1 ≡4 (mod 5)

2x_≡3 (mod 52) =_⇒x2 ≡14 (mod 52)

2x_≡3 (mod 53) =_⇒x3 ≡64 (mod 53)

2x≡3 (mod 54) =⇒x4 ≡314 (mod 54)

2x_≡3 (mod 55_{) =⇒x}

5 ≡1564 (mod 55)

✳✳✳

❋✐❝❛♠♦s ❝♦♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦❡r❡♥t❡ ❞❡ s♦❧✉çõ❡s

x= (4,14,64,314,1564, . . .)

◆♦t❡ ❛✐♥❞❛ q✉❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❝❛❞❛ xi ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ ♣♦tê♥❝✐❛s✱ ✜❝❛♥❞♦

❝♦♠✿

x1 = 4 = 4·50

x2 = 14 = 4·50 + 2·51

x3 = 64 = 4·50+ 2·51+ 2·52

x4 = 314 = 4·50+ 2·51+ 2·52+ 2·53

x5 = 1564 = 4·50+ 2·51+ 2·52+ 2·53+ 2·54

❋❛③❡♥❞♦n t❡♥❞❡r ❛♦ ✐♥✜♥✐t♦✱ t❡r❡♠♦s

xn= 4·50+ 2·51+ 2·52+ 2·53+ 2·54+. . .

P♦rt❛♥t♦ ♦ ✐♥t❡✐r♦ ✺✲á❞✐❝♦✱ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ 2x₋3 = 0 s❡rá✿ [. . . ,2,2,2,2,4]5.

❚♦♠❡♠♦s ♦ r❡s✉❧t❛❞♦ ❞❡st❡ ❡①❡♠♣❧♦✱[. . . ,2,2,2,2,4]5✳ ❱❛♠♦s ♠✉❧t✐♣❧✐❝❛r ❡❧❡ ♣♦r

[2]5✱ q✉❡ é ♦ ✐♥t❡✐r♦ ♣✲á❞✐❝♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ 2✳ ❱❡❥❛♠♦s✿ 1 1 1 1

[. . . 2, 2, 2, 2, 4]5

× [. . . 0, 0, 0, 0, 2]5

[. . . 0, 0, 0, 0, 3]5

❊♥❝♦♥tr❛♠♦s ♦ [3]5 ❝♦♠♦ r❡s✉❧t❛❞♦✳ ❖❜s❡r✈❡ q✉❡ ♦ [3]5 é ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ✺✲

á❞✐❝♦ ❞♦3✳ ▼✉❧t✐♣❧✐❝❛♠♦s ✉♠ ♥ú♠❡r♦ ♣♦r2❡ ❡♥❝♦♥tr❛♠♦s ♦3✱ ❡st❡ ♥ú♠❡r♦ só ♣♦❞❡

s❡r ♦ 3

2✱ ♦ q✉❡ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ 2x−3 = 0 ♥♦s r❛❝✐♦♥❛✐s✳

Pr♦♣♦s✐çã♦ ✶✳✸ ❙❡❥❛ a

b, a, b ∈ Z, b 6= 0 ❡ (a, b) = 1✳ ❆ ❢r❛çã♦ a

b ♣♦ss✉✐ r❡♣r❡✲

s❡♥t❛çã♦ ♣✲á❞✐❝❛ s❡ ❡ s♦♠❡♥t❡ s❡ (b, p) = 1✳

❉❡♠♦♥str❛çã♦✿ P❛r❛ ❡♥❝♦♥tr❛r♠♦s ♦ r❡♣r❡s❡♥t❛♥t❡ ♣✲á❞✐❝♦ ❞❡ a

b ❝♦♥s✐❞❡r❡♠♦s

❛ ❡q✉❛çã♦ ❧✐♥❡❛r bx₋a = 0✱ ❛❧é♠ ❞✐st♦✱ ❡st❛ ❡q✉❛çã♦ t❡♠ q✉❡ ♣♦ss✉✐r s♦❧✉çõ❡s ♥♦

s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s

bx₋a_≡0 (mod pn_).

▼❛s ♦❜s❡r✈❡ q✉❡ ❡st❡ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s só ♣♦ss✉✐ s♦❧✉çã♦ s❡ ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ t✐✈❡r s♦❧✉çã♦✳ ◆♦ ❝❛s♦✱

bx1 ≡a (mod p).

❊ ❡st❛ ❡q✉❛çã♦ só ♣♦ss✉✐ s♦❧✉çã♦ q✉❛♥❞♦ b ♣♦ss✉✐ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❡♠ Z/pZ✳

■st♦ ❛❝♦♥t❡❝❡ s❡ ❡ s♦♠❡♥t❡ s❡ (b, p) = 1✳ P♦rt❛♥t♦✱ ✜❝❛ ❝❧❛r♦ q✉❡ ♣❛r❛ t❡r♠♦s ✉♠

r❡♣r❡s❡♥t❛♥t❡ ♣✲á❞✐❝♦ ♣❛r❛ a

b ♥❡❝❡ss✐t❛♠♦s q✉❡(b, p) = 1✳

❖❜s❡r✈❛♥❞♦ ❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ✜❝❛ ❢á❝✐❧ ♣❡r❝❡❜❡r q✉❡ ♦ ❞❡♥♦♠✐♥❛❞♦r ❞❛ ❢r❛✲ çã♦✱ ♥❛ s✉❛ ❢♦r♠❛ ✐rr❡❞✉tí✈❡❧✱ ♥ã♦ ♣♦❞❡ s❡r ♠ú❧t✐♣❧♦ ❞❡p✱ ✈✐st♦ q✉❡ ♦p é ✜①♦✳

