NÚMEROS PERFEITOS

❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲ ✲ P❘❖❋▼❆❚

◆ú♠❡r♦s P❡r❢❡✐t♦s

♣♦r

❙í✈✐♦ ❖r❧❡❛♥s ❈r✉③

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛

❏❖➹❖ P❊❙❙❖❆ ✲ P❇ ✷✵✶✸

❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲ ✲ P❘❖❋▼❆❚

◆ú♠❡r♦s P❡r❢❡✐t♦s

♣♦r

❙í✈✐♦ ❖r❧❡❛♥s ❈r✉③

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ♣❡❧❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✱ s♦❜ ♦r✐❡♥t❛çã♦ ❞♦ Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛✳

❏❖➹❖ P❊❙❙❖❆ ✲ P❇ ✷✵✶✸

❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲ ✲ P❘❖❋▼❆❚

◆ú♠❡r♦s P❡r❢❡✐t♦s

♣♦r

❙í✈✐♦ ❖r❧❡❛♥s ❈r✉③

❆ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ❛❜❛✐①♦✲❛ss✐♥❛❞❛ ❛♣r♦✈❛ ♦ ❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞♦ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞♦ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❈❡rt✐✜❝❛❞♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✭P❘❖❋▼❆❚✮ ♣❡❧❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛✳

❆♣r♦✈❛❞♦ ❡♠✿ ❞❡ ✷✵✶✸

Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮

Pr♦❢✳ ❉r✳ ▲✐③❛♥❞r♦ ❙❛♥❝❤❡③ ❈❤❛❧❧❛♣❛✲❯❋P❇

Pr♦❢✳ ❉r✳ ●✐❧❜❡rt♦ ❋❡r♥❛♥❞❡s ❱✐❡✐r❛✲❯❋❈●

t❡s❡s sã♦ ❛❜s♦❧✉t❛♠❡♥t❡ ❝❡rt❛s ❡ ✐rr❡❢✉tá✈❡✐s✱ ❛♦ ♣❛ss♦ q✉❡ ❛s ♦✉tr❛s ❝✐ê♥❝✐❛s sã♦ ❝♦♥tr♦✈❡rt✐❞❛s ❛té ❝❡rt♦ ♣♦♥t♦ ❡ s❡♠♣r❡ ❡stã♦ ❡♠ ♣❡r✐❣♦ ❞❡ s❡r❡♠ ❞❡rr✉❜❛❞❛s ♣♦r ❢❛t♦s r❡❝é♠✲❞❡s❝♦❜❡rt♦s✳ ❆ ♠❛t❡♠át✐❝❛ ❣♦③❛ ❞❡st❡ ♣r❡stí❣✐♦ ♣♦rq✉❡ é ❡❧❛ q✉❡ ❞á às ♦✉tr❛s ❝✐ê♥❝✐❛s ❝❡rt❛ ♠❡❞✐❞❛ ❞❡ s❡❣✉r❛♥ç❛ q✉❡ ❡❧❛s ♥ã♦ ♣♦❞❡r✐❛♠ ❛❧❝❛♥ç❛r s❡♠ ❛ ♠❛t❡♠át✐❝❛✑✳

✭❆❧❜❡rt ❊✐♥st❡✐♥✮

❆❣r❛❞❡❝✐♠❡♥t♦s ✈✐✐

❘❡s✉♠♦ ✈✐✐✐

❆❜str❛❝t ✐①

■♥tr♦❞✉çã♦ ①

✶ ❉✐✈✐sã♦ ❡♠ N ✶

✶✳✶ ❉✐✈✐sã♦ ❡♠N ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶

✶✳✶✳✶ Pr♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ◆ú♠❡r♦s ♣r✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷✳✶ ❙♦❜r❡ ❛ ❞✐str✐❜✉✐çã♦ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ ▼❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❞♦ ♠❞❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹ ❈♦♥❣r✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✹✳✶ ❆r✐t♠ét✐❝❛ ❞♦s r❡st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✺ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✷ ◆ú♠❡r♦s P❡r❢❡✐t♦s ✷✺

✷✳✶ Pr✐♠♦s ❞❡ ❋❡r♠❛t ❡ ❞❡ ▼❡rs❡♥♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷ ❙♦♠❛ ❞♦s ❞✐✈✐s♦r❡s ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸ ◆ú♠❡r♦s P❡r❢❡✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✹ ❆❧❣✉♥s r❡s✉❧t❛❞♦s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ♥ú♠❡r♦s ♣❡r❢❡✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✺ ◆ú♠❡r♦s ♣❛r❡s ♣❡r❢❡✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✸✳✷ ❈♦♥❥❡❝t✉r❛ ❈✉♥♥✐♥❣❤❛♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✸ ❈♦♥❥❡❝t✉r❛ ❞❡ ●♦❧❞❜❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✸✳✶ ❘❡❧❛çã♦ ❞♦s ♣r✐♠❡✐r♦s ✶✵✵✵ ♣r✐♠♦s ♣♦s✐t✐✈♦s✱ ❞❡st❛❝❛❞♦s ♦s ❣ê♠❡♦s ✺✺

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✽

❆ ❉❡✉s✱ q✉❡ ❡♠ s✉❛ ✐♥✜♥✐t❛ s❛❜❡❞♦r✐❛✱ t❡♠ ♥♦s ♠♦str❛❞♦ q✉❡ ♦s ❝❛♠✐♥❤♦s ♠❛✐s ❞✐❢í❝❡✐s sã♦ ♦s ♠❛✐s ❢ért❡✐s ♣❛r❛ ♦ ♥♦ss♦ ❛♣r✐♠♦r❛♠❡♥t♦ ❡s♣✐r✐t✉❛❧✳

❆♦s ♠❡✉s ♣❛✐s ❖r❧❡❛♥s ❡ ❖❧í✈✐❛✱ q✉❡ s❡♠♣r❡ ❞❡♣♦s✐t❛r❛♠ ❡♠ ♠✐♠✱ t♦❞❛ s✉❛ ❝♦♥✜❛♥ç❛✱ ❛❝r❡❞✐t❛♥❞♦ ♥♦ ♠❡✉ s✉❝❡ss♦✳

❆ ♠✐♥❤❛ ❡s♣♦s❛ ❙✐♠♦♥❡ ❡ ♠✐♥❤❛s ✜❧❤❛s ❚❛✇❡♥♥❡ ❡ ❚❛✇❛r❛✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦ ❡♠ t♦❞❛s ❛s ❤♦r❛s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧❛ ♠✐♥❤❛ ❛✉sê♥❝✐❛ ♥♦ ❞✐❛ ❛ ❞✐❛ ❡ ♣❡❧♦s ♠❡✉s ♠♦♠❡♥t♦s ❞❡ str❡ss✳

❆♦ Pr♦❢❡ss♦r ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛✱ q✉❡ ♠❡ ❛❝❡✐t♦✉ ❝♦♠♦ s❡✉ ♦r✐❡♥t❛♥❞♦✱ s✉❣❡r✐♥❞♦ ♦ t❡♠❛ ❞♦ ♠❡✉ tr❛❜❛❧❤♦ ❡ ❛❝r❡❞✐t❛♥❞♦ q✉❡ s❡r✐❛ ♣♦ssí✈❡❧ r❡❛❧✐③á✲❧♦✳

❆ t♦❞♦s ♦s ❛❧✉♥♦s ❞♦ ❝✉rs♦ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✭P❘❖❋▼❆❚✮✱ t✉r♠❛ ✷✵✶✶✱ ❯❋P❇ ❡ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠❡✉ ❣r❛♥❞❡ ❛♠✐❣♦ ❡ ✐r♠ã♦ ❆♠❜rós✐♦ ❊❧✐❛s q✉❡ ♥ã♦ ♠❡❞✐✉ ❡s❢♦rç♦s ❡♠ ♠❡ ❛❥✉❞❛r✱ ❝♦♥tr✐❜✉✐♥❞♦ ♣❛r❛ ♦ ♠❡✉ ❛♣r❡♥❞✐③❛❞♦ ❡ s❛♥❛♥❞♦ t♦❞❛s ❛s ♠✐♥❤❛s ❞✐✜❝✉❧❞❛❞❡s✳

❆♦s Pr♦❢❡ss♦r❡s ❞❛ ❯❋P❇✱ ♣❡❧❛s ❛✉❧❛s✱ ♣❛❝✐ê♥❝✐❛✱ ❛t❡♥çã♦✱ ❡ tr♦❝❛ ❞❡ ❡①♣❡r✐ê♥❝✐❛s✳

◆❡st❛ ❞✐ss❡rt❛çã♦ ❢❛③❡♠♦s ✉♠ ❡st✉❞♦ ❞❡ ❛❧❣✉♥s tó♣✐❝♦s ❞❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ❝♦♠♦ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s ◆ú♠❡r♦s P❡r❢❡✐t♦s ❡ Pr✐♠♦s ❞❡ ▼❡rs❡♥♥❡✳ ❆♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ♥♦ss♦ ❡st✉❞♦ ❡ ❛♥❛❧✐s❛♠♦s ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s ❞♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ ❡✈✐❞❡♥❝✐❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✈ár✐♦s ♠❛t❡♠át✐❝♦s q✉❡ ♦s ♣r♦✈❛r❛♠ s♦❜ ✈ár✐♦s ❛s♣❡❝t♦s ❧ó❣✐❝♦s✳

❊✈✐❞❡♥❝✐❛♠♦s ❛❧❣✉♥s ❛s♣❡❝t♦s ❤✐stór✐❝♦s ❡ ❝♦♥❥❡❝t✉r❛s ♣❛r❛ ♦s ♥ú♠❡r♦s ♣❡r❢❡✐t♦s✱ ❛tr❛✈és ❞❡ ✉♠❛ ♥❛rr❛t✐✈❛ s✐♠♣❧❡s ❞♦s ❢❛t♦s ❡ q✉❡ ❝❡rt❛♠❡♥t❡ ♥♦s ❞ã♦ ❛ ê♥❢❛s❡ q✉❡ ♠♦t✐✈♦✉ ❡ ♠♦t✐✈❛ ✈ár✐♦s ♠❛t❡♠át✐❝♦s ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s ♣❡r❢❡✐t♦s✳

■♥ t❤✐s t❤❡s✐s ✇❡ st✉❞② s♦♠❡ t♦♣✐❝s ♦❢ t❤❡ ❚❤❡♦r② ♦❢ ◆✉♠❜❡rs ❛s ❛♥ ✐♥s♣✐r❛t✐♦♥ ❢♦r ❢✉t✉r❡ st✉❞✐❡s ♦❢ P❡r❢❡❝t ◆✉♠❜❡rs ❛♥❞ ▼❡rs❡♥♥❡ Pr✐♠❡s✳ ❲❡ ♣r❡s❡♥t s♦♠❡ ✐♠♣♦rt❛♥t r❡s✉❧ts ❢♦r ♦✉r st✉❞② ❛♥❞ ❛♥❛❧②③❡ s♦♠❡ st❛t❡♠❡♥ts ♦❢ ❋❡r♠❛t✬s ▲✐tt❧❡ ❚❤❡♦r❡♠✱ s❤♦✇✐♥❣ t❤❡ ✈❛r✐♦✉s ♠❛t❤❡♠❛t✐❝❛❧ ❞❡♠♦♥str❛t✐♦♥s t❤❛t ♣r♦✈❡❞ ✉♥❞❡r ✈❛r✐♦✉s ❧♦❣✐❝❛❧ ❛s♣❡❝ts✳

