❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
◆ú♠❡r♦s Pr✐♠♦s ❡ ♦ P♦st✉❧❛❞♦ ❞❡
❇❡rtr❛♥❞
†♣♦r
❆♥t♦♥✐♦ ❊✉❞❡s ❋❡rr❡✐r❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ✲ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❆❣♦st♦✴✷✵✶✹ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
F383n Ferreira, Antonio Eudes.
Números primos e o Postulado de Bertrand / Antonio Eudes Ferreira.- João Pessoa, 2014.
44f.
Orientador: Napoleón Caro Tuesta Dissertação (Mestrado) - UFPB/CCEN
1. Matemática. 2. Números primos. 3. Primos de Fermat. 4.Primos de Mersenne. 5. Postulado de Bertrand.
◆ú♠❡r♦s Pr✐♠♦s ❡ ♦ P♦st✉❧❛❞♦ ❞❡
❇❡rtr❛♥❞
♣♦r
❆♥t♦♥✐♦ ❊✉❞❡s ❋❡rr❡✐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❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛ ❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛ ✲ ❯❋P❇
Pr♦❢✳ ❉r✳ ❉✐♦❣♦ ❉✐♥✐③ P❡r❡✐r❛ ❞❛ ❙✐❧✈❛ ❡ ❙✐❧✈❛ ✲ ❯❋❈●
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s✱ ♣♦r s❡r ✉♠ P❛✐ tã♦ ♣r❡s❡♥t❡ ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❡ ♠❡ ♣r♦♣♦r❝✐♦♥❛r ❡st❛ ❣r❛♥❞❡ ❝♦♥q✉✐st❛✳
❆ ♠✐♥❤❛ ♠ã❡✱ ▼❛r✐❛ ❞❡ ❋át✐♠❛ ❋❡rr❡✐r❛ ❇❛r❜♦s❛✱ ♣♦r ❢❛③❡r ♣❛♣❡❧ ❞❡ ♣❛✐ ❡ ♠ã❡ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ ❡st❛♥❞♦ s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✱ ♠❡ ✐♥❝❡♥t✐✲ ✈❛♥❞♦ ❡ ❛❝r❡❞✐t❛♥❞♦ q✉❡ s♦✉ ❝❛♣❛③ ❞❡ ❢❛③❡r s❡♠♣r❡ ♦ ♠❡❧❤♦r✳ ❆❣r❛❞❡ç♦ ♣♦r s❡♠♣r❡ ❢❛③❡r s❡✉ ♣❛♣❡❧ ❞❡ ♠ã❡ ♣r♦t❡t♦r❛ ❞❛ ❢♦r♠❛ ♠❛✐s ❧✐♥❞❛ q✉❡ ❡①✐st❡ ❡ ❝♦♠♣❛r✐❧❤❛r ❞❡ t♦❞♦s ♦s s❡♥t✐♠❡♥t♦s ♣♦r ♠✐♠ ✈✐✈✐❞♦s✱ ❛❧é♠ ❞❡ s❡r ♦ ♣r✐♥❝✐♣❛❧ ♠♦t✐✈♦ ♣♦r ❡✉ q✉❡r❡r s❡♠♣r❡ ✐r ♠❛✐s ❧♦♥❣❡✳