✶✳✷✳✾ ■rr❛❝✐♦♥❛✐s ❛❧❣é❜r✐❝♦s ❝♦♠♦ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳

◆ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ❛❧❣é❜r✐❝♦s t❛♠❜é♠ ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ❝♦♠♦ ✐♥t❡✐r♦s ♣✲á❞✐❝♦s✳ P❛r❛ ✐st♦ s❡r ♣♦ssí✈❡❧ é ♥❡❝❡ssár✐♦ ❛❧❣✉♥s ❢❛t♦s✳ ■♥✐❝✐❛❧♠❡♥t❡ ❞❡✜♥✐r❡♠♦s ♦ q✉❡ ✈❡♠ ❛ s❡r ✉♠ r❡sí❞✉♦ q✉❛❞rát✐❝♦✳

❉❡✜♥✐çã♦ ✶✳✼ ❙❡♠♣r❡ q✉❡ ✉♠❛ ❡q✉❛çã♦ ❞❛ ❢♦r♠❛

X2 _≡d (mod p)

♣♦ss✉✐r s♦❧✉çã♦ ✭✐st♦ é✱ s❡d ❢♦r ✉♠ ✧q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✧ ❡♠ Z/pZ✮ ❞✐r❡♠♦s q✉❡ d é

✉♠ r❡sí❞✉♦ q✉❛❞rát✐❝♦ ♠ó❞✉❧♦p✳

❊①❡♠♣❧♦ ✶✳✾ ❉❡t❡r♠✐♥❛r ♦s r❡♣r❡s❡♥t❛♥t❡s ✶✶✲á❞✐❝♦s ❞❛s s♦❧✉çõ❡s ❞❡x2 _{−3 = 0✳}

❙♦❧✉çã♦✿ ❙❛❜✐❞❛♠❡♥t❡✱ x2_{−3 = 0 ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦ ❡♠ Z✳ ❯t✐❧✐③❛r❡♠♦s ✉♠}

♠ét♦❞♦ s❡♠❡❧❤❛♥t❡ ❛♦ ✉t✐❧✐③❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡ ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s ❛♦ ♠❡♥♦s ✉♠❛ ❡①♣❛♥sã♦ ♣✲á❞✐❝❛ ♣❛r❛ ❛s s♦❧✉çõ❡s ❞❡st❛ ❡q✉❛çã♦✳ ❚❡♠♦s p = 11✳ ◗✉❡r❡♠♦s ❡♥✲

❝♦♥tr❛r ✉♠❛ ❡①♣❛♥sã♦ ✶✶✲á❞✐❝❛ ♣❛r❛ ♣♦ssí✈❡✐s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ❞❛❞❛✳ P♦rt❛♥t♦ q✉❡r❡♠♦s ❛♥❛❧✐s❛r ♦ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s

x2 ≡3 (mod 11n).

P❛r❛ ♦ ❝❛s♦n= 1✱ t❡♠♦s q✉❡

x2

1 ≡3 (mod 11) ⇒x1 ≡ ±5 (mod 11)

x1 ≡5 (mod 11) ♦✉ x1 ≡6 (mod 11)

P❛r❛n = 2✱ ❢❛③❡♠♦s x= 5 + 11k ♦✉x= 6 + 11k ❡ r❡s♦❧✈❡♠♦s✿

(5 + 11k)2 ≡3 (mod 112) 25 + 110k+ 121k2₋3_≡0 (mod121)

22 + 110k≡0 (mod 121)

2 + 10k _≡0 (mod 11)

10k _≡9 (mod 11)

k ≡2 (mod 11)

❊♥tã♦✱ ❝❤❡❣❛♠♦s à

x= 5 + 11·2

x= 27_⇒X _≡27 (mod 121)

❈♦♠ ❛s s♦❧✉çõ❡s s❡♥❞♦

x2 ≡ ±27 (mod 121)

x2 ≡27 (mod 121) ♦✉ x2 ≡94 (mod 121)

P❛r❛ n = 3✱ ♣r♦❝❡❞❡r❡♠♦s ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛ ❛♥t❡r✐♦r✱ ♣♦rt❛♥t♦ ❝♦♥s✐❞❡r❡♠♦s x= 27 + 121j ♦✉x= 94 + 121j ❡✱ s✉❜st✐t✉✐♥❞♦✱ ❞❡t❡r♠✐♥❛♠♦s✿

(27 + 121j)2 _≡3 (mod 1331)

272_{+ 54·121j+ 11}4_j2 _{−3≡0 (mod1331)}

726 + 54·121j ≡0 (mod 1331)

6 + 54j _≡0 (mod11)