❲❡ ❤❛✈❡ ❝❧❛r✐✜❡❞ s♦♠❡ ❤✐st♦r✐❝❛❧ ❛s♣❡❝ts ❛♥❞ ❝♦♥❥❡❝t✉r❡s ❢♦r ♣❡r❢❡❝t ♥✉♠❜❡rs✱ t❤r♦✉❣❤ ❛ s✐♠♣❧❡ ♥❛rr❛t✐✈❡ ♦❢ ❢❛❝ts ❛♥❞ t❤✐s ✇✐❧❧ ❝❡rt❛✐♥❧② ❣✐✈❡ ✉s t❤❡ ❡♠♣❤❛s✐s t❤❛t ❤❛✈❡ ♠♦t✐✈❛t❡❞ ❛♥❞ st✐❧❧ ♠♦t✐✈❛t❡s ♠❛♥② ♠❛t❤❡♠❛t✐❝✐❛♥s ❢♦r t❤❡ st✉❞② ♦❢ P❡r❢❡❝t ◆✉♠❜❡rs✳

❊st❛ ❞✐ss❡rt❛çã♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ r❡✢❡t✐r s♦❜r❡ ✉♠ t❡♠❛ ❞❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ❛ s❛❜❡r✿ ◆ú♠❡r♦s P❡r❢❡✐t♦s✱ ❣r❛♥❞❡ ♣r❡♦❝✉♣❛çã♦ ❝♦♠ ✉♠❛ ❡①♣r❡ssã♦ ♠❛t❡♠át✐❝❛ q✉❡ ♥♦s ❞❡ss❡ t♦❞♦s ♦s ◆ú♠❡r♦s P❡r❢❡✐t♦s✳ ❋♦✐ ♦ ♦❜❥❡t✐✈♦ ❞♦s ♠❛t❡♠át✐❝♦s ❞❛ ❛♥t✐❣✉✐❞❛❞❡ ❝♦♠♦ ❋❡r♠❛t✱ ❊✉❧❡r ❡ t❛♥t♦s ♦✉tr♦s✳

❆✐♥❞❛ ❤♦❥❡ ♦s ♠❛t❡♠át✐❝♦s ❡ ❡st✉❞✐♦s♦s ❡stã♦ à ♣r♦❝✉r❛ ❞❡ ♥ú♠❡r♦s ♣❡r❢❡✐t♦s ❡ ❛tr❛✈és ❞❡ ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❞❡t❡r♠✐♥❛r❛♠ ♦ ✹✼♦ _{◆ú♠❡r♦ P❡r❢❡✐t♦✳}

❆❝r❡❞✐t❛♠♦s q✉❡✱ ♦r✐❡♥t❛❞♦s ♣♦r ✉♠❛ ♣❡rs♣❡❝t✐✈❛ ❝rít✐❝♦✲r❡✢❡①✐✈❛ s♦❜r❡ ♦s ◆ú♠❡r♦s P❡r❢❡✐t♦s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ s❡❥❛ ♣♦ssí✈❡❧ r❡♣❡♥s❛r ❛s ❡str✉t✉r❛s ♠❛t❡♠át✐❝❛s ❡st✉❞❛❞❛s ♥♦s ❝✉rs♦s ❞❡ ▲✐❝❡♥❝✐❛t✉r❛ ❝♦♠♣❧❡♠❡♥t❛♥❞♦ ❛ ❢♦r♠❛çã♦ ❞♦ ❞♦❝❡♥t❡ ❛tr❛✈és ❞❛s ♥♦çõ❡s ❡s♣❡❝í✜❝❛s s♦❜r❡ ❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✱ ❞❡s♣❡rt❛♥❞♦ ♦ ❛❧✉♥❛❞♦ ♣❛r❛ ✐♥✈❡st✐❣❛çõ❡s ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣♦r ❛q✉❡❧❡s ♠❛t❡♠át✐❝♦s ♥♦ ❛s♣❡❝t♦s ❤✐stór✐❝♦ ❡ ❛t✉❛❧✳

❚r❛t❛✲s❡ ❞❡ ✉♠ tr❛❜❛❧❤♦ ❞❡ ♥❛t✉r❡③❛ q✉❛❧✐t❛t✐✈❛ ❞❡ ❝✉♥❤♦ ❡①♣❧✐❝❛t✐✈♦✱ r❡s✉❧t❛♥t❡ ❞❛ ❛♥á❧✐s❡ ❞❡ ❛rt✐❣♦s s♦❜r❡ t❡♠❛s ❞❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✱ ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❞♦s ◆ú♠❡r♦s ♣❡r❢❡✐t♦s✳ ◆♦ss♦ tr❛❜❛❧❤♦ ❢♦✐ ❞✐✈✐❞✐❞♦ ❞✐❞❛t✐❝❛♠❡♥t❡ ❡♠ três ❝❛♣ít✉❧♦s✱ ♣❛r❛ ♠❡❧❤♦r ❞✐str✐❜✉✐r ❛ ❡✈♦❧✉çã♦ ❞♦ t❡♠❛ ❡♠ ❡st✉❞♦✳

◆♦ ❝❛♣ít✉❧♦ ✶✱ r❡✈❡♠♦s ❛ ❞✐✈✐sã♦ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s✱ ❚❡♦r❡♠❛ ❞❡ ❊✉❝❧✐❞❡s✱ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✱ ❉✐✈✐s♦r❡s ❞❡ ✉♠ ◆ú♠❡r♦✱ ❙♦♠❛ ❞♦s ❉✐✈✐s♦r❡s ❞❡ ✉♠ ◆ú♠❡r♦✱ ▼❛✐♦r ❉✐✈✐s♦r ❈♦♠✉♠✱ ◆ú♠❡r♦s Pr✐♠♦s✱ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ❡ ❈♦♥❣r✉ê♥❝✐❛s✳

◆♦ ❝❛♣ít✉❧♦ ✷ ❡st✉❞❛♠♦s ♦s ◆ú♠❡r♦s Pr✐♠♦s ❞❡ ▼❡rs❡♥♥❡ ❡ s✉❛s ❝♦♥❥❡❝t✉r❛s✱ ♦s ◆ú♠❡r♦s P❡r❢❡✐t♦s ❡ s❡✉s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❡ s✉❛s ❝♦♥❥❡❝t✉r❛s✱ ◆ú♠❡r♦s P❛r❡s P❡r❢❡✐t♦s✱ ❝♦♥❥❡❝t✉r❛ s♦❜r❡ ♦s ◆ú♠❡r♦s P❡r❢❡✐t♦s ❮♠♣❛r❡s ❡ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✳

❉✐✈✐sã♦ ❡♠

N

◆❡st❡ ❝❛♣ít✉❧♦ ❡st✉❞❛r❡♠♦s ❛s ❞❡✜♥✐çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞❛ ❞✐✈✐sã♦ ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐sN=_{0,1,2, . . ._}

✶✳✶ ❉✐✈✐sã♦ ❡♠

N

❉❡✜♥✐çã♦ ✶✳✶✳ ❉❛❞♦s ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s a ❡ b ❝♦♠ a ₆= 0✱ ❞✐③❡♠♦s q✉❡ a ❞✐✈✐❞❡ b✱ ❡s❝r❡✈❡♥❞♦ a _| b✱ q✉❛♥❞♦ ❡①✐st✐r ✉♠ c _∈ N t❛❧ q✉❡ b = a_·c✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s t❛♠❜é♠ q✉❡ a é ✉♠ ❞✐✈✐s♦r ♦✉ ✉♠ ❢❛t♦r ❞❡ b ♦✉✱ ❛✐♥❞❛✱ q✉❡ b é ♠ú❧t✐♣❧♦ ❞❡ a✳ ❖ ♥ú♠❡r♦ ♥❛t✉r❛❧cé ❝❤❛♠❛❞♦ ❞❡ q✉♦❝✐❡♥t❡ ❞❡ b ♣♦r a ❡ ❞❡♥♦t❛❞♦ ♣♦rc= b

a✳

✶✳✶✳✶ Pr♦♣r✐❡❞❛❞❡s

❉❡♥♦t❛r❡♠♦s ♣♦r N∗ _{={1,2,3, . . .}}

Pr♦♣♦s✐çã♦ ✶✳✶✳ ❙❡❥❛♠a, b_∈N∗ _{❡ c∈N✳ ❊♥tã♦✱ t❡♠✲s❡ q✉❡✿}

✐✮ 1_|c, a_|a, a_|0✳

✐✐✮ ❙❡ a_|b ❡ b_|c✱ ❡♥tã♦ a _|c✳ ❉❡♠♦♥str❛çã♦✿

✐✮ ❉❡❝♦rr❡ ❞❛s ✐❣✉❛❧❞❛❞❡s✿ c= 1_·c, a=a_·1❡ a_·0 = 0

✐✐✮ ❙❡ a _| b ❡ b _| c✱ ❡♥tã♦ ❡①✐st❡♠ f, g _∈ N✱ t❛✐s q✉❡ s❡✱ b = a_·f ❡ c = b_·g✱ ❡♥tã♦ c= (a_·f)_·g =a_·(f_·g) = a_·h✱ h_∈N✱ ♦ q✉❡ ♥♦s ♠♦str❛ q✉❡a_|c✳

Pr♦♣♦s✐çã♦ ✶✳✷✳ ❙❡❥❛♠ a, b, c, d _∈ N✱ ❝♦♠ a ₆= 0 ❡ c ₆= 0✱ s❡ a _| b ❡ c _| d✱ ❡♥tã♦ a_·c_|b_·d✳

❉❡♠♦♥str❛çã♦✿ ❙❡ a_| b ❡ c_|d✱ ❡♥tã♦ ❡①✐st❡♠f, g _∈ N t❛✐s q✉❡ b =a_·f ❡ d =c_·g✳ P♦rt❛♥t♦✱ s❡b_·d= (a_·c)(f_·g)✱ ❡♥tã♦a_·c_|b_·d✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡a_|b✱ ❡♥tã♦a_·c_|b_·c ♣❛r❛ t♦❞♦c_∈N∗_✳

Pr♦♣♦s✐çã♦ ✶✳✸✳ ❙❡❥❛♠ a, b, c _∈ N✱ ❝♦♠ a ₆= 0✱ t❛✐s q✉❡ a _| (b+c)✱ ❡♥tã♦ a _| b s❡✱ ❡ s♦♠❡♥t❡ s❡ a_|c✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ a _|(b+c)✱ ❡①✐st❡ f _∈N t❛❧ q✉❡ b+c=f _·a✳ ❙❡ a _| b✱ ❡①✐st❡ g _∈ N t❛❧ q✉❡ b = a_·g ❧♦❣♦ a_·g +c = f _·a = a_·f ❞♦♥❞❡ a_·f > a_·g ❧♦❣♦ f > g✳ P♦rt❛♥t♦✱ s❡c=a_·f₋a_·g =a_·(f ₋g) ❡♥tã♦ a_|c✱ ♣♦✐s f ₋g _∈N✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱

s❡a _|c✱ ❡①✐st❡ h_∈ N t❛❧ q✉❡ c=a_·h s❡ ❡♥tã♦ b+a_·h =f _·a=a_·f s❡a_·f > a_·h ❡♥tã♦ f > h✳