❆♦s ♠❡✉s ✐r♠ã♦s✱ ❊❧✐❢á❜✐♦ ❋❡rr❡✐r❛ ❇❛r❜♦s❛ ❡ ▼❛r✐❛ ❊❧✐③â♥❣❡❧❛ ❋❡rr❡✐r❛ ❇❛r❜♦s❛✱ q✉❡ ♠❡s♠♦ ❛✉s❡♥t❡s✱ s❡♠♣r❡ ♣❛rt✐❝✐♣❛r❛♠ ❛t✐✈❛♠❡♥t❡ ♥❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✱ ❛ q✉❡♠ ❞❡✈♦ t♦❞♦ ♠❡✉ ❝❛r✐♥❤♦✱ r❡s♣❡✐t♦ ❡ ❛♠♦r✳
❆♦s ♠❡✉s ♣r✐♠♦s ✲ ✐r♠ã♦s✱ ▼❡♥❡③❡s ▼❛t✐❛s ❋❡rr❡✐r❛ ❡ ❆♥t♦♥✐❛ ❊❧✐♥❛í❞❡ ❋❡rr❡✐r❛ ❉❛♥t❛s✱ ♣♦r s❡ ❢❛③❡r❡♠ s❡♠♣r❡ ♣r❡s❡♥t❡s ♥❛ ♠✐♥❤❛ ✈✐❞❛✱ ❛❧❡❣r❛♥❞♦ t♦❞♦s ♦s ♠❡✉s ❞✐❛s✱ ❛♣♦✐❛♥❞♦ ❡ ♠❡ ✐♥❝❡♥t✐✈❛♥❞♦ ❡♠ t♦❞❛s ❛s ♠✐♥❤❛s ❞❡❝✐sõ❡s✳
❆ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s ❡ ❢❛♠✐❧✐❛r❡s✳
❆ t♦❞♦s ♣r♦❢❡ss♦r❡s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❝♦♠ s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ❡ ❢♦r❛♠ ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❛♦ Pr♦❢❡ss♦r ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ❡ ♣♦r ♠❡ ♦r✐❡♥t❛r ❞❡ ❢♦r♠❛ s✐❣♥✐✜❝❛t✐✈❛ ❡ ♣❧❛✉sí✈❡❧✱ t♦r♥❛♥❞♦ ♣♦ssí✈❡❧ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ s♦♥❤♦✳
❉❡❞✐❝❛tór✐❛
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ✉♠ ❡st✉❞♦ s♦❜r❡ ♦s ♥ú♠❡r♦s ♣r✐♠♦s✱ ❝♦♠♦ ❡stã♦ ❞✐s✲ tr✐❜✉í❞♦s✱ q✉❛♥t♦s ♥ú♠❡r♦s ♣r✐♠♦s ❡①✐st❡♠ ❡♥tr❡ ✶ ❡ ✉♠ ♥ú♠❡r♦ r❡❛❧xq✉❛❧q✉❡r✱ ❢ór✲ ♠✉❧❛s q✉❡ ❣❡r❛♠ ♣r✐♠♦s✱ ❛❧é♠ ❞❡ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ♣❛r❛ ♦ P♦st✉❧❛❞♦ ❞❡ ❇❡rtr❛♥❞✳ ❙ã♦ ❛❜♦r❞❛❞❛s s❡✐s ❞❡♠♦♥str❛çõ❡s q✉❡ ♠♦str❛♠ q✉❡ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐✲ ♠♦s ✉s❛♥❞♦ r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦✱ ◆ú♠❡r♦s ❞❡ ❋❡r♠❛t✱ ◆ú♠❡r♦s ❞❡ ▼❡rs❡♥♥❡✱ ❈á❧❝✉❧♦ ❊❧❡♠❡♥t❛r ❡ ❚♦♣♦❧♦❣✐❛✳
P❛❧❛✈r❛s✲❝❤❛✈❡s✿ ◆ú♠❡r♦s Pr✐♠♦s✱ Pr✐♠♦s ❞❡ ❋❡r♠❛t✱ Pr✐♠♦s ❞❡ ▼❡rs❡♥♥❡✱ P♦s✲ t✉❧❛❞♦ ❞❡ ❇❡rtr❛♥❞✳
❆❜str❛❝t