P♦rt❛♥t♦✱ s❡b =a_·f₋a_·h=a_·(f ₋h)❡♥tã♦ a_|b✱ ♣♦✐s f₋h_∈N✳

Pr♦♣♦s✐çã♦ ✶✳✹✳ ❙❡❥❛♠ a, b, c_∈N✱ ❝♦♠ a₆= 0 ❡ b_≥c✱ t❛✐s q✉❡a _|(b₋c)✳ ❊♥tã♦a_|b s❡✱ ❡ s♦♠❡♥t❡ s❡ a_|c✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ a _|(b₋c)✱ ❡①✐st❡ f _∈N t❛❧ q✉❡ b₋c=f _·a✳ ❙❡ a _| b✱ ❡①✐st❡ g _∈Nt❛❧ q✉❡b =a_·g✱ ❡♥tã♦ a_·g₋c=f_·a=a.f ❧♦❣♦✱a_·g =a_·f+c❞♦♥❞❡✱a_·g > a.f ❧♦❣♦ g > f✳ P♦rt❛♥t♦✱ s❡ c = a_·g ₋a_·f = a_·(g ₋f) ❡♥tã♦ a _| c✱ ♣♦✐s g₋f _∈ N✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ a _| c✱ ❡①✐st❡ h _∈ N t❛❧ q✉❡ c =a_·h ❧♦❣♦✱ b₋a_·h =f _·a =a_·f ❞♦♥❞❡✱b =a_·h+a_·f =a_·(h+f)✱ ❡♥tã♦ a_|b✱ ♣♦✐s h+f _∈N✳

Pr♦♣♦s✐çã♦ ✶✳✺✳ ❙❡❥❛♠ a, b, c _∈ N✱ ❝♦♠ a ₆= 0 ❡ x, y _∈ N sã♦ t❛✐s q✉❡ a _| b ❡ a _| c✱ ❡♥tã♦ a_|(x_·b+y_·c)✱ ❡ s❡ x_·b_≥y_·c✱ ❡♥tã♦ a_|(x_·b₋y_·c)✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦a_|b❡a_|c✱ ❡♥tã♦ ❡①✐st❡♠f, g_∈Nt❛✐s q✉❡b=a_·f ❡c=a_·g✳ ▲♦❣♦ x_·b =x_·(a_·f) ❡ y_·c=y_·(a_·g)s❡ ❡♥tã♦ x_·b_±y_·c=x_·(a_·f)_±y_·(a_·g) =

a_·(x_·f)_±a_·(y_·g) =a_·(x_·f_±y_·g)❧♦❣♦✱ a_|(x_·f _±y_·g)✱ ♣♦✐s x_·f_±y_·g _∈N✳

Pr♦♣♦s✐çã♦ ✶✳✻✳ ❙❡❥❛♠a, b_∈N∗_{✱ t❡♠♦s q✉❡ s❡ a|b✱ ❡♥tã♦ a≤b✳}

❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ s❡a _|b✱ ❡①✐st❡ c_∈N∗ _{t❛❧ q✉❡✱ s❡ b =a·c❡♥tã♦ a≤b✱ ♣♦✐s}

c_≥1✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡a _|1✱ ❡♥tã♦ a_≤1 ❡✱ s❡❣✉❡✲s❡ q✉❡ a= 1✳

❖❜s❡r✈❛çã♦ ✶✳✶✳ ❆ r❡❧❛çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡♠N∗ _{é ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✱ ♣♦✐s✿}

✐✮ ➱ r❡✢❡①✐✈❛✿ P❛r❛ t♦❞♦a _∈N∗_{✱a |a✳ Pr♦♣♦s✐çã♦ ✶✳✶✳}

✐✐✮ ➱ tr❛♥s✐t✐✈❛✿ ❙❡ a_|b ❡b _|c✱ ❡♥tã♦ a_|c✳ Pr♦♣♦s✐çã♦ ✶✳✶✳ ✐✐✐✮ ➱ ❛♥t✐ss✐♠étr✐❝❛✿ ❙❡ a _|b ❡b _|a✱ ❡♥tã♦ a=b✳ Pr♦♣♦s✐çã♦ ✶✳✻✳

❉❡✜♥✐çã♦ ✶✳✷✳ ❙❡❥❛S ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ N✱ ❞✐③❡♠♦s q✉❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧a é ✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦ ❞❡S s❡ ♣♦ss✉✐r ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✐✮ a _∈S✱

✐✐✮ P❛r❛ t♦❞♦ n_∈S✱ a_≤n✳

❖❜s❡r✈❛çã♦ ✶✳✷✳ ❊st❡ ♠❡♥♦r ❡❧❡♠❡♥t♦ s❡ ❡①✐st❡✱ é ú♥✐❝♦ ❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r minS✳ ❉❡ ❢❛t♦✱ s❡ a ❡ a′ _{sã♦ ♦s ♠❡♥♦r❡s ❡❧❡♠❡♥t♦s ❞❡ S✱ ❡♥tã♦ a≤a}′ _❡a′ _{≤a ♥♦s ❧❡✈❛ ❛ q✉❡}

a=a′_{✳ ✭Pr♦♣r✐❡❞❛❞❡ ❛♥t✐ss✐♠étr✐❝❛ ❞❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✮✳}

Pr♦♣♦s✐çã♦ ✶✳✼ ✭Pr♦♣r✐❡❞❛❞❡ ❞❛ ❇♦❛ ❖r❞❡♠✮✳ ❚♦❞♦ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡N♣♦ss✉✐

✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦✳

❉❡♠♦♥str❛çã♦✿ ❆ ❞❡♠♦str❛çã♦ s❡rá ❢❡✐t❛ ♣♦r r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦✳

❙❡❥❛S ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡ N ❡ s✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ S ♥ã♦ ♣♦ss✉✐ ✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ S é ✈á③✐♦✱ ❝♦♥❞✉③✐♥❞♦ ❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦ T✱ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ S ❡♠ N✳ ◗✉❡r❡♠♦s✱ ♣♦rt❛♥t♦✱ ♠♦str❛r q✉❡

T =N✳

❉❡✜♥❛ ♦ ❝♦♥❥✉♥t♦

In ={k ∈N;k≤n},

❡ ❝♦♥s✐❞❡r❡ ❛ s❡♥t❡♥ç❛ ❛❜❡rt❛

p(n) :In ⊂T.

❈♦♠♦ 0_≤ n ♣❛r❛ t♦❞♦ n✱ s❡❣✉❡✲s❡ q✉❡ 0 _∈ T✱ ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ 0 s❡r✐❛ ✉♠ ♠❡♥♦r

❡❧❡♠❡♥t♦ ❞❡ S✳ ▲♦❣♦✱ p(0) é ✈❡r❞❛❞❡✳

❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡p(n) s❡❥❛ ✈❡r❞❛❞❡✳ s❡n+ 1 _∈S✱ ❝♦♠♦ ♥❡♥❤✉♠ ❡❧❡♠❡♥t♦ ❞❡In ❡stá

❡♠ S✱ t❡rí❛♠♦s q✉❡ n+ 1 é ✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦ ❞❡ S✱ ♦ q✉❡ ♥ã♦ é ♣❡r♠✐t✐❞♦✳ ▲♦❣♦✱ n+ 1_∈T✱ s❡❣✉✐♥❞♦ ❞❛í q✉❡

In+1 =In∪ {n+ 1} ⊂T,

♦ q✉❡ ♣r♦✈❛ q✉❡∀n, In ⊂T❀ ♣r♦t❛♥t♦✱ N⊂T ⊂N ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱T =N✳

❚❡♦r❡♠❛ ✶✳✶✳ ❙❡❥❛♠a ❡ b ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❝♦♠0< a < b✳ ❊①✐st❡♠ ❞♦✐s ú♥✐❝♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s q ❡ r t❛✐s q✉❡ b=a_·q+r✱ ❝♦♠ r < a✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ b > a ❡ ❝♦♥s✐❞❡r❡ ❡♥q✉❛♥t♦ ✜③❡r s❡♥t✐❞♦✱ ♦s ♥ú♠❡r♦s b, b₋a, b₋2a, . . . , b₋n_·a, . . .✳ P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❜♦❛ ♦r❞❡♠✱ ♦ ❝♦♥❥✉♥t♦S ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❛❝✐♠❛ t❡♠ ✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦r =b₋q_·a✳ ❱❛♠♦s ♣r♦✈❛r q✉❡ r < a✳ ❙❡a_|b✱ ❡♥tã♦ r= 0 ❡ ♥❛❞❛ ♠❛✐s t❡♠♦s ❛ ♣r♦✈❛r✳

❙❡a∤b✱ ❡♥tã♦r₆= 0✱ ❡ ♣♦rt❛♥t♦✱ ❜❛st❛ ♠♦str❛r q✉❡ ♥ã♦ ♣♦❞❡ ♦❝♦rr❡rr > a✳ ❉❡ ❢❛t♦✱ s❡

✐ss♦ ♦❝♦rr❡ss❡✱ ❡①✐st✐r✐❛ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧c < r t❛❧ q✉❡r =c+a✳ ❈♦♠♦ r=b₋q_·a✱ t❡♠♦s✱c+a=b₋q_·a❧♦❣♦c=b₋q_·a₋a=b₋(q+ 1)_·a_∈S❝♦♠c < r✱ ❝♦♥tr❛❞✐③❡♥❞♦ ♦ ❢❛t♦ ❞❡ s❡rr é ♦ ♠❡♥♦r ❡❧❡♠❡♥t♦ ❞❡ S✳ P♦rt❛♥t♦ b=a_·q+r ❝♦♠ r < a✱ ♣r♦✈❛♥❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ q ❡r✳

Pr♦✈❡♠♦s ❛ ✉♥✐❝✐❞❛❞❡ ❞❡q ❡r✳

❉❛❞♦s ❞♦✐s ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s ❞❡ S✱ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ♠❛✐♦r ❡ ♦ ♠❡♥♦r ❞❡ss❡s ❡❧❡♠❡♥t♦s✱ s❡♥❞♦ ✉♠ ♠ú❧t✐♣❧♦ ❞❡a✱ é ♣❡❧♦ ♠❡♥♦sa✳ ▲♦❣♦✱ s❡r=b₋a_·q ❡r′ _=b−a·q′

❝♦♠r < r′ _{< a✱ t❡rí❛♠♦sr}′_{−r=b−a·q}′_{−(b−a·q) =b−a·q}′_{−b+a·q=a(q−q}′_)✳

P♦rt❛♥t♦a_|r′_{−r❧♦❣♦a≤r}′_{−r✐♠♣❧✐❝❛♥❞♦ q✉❡r}′ _{≥a+r ≥a✭❛❜s✉r❞♦✮✳ ▲♦❣♦r}′ _=r✳

❈♦♠♦r′ _{=r s❡❣✉❡ q✉❡b−a·q=b−a·q}′ _{⇒ −a·q =−a·q}′ _{⇒a·q =a·q}′ _⇒q=q′_✳

◆❛s ❝♦♥❞✐çõ❡s ❞♦ t❡♦r❡♠❛ ❛❝✐♠❛✱ ♦s ♥ú♠❡r♦sq ❡r sã♦ ❝❤❛♠❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ q✉♦❝✐❡♥t❡ ❡ ❞❡ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❜ ♣♦r ❛✳

❈♦r♦❧ár✐♦ ✶✳✶✳ ❉❛❞♦s ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s a ❡ b ❝♦♠ 1< a_≤b✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ n t❛❧ q✉❡

na_≤b <(n+ 1)a.

❉❡♠♦♥str❛çã♦✿ P❡❧❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✱ t❡♠♦s q✉❡ ❡①✐st❡♠ q, r _∈ N ❝♦♠ r < a✱

✉♥✐✈♦❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦s✱ t❛✐s q✉❡b=a_·q+r✳ ❇❛st❛ ❛❣♦r❛ t♦♠❛rn =q

❊①❡♠♣❧♦ ✶✳✶✳ ❱❛♠♦s ♠♦str❛r ❛q✉✐ q✉❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ 10n _{♣♦r 9 é s❡♠♣r❡ 1✱}

q✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ ♥ú♠❡r♦ ♥❛t✉r❛❧n✳

■st♦ s❡rá ❢❡✐t♦ ♣♦r ✐♥❞✉çã♦✳ P❛r❛ n = 0✱ t❡♠♦s 100 _{= 9·0 + 1❀ ♣♦rt❛♥t♦✱ ♦ r❡s✉❧t❛❞♦}

✈❛❧❡✳

❙✉♣♦♥❤❛✱ ❛❣♦r❛✱ ♦ r❡s✉❧t❛❞♦ ✈á❧✐❞♦ ♣❛r❛ ✉♠ ❞❛❞♦ n✱ ✐st♦ é 10n _{= 9}

·q+ 1✳ ❈♦♥s✐❞❡r❡

❛ ✐❣✉❛❧❞❛❞❡

10n+1 = 10_·10n= (9 + 1)10n = 9_·10n+ 10n = 9_·10n+ 9_·q+ 1 = 9(10n+q) + 1, ♣r♦✈❛♥❞♦ q✉❡ ♦ r❡s✉❧t❛❞♦ ✈❛❧❡ ♣❛r❛ n+ 1 ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ✈❛❧❡ ♣❛r❛ t♦❞♦ n_∈N✳

❊①❡♠♣❧♦ ✶✳✷✳ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧n _∈N∗ _{q✉❛❧q✉❡r✱ t❡♠♦s ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿}

✐✮ ❖ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ n ♣♦r 2 é0✱ ✐st♦ é✱ ❡①✐st❡ q_∈N t❛❧ q✉❡ n= 2_·q❀ ♦✉ ✐✐✮ ❖ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ n ♣♦r 2 é1✱ ✐st♦ é✱ ❡①✐st❡ q_∈N t❛❧ q✉❡ n= 2_·q+ 1✳

P♦rt❛♥t♦✱ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s s❡ ❞✐✈✐❞❡♠ ❡♠ ❞✉❛s ❝❧❛ss❡s✱ ❛ ❞♦s ♥ú♠❡r♦s ❞❛ ❢♦r♠❛

2q ♣❛r❛ ❛❧❣✉♠ q _∈N✱ ❝❤❛♠❛♠♦s ❞❡ ♥ú♠❡r♦s ♣❛r❡s✱ ❡ ❛ ❞♦s ♥ú♠❡r♦s ❞❛ ❢♦r♠❛ 2q+ 1✱

❝❤❛♠❛❞♦s ❞❡ ♥ú♠❡r♦s í♠♣❛r❡s✳ ❖s ♥❛t✉r❛✐s sã♦ ❝❧❛ss✐✜❝❛❞♦s ❡♠ ♣❛r❡s ❡ í♠♣❛r❡s✱ ♣❡❧♦ ♠❡♥♦s✱ ❞❡s❞❡ P✐tá❣♦r❛s✱ ✺✵✵ ❛♥♦s ❛♥t❡s ❞❡ ❈r✐st♦✳

❊①❡♠♣❧♦ ✶✳✸✳ ▼❛✐s ❣❡r❛❧♠❡♥t❡✱ ✜①❛❞♦ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ m _≥ 2✱ ♣♦❞❡✲s❡ s❡♠♣r❡

❡s❝r❡✈❡r ✉♠ ♥ú♠❡r♦ q✉❛❧q✉❡rn✱ ❞❡ ♠♦❞♦ ú♥✐❝♦✱ ♥❛ ❢♦r♠❛ n =mk+r✱ ♦♥❞❡ k, r_∈ N

❡r < m✳

❊①❡♠♣❧♦ ✶✳✹✳ ❉❛❞♦sa, n_∈N∗_{✱ ❝♦♠ a >2❡ í♠♣❛r✱ ❞❡t❡r♠✐♥❛r ❛ ♣❛r✐❞❛❞❡ ❞❡} an−1

2 ✳

❈♦♠♦a é í♠♣❛r✱an _{é ✐♠♣❛r ❡ a}n

−1 é ♣❛r✱ ❡✱ ♣♦rt❛♥t♦ a

n

−1

2 é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✱

♣♦rt❛♥t♦ é ❧❡❣ít✐♠♦ ❞❡t❡r♠✐♥❛r s✉❛ ♣❛r✐❞❛❞❡✳ ❖❜s❡r✈❡ q✉❡

an₋₁

2 =

a₋1 2 ·(a

n

−1 +. . .+a+ 1)

❙❡♥❞♦ ❛ í♠♣❛r✱ t❡♠♦s q✉❡ an_{−1 +. . .+a+ 1é ♣❛r ♦✉ í♠♣❛r✱ s❡❣✉♥❞♦ n s❡❥❛ ♣❛r ♦✉}

í♠♣❛r✳ P♦rt❛♥t♦✱ ❜❛st❛ ✈❡r✐✜❝❛r ❛ ♣❛r✐❞❛❞❡ ❞❡ a−1

2 ✳

❙❡♥❞♦a í♠♣❛r✱ ❡❧❡ é ❞❛ ❢♦r♠❛ 4k+ 1 ♦✉ 4k+ 3✳ ❙❡ a= 4k+ 1✱ ❡♥tã♦

a₋1 2 =

4k+ 1₋1 2 =

4k

2 = 2k

é ♣❛r✳ ❊♥q✉❛♥t♦ q✉❡

a₋1 2 =

4k+ 3₋1 2 =

4k+ 2

2 = 2k+ 1

é í♠♣❛r✳ P♦rt❛♥t♦✱ an−1

2 é ♣❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ n é ♣❛r ♦✉ a é ❞❛ ❢♦r♠❛ 4k+ 1✳

✶✳✷ ◆ú♠❡r♦s ♣r✐♠♦s

❉❡✜♥✐çã♦ ✶✳✸✳ ❯♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♠❛✐♦r ❞♦ q✉❡1 ❡ q✉❡ só é ❞✐✈✐sí✈❡❧ ♣♦r 1 ❡ ♣♦r s✐

♣ró♣r✐♦ é ❝❤❛♠❛❞♦ ❞❡ ♥ú♠❡r♦ ♣r✐♠♦✳

❖❜s❡r✈❛çã♦ ✶✳✸✳ ❉❛❞♦s ❞♦✐s ♥ú♠❡r♦s ♣r✐♠♦sp ❡q ❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ a q✉❛❧q✉❡r✱ ♦❜s❡r✈❛♠✲s❡ ♦s s❡❣✉✐♥t❡s ❝❛s♦s✿

✐✮ ❙❡ p _| q✱ ❡♥tã♦ p =q✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ p_| q ❡ s❡♥❞♦ q ♣r✐♠♦✱ t❡♠♦s q✉❡ p = 1 ♦✉

p=q✳ ❙❡♥❞♦ p♣r✐♠♦✱ t❡♠✲s❡ q✉❡✱ s❡ p >1 ❡♥tã♦ p=q✳

✐✐✮ ❙❡ p_|a✱ ❡♥tã♦ ▼✳❉✳❈(p, a) = (p, a) = 1✳ ❉❡ ❢❛t♦✱ s❡ (p, a) = d✱ t❡♠♦s q✉❡ d_|p ❡ d _|a✱ ♣♦rt❛♥t♦d =p♦✉d= 1✳ ▼❛s d₆=p✱ ♣♦✐s p∤a ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱d= 1✳

❉❡✜♥✐çã♦ ✶✳✹✳ ❈❤❛♠❛✲s❡ ♥ú♠❡r♦ ❝♦♠♣♦st♦ ❛ t♦❞♦ ♥ú♠❡r♦ ♠❛✐♦r ❞♦ q✉❡ 1❡ q✉❡ ♥ã♦

é ♣r✐♠♦✳ P♦rt❛♥t♦✱ s❡ ✉♠ ♥ú♠❡r♦ n é ❝♦♠♣♦st♦✱ ❡①✐st✐rá ✉♠ ❞✐✈✐s♦r n1 ❞❡ n t❛❧ q✉❡

n1 6= 1 ❡ n1 6= n✳ ❆ss✐♠✱ ❡①✐st✐rá ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ n2 t❛❧ q✉❡ n = n1 ·n2✱ ❝♦♠

1< n1 < n❡ 1< n2 < n✳

❖❜s❡r✈❛çã♦ ✶✳✹✳ ❙♦❜ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛ ❡str✉t✉r❛ ♠✉❧t✐♣❧✐❝❛t✐✈❛ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ♦s ♥ú♠❡r♦s ♣r✐♠♦s sã♦ ♦s ♠❛✐s s✐♠♣❧❡s ❡ ❛♦ ♠❡s♠♦ t❡♠♣♦ sã♦ s✉✜❝✐❡♥t❡s ♣❛r❛ ❣❡r❛r t♦❞♦s ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❝♦♥❢♦r♠❡ ✈❡r❡♠♦s ♥♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✳

Pr♦♣♦s✐çã♦ ✶✳✽✳ ❙❡❥❛♠a, b, p _∈N∗_{✱ ❝♦♠ p ♣r✐♠♦✳ ❙❡ p|a·b✱ ❡♥tã♦ p|a ♦✉ p|b✳}

❉❡♠♦♥str❛çã♦✿ ❇❛st❛ ♣r♦✈❛r q✉❡✱ s❡p_|a_·b ❡ p∤a✱ ❡♥tã♦ p_| b✳ ▼❛s s❡ p∤ a✱ t❡♠♦s

q✉❡(p, a) = 1❡✱ t❡♠♦s p_|a_·b ❡(p, a) = 1 ♥♦s ❧❡✈❛ ❛p_|b✳ ✭Pr♦♣r✐❡❞❛❞❡ ❞♦ ▼✳❉✳❈✳✮ ❆①✐♦♠❛ ❞❡ ■♥❞✉çã♦

❙❡❥❛S ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ N t❛❧ q✉❡

✐✮ 0_∈S

✐✐✮ S é ❢❡❝❤❛❞♦ ❝♦♠ r❡❧❛çã♦ à ♦♣❡r❛çã♦ ❞❡ s♦♠❛r 1 ❛ s❡✉s ❡❧❡♠❡♥t♦s✱ ♦✉ s❡❥❛✱ ♣❛r❛

t♦❞♦ n_∈S ✐♠♣❧✐❝❛n+ 1_∈S✳ ❊♥tã♦✱ S =N✳

❙❡ A_⊂N ❡a _∈N✱ a+A=_{a+x;x_∈A_}❡ a+N=_{m _∈N;m _≥a_}✳

Pr♦♣♦s✐çã♦ ✶✳✾ ✭Pr✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛ ✶❛ _{❢♦r♠❛✮✳ ❙❡❥❛ a ∈N ❡ s❡❥❛ p(n)} ✉♠❛ s❡♥t❡♥ç❛ ❛❜❡rt❛ ❡♠ n✳ ❙✉♣♦♥❤❛ q✉❡

✐✮ p(a) é ✈❡r❞❛❞❡✐r❛✱ ❡ q✉❡

✐✐✮ P❛r❛ t♦❞♦n_≥a✱ p(n)✐♠♣❧✐❝❛ p(n+ 1)é ✈❡r❞❛❞❡✱ ❡♥tã♦ p(n) é ✈❡r❞❛❞❡ ♣❛r❛ t♦❞♦

n _≥a✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ V = _{n _∈ N;p(u)_}✱ ✐st♦ é✱ V é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞♦s ♥❛t✉r❛✐s ♣❛r❛ ♦s q✉❛✐s p(n)é ✈❡r❞❛❞❡✳

❈♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦S =_{m _∈N; a+m _∈V_}✱ q✉❡ tr✐✈✐❛❧♠❡♥t❡ ♥♦s ❧❡✈❛ a+S _⊂V✳ ❈♦♠♦ ♣❡❧❛ ❝♦♥❞✐çã♦ ✭✐✮✱ t❡♠♦s q✉❡ a+ 0 =a_∈V✱ s❡❣✉❡✲s❡ q✉❡ 0_∈S✳