❚❤✐s ✇♦r❦ ♣r❡s❡♥ts ❛ st✉❞② ♦❢ ♣r✐♠❡ ♥✉♠❜❡rs✱ ❤♦✇ t❤❡② ❛r❡ ❞✐str✐❜✉t❡❞✱ ❤♦✇ ♠❛♥② ♣r✐♠❡ ♥✉♠❜❡rs ❛r❡ t❤❡r❡ ❜❡t✇❡❡♥ ✶ ❛♥❞ ❛ r❡❛❧ ♥✉♠❜❡r x✱ ❢♦r♠✉❧❛s t❤❛t ❣❡♥❡✲ r❛t❡ ♣r✐♠❡s✱ ❛♥❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ ❇❡rtr❛♥❞✬s P♦st✉❧❛t❡✳ ❙✐① ♣r♦♦❢s t❤❛t t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡s ✉s✐♥❣ r❡❞✉❝t✐♦ ❛❞ ❛❜s✉r❞✉♠✱ ❋❡r♠❛t ♥✉♠❜❡rs✱ ▼❡rs❡♥♥❡ ♥✉♠❜❡rs✱ ❊❧❡♠❡♥t❛r② ❈❛❧❝✉❧✉s ❛♥❞ ❚♦♣♦❧♦❣② ❛r❡ ❞✐s❝✉ss❡❞✳
❑❡②✇♦r❞s✿ Pr✐♠❡ ◆✉♠❜❡rs✱ ❋❡r♠❛t ♣r✐♠❡s✱ ▼❡rs❡♥♥❡ ♣r✐♠❡s✱ ❇❡rtr❛♥❞ P♦st✉❧❛t❡✳
❙✉♠ár✐♦
✶ ◆ú♠❡r♦s Pr✐♠♦s ✶
✶✳✶ Pr♦✈❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ Pr♦✈❛ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸ Pr♦✈❛ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✹ Pr♦✈❛ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✺ Pr♦✈❛ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✻ Pr♦✈❛ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✷ P♦st✉❧❛❞♦ ❞❡ ❇❡rtr❛♥❞ ✶✶
✸ ❈♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s ✭q✉❛s❡✮ ♥✉♥❝❛ sã♦ ♣♦tê♥❝✐❛s ✶✻ ❆ Pr✐♠♦s ❡♠ ❝❡rt❛s ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s ✷✶
❇ ❊✉❧❡r✱ ✉♠ ●✐❣❛♥t❡ ❞❛ ▼❛t❡♠át✐❝❛ ✷✸
❈ ▲❡❣❡♥❞r❡ ✷✹
❈✳✶ ❚❡♦r❡♠❛ ❞❡ ▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
❉ ❊st✐♠❛t✐✈❛ ✈✐❛ ✐♥t❡❣r❛✐s ✷✽
❉✳✶ ❊st✐♠❛t✐✈❛s ♣❛r❛ ❢❛t♦r✐❛✐s ✲ ❢ór♠✉❧❛ ❞❡ ❙t✐r❧✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
❊ ❚♦♣♦❧♦❣✐❛ ✸✶
❋ ❉❡♠♦♥str❛çã♦ ❞♦ P♦st✉❧❛❞♦ ❞❡ ❇❡rtr❛♥❞ ✈✐❛ ❘❛♠❛♥✉❥❛♥ ✸✸ ● ❊st✐♠❛t✐✈❛s ♣❛r❛ ❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s ✸✼
❍ ❆ ❢✉♥çã♦ ③❡t❛ ❞❡ ❘✐❡♠❛♥♥ ✸✾
■ ❈r✐♣t♦❣r❛✜❛ ✹✶