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ m _∈ S✱ ❡♥tã♦ a+m _∈ V ❡ ♣♦r ✭✐✐✮✱ t❡♠♦s q✉❡ a+m+ 1 _∈ V✱ ❧♦❣♦ m + 1 _∈ S✳ ❆ss✐♠✱ ♣❡❧♦ ❛①✐♦♠❛ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦s S = N✳ P♦rt❛♥t♦✱

{m_∈N; m_≥a_}=a+N_⊂V ♦ q✉❡ ♣r♦✈❛ ♦ r❡s✉❧t❛❞♦✳

Pr♦♣♦s✐çã♦ ✶✳✶✵ ✭Pr✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛ ✷❛_{❢♦r♠❛✮✳ ❙❡❥❛p(n)✉♠❛ s❡t❡♥ç❛} ❛❜❡rt❛ t❛❧ q✉❡

✐✮ p(a) é ✈❡r❞❛❞❡✱ ❡ q✉❡

✐✐✮ P❛r❛ t♦❞♦ n✱ p(a) ❡ p(a+ 1) ❡ . . . p(n) ✐♠♣❧✐❝❛ p(n+ 1) ✈❡r❞❛❞❡✳

❊♥tã♦✱ p(n) é ✈❡r❞❛❞❡ ♣❛r❛ t♦❞♦ n_≥a✳

❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦V =_{n _∈a+N; p(n)_}✳ ◗✉❡r❡♠♦s ♣r♦✈❛r q✉❡

♦ ❝♦♥❥✉♥t♦W = (a+N)₋V é ✈á③✐♦✳

❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ ✈❛❧❡ ♦ ❝♦♥trár✐♦✳ ▲♦❣♦✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❜♦❛ ♦r❞❡♠✱W t❡r✐❛ ✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦k✱ ❡✱ ❝♦♠♦ s❛❜❡♠♦s ❞❡ ✭✐✮ q✉❡ a _6∈W✱ s❡❣✉❡✲s❡ q✉❡ ❡①✐st❡♠ t❛❧ q✉❡k =a+n > a✳ P♦rt❛♥t♦✱ a, a+ 1, . . . , k₋1_6∈W❀ ▲♦❣♦ a, a+ 1, . . . , k₋1_6∈V✳ P♦r ✭✐✐✮ ❝♦♥❝❧✉✐✲s❡ q✉❡k =k₋1 + 1 _∈V✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ ❢❛t♦ ❞❡ k _∈W✳

❈♦r♦❧ár✐♦ ✶✳✷✳ ❙❡ p, p1, p2, . . . , pn sã♦ ♥ú♠❡r♦s ♣r✐♠♦s ❡✱ s❡ p|p1 ·p2·. . .·pn✱ ❡♥tã♦

p=pi✱ ♣❛r❛ ❛❧❣✉♠ i= 1,2, . . . , n✳

❉❡♠♦♥str❛çã♦✿ ❯s❛♥❞♦ ❛ ✐♥❞✉çã♦ s♦❜r❡n✳ P❛r❛n = 1✱ p=p1 ♦❦✦

P❛r❛n = 2✱ s❡❥❛♠ p, p1 ❡p2 t❛✐s q✉❡ (p1, p) = 1✱(p2, p) = 1✳

❙❡p_|p1p2✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✽✱p|p1 ♦✉p|p2✱ ♠❛spé ♣r✐♠♦✳ ▲♦❣♦ p=p1 ♦✉p=p2

♣❛r❛ i= 1,2✳

❆❣♦r❛ n = k✱ ♣❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦s✱ s❡ p _| p1p2. . . pk✱ ❡♥tã♦ p = pi ♣❛r❛

i= 1,2, . . . , k✱ ♣♦✐s ❡❧❡s sã♦ ♣r✐♠♦s✳

P❛r❛ n = k + 1✱ ❝♦♥s✐❞❡r❡ ♦s ♥ú♠❡r♦s ♣r✐♠♦s p, p1, p2, . . . , pk, pk+1✳ ❙❡ p |

p1, p2, . . . , pk, pk+1✱ ❡♥tã♦ p | p1, p2, . . . , pk ♦✉ p | pk+1✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✽✱ p = pi✱

i= 1,2, . . . , k ♦✉p=pk+ 1✳

❚❡♦r❡♠❛ ✶✳✷ ✭❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✮✳ ❚♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♠❛✐♦r ❞♦ q✉❡ 1 ♦✉ é ♣r✐♠♦ ♦✉ s❡ ❡s❝r❡✈❡ ❞❡ ♠♦❞♦ ú♥✐❝♦ ✭❛ ♠❡♥♦s ❞❛ ♦r❞❡♠ ❞♦s ❢❛t♦r❡s✮ ❝♦♠♦

♣r♦❞✉t♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✳

❉❡♠♦♥str❛çã♦✿ ❯s❛r❡♠♦s ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦✳ ❙❡ n = 2✱ ♦

r❡s✉❧t❛❞♦ é ♦❜✈✐❛♠❡♥t❡ ✈❡r✐✜❝❛❞♦✳

❙✉♣♦♥❤❛♠♦s ♦ r❡s✉❧t❛❞♦ ✈á❧✐❞♦ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♠❡♥♦r ❞♦ q✉❡ n ❡ ✈❛♠♦s ♣r♦✈❛r q✉❡ ✈❛❧❡ ♣❛r❛n✳ ❙❡ ♦ ♥ú♠❡r♦né ♣r✐♠♦✱ ♥❛❞❛ t❡♠♦s ❛ ❞❡♠♦♥str❛r✳ ❙✉♣♦♥❤❛♠♦s✱ ❡♥tã♦✱ q✉❡ns❡❥❛ ❝♦♠♣♦st♦✳ ▲♦❣♦✱ ❡①✐st❡♠ ♥ú♠❡r♦s ♥❛t✉r❛✐sn1 ❡n2 t❛✐s q✉❡n =n1n2✱

❝♦♠ 1< n1 < n❡ 1 < n2 < n✳ P❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦s q✉❡ ❡①✐st❡♠ ♥ú♠❡r♦s

♣r✐♠♦s p1p2· · ·pr ❡ q1q2· · ·qs t❛✐s q✉❡ n1 = p1p2· · ·pr ❡ n2 = q1q2· · ·qs✳ P♦rt❛♥t♦✱

n=p1p2· · ·prq1q2· · ·qs✳

❱❛♠♦s✱ ❛❣♦r❛✱ ♣r♦✈❛r ❛ ✉♥✐❝✐❞❛❞❡ ❞❛ ❡s❝r✐t❛✳ ❙✉♣♦♥❤❛✱ ❛❣♦r❛✱ q✉❡n=p1·pr =q1· · ·qs✱

♦♥❞❡ ♦s pi ❡ ♦s pj sã♦ ♥ú♠❡r♦s ♣r✐♠♦s ✳ ❈♦♠♦p1 |q1q2· · ·qs P❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✷✱ t❡♠♦s

q✉❡ p1 = pj ♣❛r❛ ❛❧❣✉♠ j✱ q✉❡ ❛♣ós r❡♦r❞❡♥❛♠❡♥t♦ ❞❡ q1q2· · ·qs✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡

s❡❥❛q1✳ P♦rt❛♥t♦✱

p2· · ·pr =q2· · ·qs.

❈♦♠♦p2· · ·pr < n✱ ❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ❛❝❛rr❡t❛ q✉❡ r=s ❡ ♦spi ❡qj sã♦ ✐❣✉❛✐s ❛♦s

♣❛r❡s✳

Pr♦♣♦s✐çã♦ ✶✳✶✶✳ ❙❡❥❛ n =pα1

1 · · ·pαrr ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ❙❡ n′ é ✉♠ ❞✐✈✐s♦r ❞❡ n✱

❡♥tã♦

n′ =pβ1

1 · · ·pβrr,

♦♥❞❡ 0_≤βi ≤αi✱ ♣❛r❛ i= 1, . . . , r✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛n′ _{✉♠ ❞✐✈✐s♦r ❞❡n ❡ s❡❥❛p}β _{❛ ♣♦tê♥❝✐❛ ❞❡ ✉♠ ♣r✐♠♦pq✉❡ ✜❣✉r❛}

♥❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡n′ _{❡♠ ❢❛t♦r❡s ♣r✐♠♦s✳ ❈♦♠♦ p}β

|n✱ s❡❣✉❡ q✉❡pβ _{❞✐✈✐❞❡ ❛❧❣✉♠p}αi

i

♣♦r s❡r ♣r✐♠♦ ❝♦♠ ♦s ❞❡♠❛✐s pαj

j ✱ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ p=pi ❡ β ≤αi✳

❉❡♥♦t❛♥❞♦ ♣♦r d(n) ♦ ♥ú♠❡r♦ ❞❡ ❞✐✈✐s♦r❡s ❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ n✱ s❡❣✉❡✱ ♣♦r ✉♠❛ ❝♦♥t❛❣❡♠ ❢á❝✐❧✱ q✉❡ s❡ n = pα1

1 · · ·pαrr✱ ♦♥❞❡ p1, . . . , pr sã♦ ♥ú♠❡r♦s ♣r✐♠♦s ❡

α1, . . . , αr ∈N✱ ❡♥tã♦

d(n) = (α1+ 1)(α2+ 1)· · ·(αr+ 1).

❊①❡♠♣❧♦ ✶✳✺✳ ❆ ❢ór♠✉❧❛ ❛❝✐♠❛ ♥♦s ♠♦str❛ q✉❡ ✉♠ ♥ú♠❡r♦n=pα1

1 · · ·pαrr ♣♦ss✉✐ ✉♠❛

q✉❛♥t✐❞❛❞❡ í♠♣❛r ❞❡ ❞✐✈✐s♦r❡s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❝❛❞❛ αi é ♣❛r✱ ♦✉ s❡❥❛✱ s❡✱ ❡ s♦♠❡♥t❡

s❡✱n é ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✳

❊①❡♠♣❧♦ ✶✳✻✳ ❙❡❥❛n >4✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❝♦♠♣♦st♦❀ ✈❛♠♦s ♣r♦✈❛r q✉❡n_|(n₋2)!✳

Pr♦✈❛r❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡n_|(n₋1)!✳ ❉❡ ❢❛t♦✱ s❡❥❛ n=n1n2 ❝♦♠ n1 < n ❡n2 < n✳

❙❡n1 6=n2✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ n1 < n2✱ ❡ ♣♦rt❛♥t♦✱

(n₋1)! = 1_{· · ·}n1· · ·n2· · ·(n−1),

♦ q✉❡ ♠♦str❛ q✉❡n _|(n₋1)!✱ ♥❡st❡ ❝❛s♦✳

❙✉♣♦♥❤❛♠♦s q✉❡n1 =n2 >2✳ ▲♦❣♦✱ n =n21 >2n1❀ ❡♥tã♦

(n₋1)! = 1_{· · ·}n1· · ·2n1· · ·(n−1),

♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡n=n2

1 ❞✐✈✐❞❡ (n−1)!✳

❆❣♦r❛✱ ♥♦t❡ q✉❡(n, n₋1) = 1 ❡ q✉❡ n _|(n₋2)!(n₋1)❀ ♣♦rt❛♥t♦✱ n_|(n₋p)!

❉❡ ❢❛t♦✱ t❡♠♦s q✉❡ (n₋1, n) = 1 ❡ q✉❡ n _|(n₋2)!(n₋1)❀ P♦rt❛♥t♦✱ n _|(n₋2)!

❖❜s❡r✈❛çã♦ ✶✳✺✳ ❆ ♣r♦♣r✐❡❞❛❞❡ ❛❝✐♠❛ ♣♦❞❡ s❡r ❣❡♥❡r❛❧✐③❛❞❛ ❝♦♠♦ s❡❣✉❡✿

❙❡❥❛♠ n >4❝♦♠♣♦st♦ ❡ ♦p♦ ♠❡♥♦r ♥ú♠❡r♦ ♣r✐♠♦ q✉❡ ❞✐✈✐❞❡n ❡♥tã♦ n_|(n₋p)!✳

❉❡ ❢❛t♦✱ t❡♠♦s q✉❡ (n₋1, n) = 1, . . . ,(n₋2, n) = 1, . . . ,(n₋(p₋1), n) = 1✳ ▲♦❣♦✱

s❡❣✉❡ q✉❡((n₋1)(n₋2)_{· · ·}(n₋p+ 1), n) = 1✱ ♦ q✉❡✱ ❡♠ ✈✐st❛ ❞❡ n_|(n₋1)!✱ ♦ q✉❡

❛❝❛rr❡t❛ n_|(n₋p)!✳

✶✳✷✳✶ ❙♦❜r❡ ❛ ❞✐str✐❜✉✐çã♦ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s

◗✉❛♥t♦s s❡rã♦ ♦s ♥ú♠❡r♦s ♣r✐♠♦s❄ ❊ss❛ ♣❡r❣✉♥t❛ ❢♦✐ r❡s♣♦♥❞✐❞❛ ♣♦r ❊✉❝❧✐❞❡s ♥♦ ▲✐✈r♦ ■❳ ❞♦s ❊❧❡♠❡♥t♦s✳ ❯t✐❧✐③❛r❡♠♦s ❛ ♠❡s♠❛ ♣r♦✈❛ ❞❛❞❛ ♣♦r ❊✉❝❧✐❞❡s✱ ♦♥❞❡ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ s❡ r❡❣✐str❛ ♦ ✉s♦ ❞❡ ✉♠❛ ❞❡♠♦♥str❛çã♦ ♣♦r r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦ ❡♠ ♠❛t❡♠át✐❝❛✳ ❊ss❛ ♣r♦✈❛ é ❝♦♥s✐❞❡r❛❞❛ ✉♠❛ ❞❛s ♣ér♦❧❛s ❞❛ ♠❛t❡♠át✐❝❛✳

❚❡♦r❡♠❛ ✶✳✸✳ ❊①✐st❡♠ ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐♠♦s✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s p1, . . . , pr✳ ❈♦♥s✐❞❡r❡ ♦ ♥ú♠❡r♦ ♥❛t✉r❛❧

n =p1p2· · ·pr+ 1.

P❡❧♦ ❚❡♦r❡♠❛ ✶✳✷✱ ♦ ♥ú♠❡r♦ n ♣♦ss✉✐ ✉♠ ❢❛t♦r ♣r✐♠♦ p q✉❡✱ ♣♦rt❛♥t♦✱ ❞❡✈❡ s❡r ✉♠ ❞♦s p1, . . . , pr ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❞✐✈✐❞❡ ♦ ♣r♦❞✉t♦ p1p2· · ·pr✳ ▼❛s ✐st♦ ✐♠♣❧✐❝❛ q✉❡

p ❞✐✈✐❞❡ n = p1p2· · ·pr + 1 ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ p ❞✐✈✐❞❡ 1✱ ♦ q✉❡ é ❛❜s✉r❞♦✳ ▲♦❣♦✱

❡①✐st❡♠ ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐♠♦s✳

❆❣♦r❛ q✉❡ s❛❜❡♠♦s q✉❡ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐♠♦s✱ ♥♦s ♣❡r❣✉♥t❛♠♦s✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ❝♦♠♦ ♣♦❞❡♠♦s ♦❜t❡r ✉♠❛ ❧✐st❛ ❝♦♥t❡♥❞♦ ♦s ♥ú♠❡r♦s ♣r✐♠♦s ❛té ✉♠❛ ❞❛❞❛ ♦r❞❡♠✳ ❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ ❞♦s ♠❛✐s ❛♥t✐❣♦s ♠ét♦❞♦s ♣❛r❛ ❡❧❛❜♦r❛r t❛❜❡❧❛s ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ♦ ❝❤❛♠❛❞♦ ❈r✐✈♦ ❞❡ ❊r❛tóst❡❧❡s✱ ❞❡✈✐❞♦ ❛♦ ♠❛t❡♠át✐❝♦ ❣r❡❣♦ ❊r❛tóst❡♥❡s✱ q✉❡ ✈✐✈❡✉ ♣♦r ✈♦❧t❛ ❞❡ ✷✸✵ ❛♥♦s ❛♥t❡s ❞❡ ❈r✐st♦❀ ♣❡r♠✐t❡ ❞❡t❡r♠✐♥❛r t♦❞♦s ♦s ♥ú♠❡r♦s ♣r✐♠♦s ❛té ❛ ♦r❞❡♠ q✉❡ s❡ ❞❡s❡❥❛r✱ ♠❛s ♥ã♦ é ♠✉✐t♦ ❡✜❝✐❡♥t❡ ♣❛r❛ ♦r❞❡♥s ♠✉✐t♦ ❡❧❡✈❛❞❛s✳

P♦r ❡①❡♠♣❧♦✱ ✈❛♠♦s ❡❧❛❜♦r❛r ❛ t❛❜❡❧❛ ❞❡ t♦❞♦s ♦s ♥ú♠❡r♦s ♣r✐♠♦s ✐♥❢❡r✐♦r❡s ❛ ✶✷✵✳ ❊s❝r❡✈❡♠✲s❡ t♦❞♦s ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❞❡ ✷ ❛ ✶✷✵✳ ❘✐s❝❛♠✲s❡✱ ❞❡ ♠♦❞♦ s✐st❡♠át✐❝♦✱ t♦❞♦s ♦s ♥ú♠❡r♦s ❝♦♠♣♦st♦s ❞❛ t❛❜❡❧❛✱ s❡❣✉✐♥❞♦ ♦ r♦t❡✐r♦ ❛❜❛✐①♦✳

❘✐sq✉❡ t♦❞♦s ♦s ♠ú❧t✐♣❧♦s ❞❡ ✷ ❛❝✐♠❛ ❞❡ ✷✱ ❥á q✉❡ ♥❡♥❤✉♠ ❞❡❧❡s é ♣r✐♠♦✳

❖ s❡❣✉♥❞♦ ♥ú♠❡r♦ ♥ã♦ r✐s❝❛❞♦ é ✸✱ q✉❡ é ♣r✐♠♦✳ ❘✐sq✉❡ t♦❞♦s ♦s ♠ú❧t✐♣❧♦s ❞❡ ✸ ♠❛✐♦r❡s ❞♦ q✉❡ ✸ ♣♦✐s ❡ss❡s ♥ã♦ sã♦ ♣r✐♠♦s✳

❖ t❡r❝❡✐r♦ ♥ú♠❡r♦ ♥ã♦ r✐s❝❛❞♦ q✉❡ ❛♣❛r❡❝❡ é ✺✱ q✉❡ é ♣r✐♠♦✳ ❘✐sq✉❡ t♦❞♦s ♦s ♠ú❧t✐♣❧♦s ❞❡ ✺ ♠❛✐♦r❡s ❞♦ q✉❡ ✺ ♣♦✐s ❡ss❡s ♥ã♦ sã♦ ♣r✐♠♦s✳

❖ q✉❛rt♦ ♥ú♠❡r♦ ♥ã♦ r✐s❝❛❞♦ q✉❡ ♦r❛ ❛♣❛r❡❝❡ é ✼✱ q✉❡ é ♣r✐♠♦✳ ❘✐sq✉❡ t♦❞♦s ♦s ♠ú❧t✐♣❧♦s ❞❡ ✼ ♠❛✐♦r❡s ❞♦ q✉❡ ✼ ♣♦✐s ❡ss❡s ♥ã♦ sã♦ ♣r✐♠♦s✳

❙❡rá ♥❡❝❡ssár✐♦ ♣r♦ss❡❣✉✐r ❝♦♠ ❡st❡ ♣r♦❝❡❞✐♠❡♥t♦ ❛té ❝❤❡❣❛r ❛ ✶✷✵❄ ❆ r❡s♣♦st❛ é ♥ã♦ ❡ s❡ ❜❛s❡✐❛ ♥♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❞❡✈✐❞♦ ❛♦ ♣ró♣r✐♦ ❊r❛tóst❡♥❡s✳

▲❡♠❛ ✶✳✶✳ ❙❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ n >1 ♥ã♦ é ❞✐✈✐sí✈❡❧ ♣♦r ♥❡♥❤✉♠ ♥ú♠❡r♦ ♣r✐♠♦ p t❛❧ q✉❡ p2 _{≤n✱ ❡♥tã♦ ❡❧❡ é ♣r✐♠♦✳}

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛♠♦s✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ n ♥ã♦ s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r ♥❡♥❤✉♠ ♥ú♠❡r♦ ♣r✐♠♦ p t❛❧ q✉❡ p2 _{≤ n ❡ q✉❡ ♥ã♦ s❡❥❛ ♣r✐♠♦✳ ❙❡❥❛ q ♦ ♠❡♥♦r ♥ú♠❡r♦ ♣r✐♠♦}

q✉❡ ❞✐✈✐❞❡ n❀ ❡♥tã♦✱ n = qn1✱ ❝♦♠ q ≤ n1✳ ❙❡❣✉❡ ❞❛í q✉❡ q2 ≤ qn1 = n✳ ▲♦❣♦✱ n é

❞✐✈✐sí✈❡❧ ♣♦r ✉♠ ♥ú♠❡r♦ ♣r✐♠♦q t❛❧ q✉❡ q2 _{≤n✱ ❛❜s✉r❞♦✳}

P♦rt❛♥t♦✱ ♥❛ t❛❜❡❧❛ ❞❡ ♥ú♠❡r♦s ❞❡ ✷ ❛ ✶✷✵✱ ❞❡✈❡♠♦s ✐r ❛té ❛❧❝❛♥ç❛r♠♦s ♦ ♣r✐♠♦ ✼✱ ♣♦✐s ♦ ♣ró①✐♠♦ ♣r✐♠♦ é ✶✶✱ ❝✉❥♦ q✉❛❞r❛❞♦ s✉♣❡r❛ ✶✷✵✳

✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷

✶✸ ✶✹ ✶✺ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ✷✶ ✷✷ ✷✸ ✷✹

✷✺ ✷✻ ✷✼ ✷✽ ✷✾ ✸✵ ✸✶ ✸✷ ✸✸ ✸✹ ✸✺ ✸✻

✸✼ ✸✽ ✸✾ ✹✵ ✹✶ ✹✷ ✹✸ ✹✹ ✹✺ ✹✻ ✹✼ ✹✽

✹✾ ✺✵ ✺✶ ✺✷ ✺✸ ✺✹ ✺✺ ✺✻ ✺✼ ✺✽ ✺✾ ✻✵

✻✶ ✻✷ ✻✸ ✻✹ ✻✺ ✻✻ ✻✼ ✻✽ ✻✾ ✼✵ ✼✶ ✼✷

✼✸ ✼✹ ✼✺ ✼✻ ✼✼ ✼✽ ✼✾ ✽✵ ✽✶ ✽✷ ✽✸ ✽✹

✽✺ ✽✻ ✽✼ ✽✽ ✽✾ ✾✵ ✾✶ ✾✷ ✾✸ ✾✹ ✾✺ ✾✻

✾✼ ✾✽ ✾✾ ✶✵✵ ✶✵✶ ✶✵✷ ✶✵✸ ✶✵✹ ✶✵✺ ✶✵✻ ✶✵✼ ✶✵✽ ✶✵✾ ✶✶✵ ✶✶✶ ✶✶✷ ✶✶✸ ✶✶✹ ✶✶✺ ✶✶✻ ✶✶✼ ✶✶✽ ✶✶✾ ✶✷✵

◆♦t❡ q✉❡ ♦ ▲❡♠❛ ✶✳✶ t❛♠❜é♠ ♥♦s ❢♦r♥❡❝❡ ✉♠ t❡st❡ ❞❡ ♣r✐♠❛❧✐❞❛❞❡✱ ♣♦✐s✱ ♣❛r❛ ✈❡r✐✜❝❛r s❡ ✉♠ ❞❛❞♦ ♥ú♠❡r♦n é ♣r✐♠♦✱ ❜❛st❛ ✈❡r✐✜❝❛r q✉❡ ♥ã♦ é ❞✐✈✐sí✈❡❧ ♣♦r ♥❡♥❤✉♠ ♣r✐♠♦p q✉❡ ♥ã♦ s✉♣❡r❡ √n✳

❚❛♥t♦ ♦ ❈r✐✈♦ ❞❡ ❊r❛tóst❡♥❡s ♣❛r❛ ❣❡r❛r ♥ú♠❡r♦s ♣r✐♠♦s✱ q✉❛♥t♦ ♦ t❡st❡ ❞❡ ♣r✐♠❛❧✐❞❛❞❡ ❛❝✐♠❛ ❞❡s❝r✐t♦✱ sã♦ ❡①tr❡♠❛♠❡♥t❡ ❧❡♥t♦s ❡ tr❛❜❛❧❤♦s♦s✳ ▼✉✐t♦s ♣r♦❣r❡ss♦s tê♠ s✐❞♦ ❢❡✐t♦s ♥❡ss❛ ❞✐r❡çã♦✳

❯♠❛ q✉❡stã♦ ✐♠♣♦rt❛♥t❡ q✉❡ s❡ ❝♦❧♦❝❛ é ❞❡ ❝♦♠♦ ♦s ♥ú♠❡r♦s ♣r✐♠♦s s❡ ❞✐str✐❜✉❡♠ ❞❡♥tr♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ q✉❛❧ ♣♦❞❡ s❡r ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣r✐♠♦s ❝♦♥s❡❝✉t✐✈♦s❄ ◗✉❛❧ é ❛ s✉❛ ❢r❡q✉ê♥❝✐❛❄

❖❧❤❛♥❞♦ ♣❛r❛ t❛❜❡❧❛ ❛❝✐♠❛ ✱ ♥♦t❛✲s❡ q✉❡ ❤á ✈ár✐♦s ♣❛r❡s ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s q✉❡ ❞✐❢❡r❡♠ ❞❡ ❞✉❛s ✉♥✐❞❛❞❡s✳ ❊ss❡s sã♦✿ ✭✸✱✺✮✱ ✭✺✱✼✮✱ ✭✶✶✱✶✸✮✱ ✭✶✼✱✶✾✮✱ ✭✹✶✱✹✸✮✱ ✭✺✾✱✻✶✮✱ ✭✼✶✱✼✸✮✱ ✭✶✵✶✱ ✶✵✸✮✱ ✭✶✵✼✱ ✶✵✾✮✳

P❛r❡s ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ❝♦♠ ❡st❛ ♣r♦♣r✐❡❞❛❞❡ sã♦ ❝❤❛♠❛❞♦s ❞❡ ♣r✐♠♦s ❣ê♠❡♦s✳ ❆té ♦ ♣r❡s❡♥t❡ ♠♦♠❡♥t♦✱ ❛✐♥❞❛ ♥ã♦ s❡ s❛❜❡ s❡ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♣❛r❡s ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ❣ê♠❡♦s✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ❡♠ ❝♦♥tr❛st❡ ❝♦♠ ❡ss❡s ♣❛r❡s ❞❡ ♣r✐♠♦s ❝♦♥s❡❝✉t✐✈♦s ♠✉✐t♦ ♣ró①✐♠♦s✱ ❡①✐st❡♠ ♣r✐♠♦s ❝♦♥s❡❝✉t✐✈♦s ❛r❜✐tr❛r✐❛♠❡♥t❡ ❛❢❛st❛❞♦s✳

❉❡ ❢❛t♦✱ ❞❛❞♦n✱ ❛ s❡q✉ê♥❝✐❛

(n+ 1)! + 2,(n+ 1)! + 3, . . . ,(n+ 1)! +n+ 1

❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s é ❢♦r♠❛❞❛ ♣♦rn ♥ú♠❡r♦s ❝♦♥s❡❝✉t✐✈♦s ❝♦♠♣♦st♦s✳

P♦rt❛♥t♦✱ ❛ r❡s♣♦st❛ à ♣r✐♠❡✐r❛ ♣❡r❣✉♥t❛ é q✉❡ ♥ã♦ ❤á ♥❡♥❤✉♠ ♣❛❞rã♦ q✉❡ ❞❡s❝r❡✈❛ ♦ q✉❛♥t♦ ❞♦✐s ♣r✐♠♦s ❝♦♥s❡❝✉t✐✈♦s ❡stã♦ ❧♦♥❣❡ ✉♠ ❞♦ ♦✉tr♦✳

◗✉❛♥t♦ à s❡❣✉♥❞❛ ♣❡r❣✉♥t❛✱ é ♥❡❝❡ssár✐♦ ❢♦r♠❛❧✐③❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢r❡q✉ê♥❝✐❛ ❞❡ ♣r✐♠♦s✱ q✉❡ é ❛ ♠❡s♠❛ ❝♦✐s❛ q✉❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ❉❡♥♦t❛r❡♠♦s✱ ♣♦rπ(x)✱ ❛ q✉❛♥t✐❞❛❞❡

❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ♠❡♥♦r❡s ♦✉ ✐❣✉❛✐s ❛ x✳ P♦rt❛♥t♦✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ q✉❡ ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ {1, . . . , x_} s❡❥❛ ♣r✐♠♦ é ❞❛❞❛ ♣♦r

π(x)

x .

❈♦♠♦ ❡st❡ q✉♦❝✐❡♥t❡ é ✉♠❛ ❢✉♥çã♦ ❜❛st❛♥t❡ ❝♦♠♣❧❡①❛✱ ♦ q✉❡ s❡ ❣♦st❛r✐❛ ❞❡ ❢❛③❡r é ❛❝❤❛r ✉♠❛ ❢✉♥çã♦ ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❜❡♠ ❝♦♥❤❡❝✐❞♦ q✉❡ s❡ ❛♣r♦①✐♠❛ ❞♦ q✉♦❝✐❡♥t❡ ❛❝✐♠❛ ♣❛r❛n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳

▲❡❣❡♥❞r❡ ❡ ●❛✉ss✱ ❛♥❛❧✐s❛♥❞♦ t❛❜❡❧❛s✱ ❝❤❡❣❛r❛♠ à ❝♦♥❝❧✉sã♦ ❞❡ q✉❡ ❡st❡ q✉♦❝✐❡♥t❡ t❡♠ ❛ ✈❡r ❝♦♠ 1

lnx✳ P♦r ✈♦❧t❛ ❞❡ ✶✾✵✵✱ ❏✳ ❍❛❞❛♠❛r❞ ❡ ❈❤✳ ❞❡ ❧❛ ❱❛❧❧❡è✲ P♦✉ss✐♥✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ♣r♦✈❛r❛♠ ♦ ♣r♦❢✉♥❞♦ r❡s✉❧t❛❞♦ ❝❤❛♠❛❞♦ ❞❡ ❚❡♦r❡♠❛ ❞♦s ◆ú♠❡r♦s Pr✐♠♦s ❡ ❝✉❥♦ ❡♥✉♥❝✐❛❞♦ s✐♠♣❧❡s♠❡♥t❡ é

lim

x→∞

π(x)

x

1 lnx

−1

= 1.

❊♠ ✶✾✹✾✱ ❆✳ ❙❡❧❜❡r❣ s✐♠♣❧✐✜❝♦✉ s✉❜st❛♥❝✐❛❧♠❡♥t❡ ❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ❞♦s ◆ú♠❡r♦s Pr✐♠♦s✱ ♠❡r❡❝❡♥❞♦ ♣♦r ❡ss❡ s❡✉ tr❛❜❛❧❤♦ ❛ ▼❡❞❛❧❤❛ ❋✐❡❧❞s✳

❆ ❞✐str✐❜✉✐çã♦ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s é ❛❧❣♦ ❛✐♥❞❛ ❜❛st❛♥t❡ ♠✐st❡r✐♦s♦ ❡ ❛ ❡❧❛ ❡stã♦ ❛ss♦❝✐❛❞♦s ♠✉✐t♦s ♣r♦❜❧❡♠❛s ❡♠ ❛❜❡rt♦✳ P♦r ❡①❡♠♣❧♦✱ ♦ ❥á ❝✐t❛❞♦ ♣r♦❜❧❡♠❛ ❞❡ s❛❜❡r s❡ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐♠♦s ❣ê♠❡♦s✳

✶✳✸ ▼❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠

❉❛❞♦s ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s a ❡ b✱ ♥ã♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♥✉❧♦s✱ ❞✐r❡♠♦s q✉❡ ♦ ♥ú♠❡r♦ ♥❛t✉r❛❧d_∈N∗ _{é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b s❡d |a ❡d|b✳}

P♦r ❡①❡♠♣❧♦✱ ♦s ♥ú♠❡r♦s 1,2,3 ❡6 sã♦ ♦s ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡ 12 ❡18✳

❆ ❞❡✜♥✐çã♦ q✉❡ s❡ s❡❣✉❡ é ❡①❛t❛♠❡♥t❡ ❛ ❞❡✜♥✐çã♦ ❞❛❞❛ ♣♦r ❊✉❝❧✐❞❡s ♥♦s ❊❧❡♠❡♥t♦s ❡ s❡ ❝♦♥st✐t✉✐ ❡♠ ✉♠ ❞♦s ♣✐❧❛r❡s ❞❛ s✉❛ ❛r✐t♠ét✐❝❛✳

❉✐r❡♠♦s q✉❡ d é ✉♠ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ✭♠❞❝✮ ❞❡a ❡ b s❡ ♣♦ss✉✐r ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✐✮ d é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ ❞❡ b✱ ❡

✐✐✮ d é ❞✐✈✐sí✈❡❧ ♣♦r t♦❞♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡b✳

❆ ❝♦♥❞✐çã♦ ✭✐✐✮ ❛❝✐♠❛ ♣♦❞❡ s❡r r❡❡♥✉♥❝✐❛❞❛ ❝♦♠♦ s❡ s❡❣✉❡✿ ✐✐✬✮ ❙❡ cé ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡b✱ ❡♥tã♦ c_|d✳

P♦rt❛♥t♦✱ s❡ d é ✉♠ ♠❞❝ ❞❡ a ❡ b ❡ c é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ss❡s ♥ú♠❡r♦s✱ ❡♥tã♦ c_≤d✳ ■st♦ ♥♦s ♠♦str❛ q✉❡ ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ ❞♦✐s ♥ú♠❡r♦s é ❡❢❡t✐✈❛♠❡♥t❡ ♦ ♠❛✐♦r ❞❡♥tr❡ t♦❞♦s ♦s ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡ss❡s ♥ú♠❡r♦s✳

❊♠ ♣❛rt✐❝✉❧❛r✱ ✐st♦ ♥♦s ♠♦str❛ q✉❡✱ s❡ d ❡ d′ _{sã♦ ❞♦✐s ♠❞❝ ❞❡ ✉♠ ♠❡s♠♦ ♣❛r ❞❡}

♥ú♠❡r♦s✱ ❡♥tã♦ d _≤d′ _{❡ d}′ _{≤ d✱ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ d =d}′_{✳ ❖✉ s❡❥❛✱ ♦ ♠❞❝ ❞❡ ❞♦✐s}

♥ú♠❡r♦s✱ q✉❛♥❞♦ ❡①✐st❡✱ é ú♥✐❝♦✳

❖ ♠❞❝ ❞❡ a ❡ b✱ q✉❛♥❞♦ ❡①✐st❡ ✭✈❡r❡♠♦s ♠❛✐s ❛❞✐❛♥t❡ q✉❡ s❡♠♣r❡ ❡①✐st❡ ♦ ♠❞❝ ❞❡ ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♥ã♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♥✉❧♦s✮✱ ❡stá s❡♥❞♦ ❞❡♥♦t❛❞♦ ♣♦r (a, b)✳

❈♦♠♦ ♦ ♠❞❝ ❞❡a ❡ b ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ♦r❞❡♠ ❡♠ q✉❡ a ❡ b sã♦ t♦♠❛❞♦s✱ t❡♠♦s q✉❡

(a, b) = (b, a).

❊♠ ❛❧❣✉♥s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✱ é ❢❛❝✐❧ ✈❡r✐✜❝❛r ❛ ❡①✐stê♥❝✐❛ ❞♦ ♠❞❝✳ P♦r ❡①❡♠♣❧♦✱ s❡ a ❡ b sã♦ ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ t❡♠✲s❡ ❝❧❛r❛♠❡♥t❡ q✉❡ (0, a) = a✳ (1, a) = 1 ❡ q✉❡ (a, a) =a✳ ▼❛✐s ❛✐♥❞❛✱ t❡♠♦s q✉❡

a_|b _⇐⇒(a, b) =a. ✭✶✳✶✮

❉❡ ❢❛t♦✱ s❡ a _| b✱ t❡♠♦s q✉❡ a é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b✱ ❡✱ s❡ c é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡a ❡ b✱ ❡♥tã♦ c ❞✐✈✐❞❡ a✱ ♦ q✉❡ ♠♦str❛ q✉❡a= (a, b)✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡(a, b) =a✱ s❡❣✉❡✲s❡ q✉❡ a_|b✳

❆ ❞❡♠♦♥str❛çã♦ ❞❛ ❡①✐stê♥❝✐❛ ❞♦ ♠❞❝ ❞❡ q✉❛❧q✉❡r ♣❛r ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ♥ã♦ ❛♠❜♦s ♥✉❧♦s✱ é ❜❡♠ ♠❛✐s s✉t✐❧✳ P♦❞❡r✲s❡✲✐❛✱ ❝♦♠♦ s❡ ❢❛③ ✉s✉❛❧♠❡♥t❡ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✱ ❞❡✜♥✐r ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ ❞♦✐s ♥ú♠❡r♦s a ❡ b ❝♦♠♦ s❡♥❞♦ ♦ ♠❛✐♦r ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡ss❡s ♥ú♠❡r♦s✱ ♦ q✉❡ ❞❡ ✐♠❡❞✐❛t♦ ❣❛r❛♥t✐❛ ❛ s✉❛ ❡①✐stê♥❝✐❛✳ ❉❡ q✉❛❧q✉❡r ♠♦❞♦✱ s❡r✐❛ ♥❡❝❡ssár✐♦ ♣r♦✈❛r ❛ ♣r♦♣r✐❡❞❛❞❡ ✭✐✐✮ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ♠❞❝❀ ♣♦✐s é ❡❧❛ q✉❡ ♣♦ss✐❜✐❧✐t❛ ♣r♦✈❛r ♦s r❡s✉❧t❛❞♦s s✉❜s❡q✉❡♥t❡s✱ ❡ ♥ã♦ ♦ ❢❛t♦ ❞♦ ♠❞❝ s❡r ♦ r❡s✉❧t❛❞♦ ❛❜❛✐①♦✳

▲❡♠❛ ✶✳✷ ✭▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s✮✳ ❙❡❥❛♠a, b, n_∈N❝♦♠ a < na < b✳ ❙❡ ❡①✐st❡(a, b₋na)✱

❡♥tã♦ (a, b) ❡①✐st❡✱ ❡

(a, b) = (a, b₋na).

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ d = (a, b₋na)✳ ❈♦♠♦ d _| a ❡ d _| (b₋na)✱ s❡❣✉❡ q✉❡ d ❞✐✈✐❞❡ b=b₋na+na✳ ▲♦❣♦✱ d é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡a ❡b✳ ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ cs❡❥❛ ✉♠

❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b❀ ❧♦❣♦✱ c é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b₋na ❡✱ ♣♦rt❛♥t♦✱ c _| d✳ ■ss♦ ♣r♦✈❛ q✉❡ d= (a, b)✳

❖❜s❡r✈❛çã♦ ✶✳✻✳ ❈♦♠ ❛ ♠❡s♠❛ té❝♥✐❝❛ ✉s❛❞❛ ♥❛ ♣r♦✈❛ ❞♦ ▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s✱ ♣♦❞❡r✲ s❡✲✐❛ ♣r♦✈❛r q✉❡✱ ♣❛r❛ t♦❞♦sa, b, n_∈N✱

(a, b) = (a, b+na), ♦✉ q✉❡✱ s❡na > b✱ ❡♥tã♦

(a, b) = (a, na₋b).

❖ ▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s é ❡❢❡t✐✈♦ ♣❛r❛ ❝❛❧❝✉❧❛r ♠❞❝✱ ❝♦♥❢♦r♠❡ ✈❡r❡♠♦s ♥♦s ❡①❡♠♣❧♦s ❛ s❡❣✉✐r✱ ❡ s❡rá ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❡st❛❜❡❧❡❝❡r♠♦s ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✱ q✉❡ ♣❡r♠✐t✐rá✱ ❝♦♠ ♠✉✐t❛ ❡✜❝✐ê♥❝✐❛✱ ❝❛❧❝✉❧❛r ♦ ♠❞❝ ❞❡ ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s q✉❛✐sq✉❡r✳

❊①❡♠♣❧♦ ✶✳✼✳ ❉❛❞♦sa, m _∈N ❝♦♠ a >1✱ t❡♠♦s q✉❡

am₋₁

a₋1 , a−1

= (a₋1, m).

❉❡ ❢❛t♦✱ ❝❤❛♠❛❞♦ ❞❡d ♦ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ ❞❛ ✐❣✉❛❧❞❛❞❡✱ t❡♠♦s q✉❡

d= (am−1+am−2+. . .+a+1, a₋1) = ((am−1₋1)+(am−2₋1)+. . .+(a₋1)+m, a₋1). ❈♦♠♦✱

a₋1_|(am−1₋1) + (am−2₋1) +. . .+ (a₋1)

s❡❣✉❡✲s❡ q✉❡ (am−1_{−1) + (a}m−2_{−1) +. . .+ (a−1) = n(a−1) ♣❛r❛ ❛❧❣✉♠ n∈N✱ ❡✱}

♣♦rt❛♥t♦✱ ♣❡❧❛ ❖❜s❡r✈❛çã♦ ✶✳✻✱ t❡♠✲s❡ q✉❡

d= (n(a₋1) +m, a₋1) = (a₋1, n(a₋1) +m) = (a₋1, m).

❊①❡♠♣❧♦ ✶✳✽✳ ❱❛♠♦s✱ ♥❡st❡ ❡①❡♠♣❧♦✱ ❞❡t❡r♠✐♥❛r ♦s ✈❛❧♦r❡s ❞❡ a ❡ n ♣❛r❛ ♦s q✉❛✐s a+ 1 ❞✐✈✐❞❡ a2n_{+ 1✳}

◆♦t❡ ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡

a+ 1_|a2n+ 1_⇐⇒(a+ 1, a2n+ 1) =a+ 1. ❈♦♠♦a2n_{+ 1 = (a}2n

−1) + 2✱ ❡✱ s❡❣✉❡✲s❡✱ ♣❡❧❛ ❖❜s❡r✈❛çã♦ ✶✳✻✱ q✉❡ ♣❛r❛ t♦❞♦ n✱ (a+ 1, a2n_{+ 1) = (a+ 1,(a}2n

P♦rt❛♥t♦✱ a+ 1_|a2n_{+ 1✱ ♣❛r❛ ❛❧❣✉♠ n}

∈N✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱a+ 1 = (a+ 1,2)✱ ♦ q✉❡

♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱a= 0 ♦✉a= 1✳

❊①❡♠♣❧♦ ✶✳✾✳ ❱❛♠♦s✱ ♥❡st❡ ❡①❡♠♣❧♦✱ ❞❡t❡r♠✐♥❛r ♦s ✈❛❧♦r❡s ❞❡ a ❡ n ♣❛r❛ ♦s q✉❛✐s a+ 1 ❞✐✈✐❞❡ a2n+1_−1✳

◆♦t❡ q✉❡

(a+ 1, a2n+1_{−1) = (a+ 1, a(a}2n

−1) +a₋1) = (a+ 1, a₋1). P♦rt❛♥t♦✱ a+ 1_|a2n+1_{−1✱ ♣❛r❛ ❛❧❣✉♠ n∈N✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱}

a+ 1 = (a+ 1, a2n+1₋1) = (a+ 1, a₋1), ♦ q✉❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱a = 1✳

❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s

❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ♣r♦✈❛ ❝♦♥str✉t✐✈❛ ❞❛ ❡①✐stê♥❝✐❛ ❞♦ ♠❞❝ ❞❛❞❛ ♣♦r ❊✉❝❧✐❞❡s ✭❖s ❊❧❡♠❡♥t♦s✱ ▲✐✈r♦ ❱■■✱ Pr♦♣♦s✐çã♦ ✷✮✳ ❖ ♠ét♦❞♦✱ ❝❤❛♠❛❞♦ ❞❡ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✱ é ✉♠ ♣r✐♠♦r ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❡ ♣♦✉❝♦ ❝♦♥s❡❣✉✐✉✲s❡ ❛♣❡r❢❡✐ç♦á✲ ❧♦ ❡♠ ♠❛✐s ❞❡ ❞♦✐s ♠✐❧ê♥✐♦s✳

❉❛❞♦s a, b_∈N✱ ♣♦❞❡♠♦s s✉♣♦r a _≤b✳ ❙❡ a = 1 ♦✉a =b✱ ♦✉ ❛✐♥❞❛ a _|b✱ ❥á ✈✐♠♦s q✉❡ (a, b) = a✳ ❙✉♣♦♥❤❛♠♦s✱ ❡♥tã♦✱ q✉❡ 1 < a < b ❡ q✉❡ ❛ a ∤ b✳ ▲♦❣♦✱ ♣❡❧❛ ❞✐✈✐sã♦

❡✉❝❧✐❞✐❛♥❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

b=aq1+r1, ❝♦♠ r1 < a.

❚❡♠♦s ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿

❛✮ r1 |a✱ ❡✱ ❡♠ t❛❧ ❝❛s♦✱ ♣♦r ✭✶✳✶✮ ❡ ♣❡❧♦ ▲❡♠❛ ✶✳✷✱

r1 = (a, r1) = (a, b−q1a) = (a, b),

❡ t❡r♠✐♥❛ ♦ ❛❧❣♦r✐t♠♦✱ ♦✉

❜✮ r1 ∤a✱ ❡✱ ❡♠ t❛❧ ❝❛s♦✱ ♣♦❞❡♠♦s ❡❢❡t✉❛r ❛ ❞✐✈✐sã♦ ❞❡a ♣♦r r1✱ ♦❜t❡♥❞♦

a=r1q2 +r2, ❝♦♠ r2 < r1.

◆♦✈❛♠❡♥t❡✱ t❡♠♦s ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