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✹
■♥tr♦❞✉çã♦
❖s ♥ú♠❡r♦s ♣r✐♠♦s sã♦ ❝♦♥❤❡❝✐❞♦s ♣❡❧❛ ❤✉♠❛♥✐❞❛❞❡ ❤á ♠✉✐t♦ t❡♠♣♦✳ ❊ss❡s ♥ú♠❡r♦s ❞❡s❡♠♣❡♥❤❛♠ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ♥❛ ❆r✐t♠ét✐❝❛ ❡ ❛ ❡❧❡s ❡stã♦ ❛ss♦❝✐❛❞♦s ♠✉✐t♦s ♣r♦❜❧❡♠❛s ❢❛♠♦s♦s ❝✉❥❛s s♦❧✉çõ❡s tê♠ r❡s✐st✐❞♦ ❛♦s ❡s❢♦rç♦s ❞❡ ✈ár✐❛s ❣❡r❛✲ çõ❡s ❞❡ ♠❛t❡♠át✐❝♦s✳ ❖s ❣r❡❣♦s ❢♦r❛♠ ♦s ♣✐♦♥❡✐r♦s ♥♦ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s ❡ ❛s s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❊♥tr❡ ♦s ❣r❡❣♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♥tr❡ ❣r❡❣♦s ♣✐t❛❣ór✐❝♦s ❞❡ ✈ár✐❛s ❣❡r❛çõ❡s ❞❡♣♦✐s ❞❡ P✐tá❣♦r❛s✱ s✉r❣✐r❛♠ ♦✉tr❛s ❞❡♥♦♠✐♥❛çõ❡s ♣❛r❛ ♦s ♥ú♠❡r♦s ♣r✐♠♦s✱ ❝♦♠♦✿ r❡t✐❧í♥❡♦s✱ ❧✐♥❡❛r❡s ❡ ❡✉t✐♠étr✐❝♦s✳ ❈♦♥t✉❞♦✱ ❡❧❛s t✐✈❡r❛♠ ✉s♦ ♠✉✐t♦ r❡str✐t♦ ❡ ❝❛✐r❛♠ ♥♦ ❞❡s✉s♦✳ ▼❛t❡♠át✐❝♦s ❞❛ ❡s❝♦❧❛ ❞❡ P✐tá❣♦r❛s ✭✺✵✵ ❛ ✸✵✵ ❆✳❈✳✮ t✐♥❤❛♠ ❜❛st❛♥t❡ ✐♥t❡✲r❡ss❡ ♥♦s ♥ú♠❡r♦s ♣❡❧❛s s✉❛s ♣r♦♣r✐❡❞❛❞❡s ♥✉♠❡r♦❧ó❣✐❝❛s ❡ ♠ís✲ t✐❝❛s✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s ♣♦ss✉✐ ✉♠❛ ❞❡✜♥✐çã♦ ❜❛st❛♥t❡ s✐♠♣❧❡s✱ ♣♦ré♠ ❝♦♠ ✉♠❛ ❛r✐t♠ét✐❝❛ ♠✉✐t♦ ❝♦♠♣❧❡①❛✳ ❖ ♠❛t❡♠át✐❝♦ ❣r❡❣♦ ❊✉❝❧✐❞❡s ♣r♦✈♦✉ q✉❡ ♦s ♥ú♠❡r♦s ♣r✐♠♦s ❡r❛♠ ✐♥✜♥✐t♦s✱ ♠❡s♠♦ ❛ss✐♠✱ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ♥ú♠❡r♦s ♣r✐✲ ♠♦s ♠❛♥t✐✈❡r❛♠ ♦❝✉♣❛❞♦s q✉❛s❡ t♦❞♦s ♦s ♠❛t❡♠át✐❝♦s ❞❡s❞❡ ❛ ❛♥t✐❣✉✐❞❛❞❡✿ ❝♦♠♦ s❛❜❡r s❡ ✉♠ ♥ú♠❡r♦ é ♣r✐♠♦ ♦✉ ♥ã♦✱ ♦✉ ♣r❡✲✈❡r ❛ s✉❛ ❡①✐stê♥❝✐❛ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s✱ ♦✉ ❛✐♥❞❛ ❡♥❝♦♥tr❛r ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ❞❡✜♥í✲❧♦s✳ ▼✉✐t❛s ❞❡ss❛s q✉❡stõ❡s ❝♦♥t✐♥✉❛♠ s❡♠ r❡s♣♦st❛✳ ◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛r❡♠♦s s❡✐s ❞❡♠♦♥str❛çõ❡s ❞❛ ✐♥✲ ✜♥✐t✉❞❡ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s✳ ❆ ♣r✐♠❡✐r❛✱ ♠❛✐s ❛♥t✐❣❛ ❡ ❝❧áss✐❝❛✱ ❢♦✐ ❛♣r❡s❡♥t❛❞❛ ♣♦r ❊✉❝❧✐❞❡s✱ ❛ s❡❣✉♥❞❛ ✉s❛♥❞♦ ♥ú♠❡r♦s ❞❡ ❋❡r♠❛t✱ ❛ t❡r❝❡✐r❛ ✉s❛♥❞♦ ♥ú♠❡r♦s ❞❡ ▼❡rs❡♥♥❡✱ ❛ q✉❛rt❛ ✉s❛♥❞♦ ❈á❧❝✉❧♦ ❊❧❡♠❡♥t❛r✱ ❛ q✉✐♥t❛ ✉s❛♥❞♦ ❚♦♣♦❧♦❣✐❛ ❡ ❛ ú❧t✐♠❛✱ ❢♦✐ ❣r❛ç❛s ❛♦ ❣r❛♥❞❡ P❛✉❧ ❊r❞➤s✳ ❊st✉❞❛r❡♠♦s ♦ P♦st✉❧❛❞♦ ❞❡ ❇❡rt❛♥❞ ❡ ♠♦str❛r❡♠♦s q✉❡ ❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s ✭q✉❛s❡✮ ♥✉♥❝❛ sã♦ ♣♦tê♥❝✐❛s✳
❈❛♣ít✉❧♦ ✶
◆ú♠❡r♦s Pr✐♠♦s
✏➚ ▼❛t❡♠át✐❝❛ é ❛ r❛✐♥❤❛ ❞❛s ❝✐ê♥❝✐❛s ❡ ❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s é ❛ r❛✐♥❤❛ ❞❛ ▼❛t❡♠át✐❝❛✑✳
❑❛r❧✳ ❋✳ ●❛✉ss✳ ◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❞❛❞❛s s❡✐s ❞❡♠♦♥str❛çõ❡s s♦❜r❡ ❛ ✐♥✜♥✐t✉❞❡ ❞♦s ♥ú♠❡r♦s ♣r✐✲ ♠♦s✳ ▼❛s✱ ❛♥t❡s ❧❡♠❜r❡♠♦s ❛❧❣✉♠❛s ♥♦çõ❡s ❜ás✐❝❛s✳
❉❡✜♥✐çã♦✿ ❯♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♠❛✐♦r ❞♦ q✉❡ ✶ ❡ q✉❡ só é ❞✐✈✐sí✈❡❧ ♣♦r ✶ ❡ ♣♦r s✐ ♣ró♣r✐♦ é ❝❤❛♠❛❞♦ ❞❡ ♥ú♠❡r♦ ♣r✐♠♦✳
❉❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ ❞❛❞♦s ❞♦✐s ♥ú♠❡r♦s ♣r✐♠♦s p ❡ q ❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ a q✉❛❧✲ q✉❡r✱ ❞❡❝♦rr❡♠ ♦s s❡❣✉✐♥t❡s ❢❛t♦s✿
■✮ ❙❡ p|q✱ ❡♥tã♦ p=q✳
❉❡ ❢❛t♦✱ ❝♦♠♦ p | q ❡ s❡♥❞♦ q ♣r✐♠♦✱ t❡♠♦s q✉❡ p = 1 ♦✉ p = q✳ ❙❡♥❞♦ p ♣r✐♠♦✱ t❡♠✲s❡ q✉❡ p >1✱ ♦ q✉❡ ❛❝❛rr❡t❛ p=q✳
■■✮ ❙❡ p∤a✱ ❡♥tã♦ ♠❞❝(p, a) = 1✳
❉❡ ❢❛t♦✱ s❡ ♠❞❝(p, a) = d✱ t❡♠♦s q✉❡ d | p ❡ d |a✳ P♦rt❛♥t♦✱ d =p ♦✉ d = 1✳ ▼❛s d6=p✱ ♣♦✐sp∤a ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱d = 1✳
❯♠ ♥ú♠❡r♦ ♠❛✐♦r ❞♦ q✉❡ ✶ ❡ q✉❡ ♥ã♦ é ♣r✐♠♦ s❡rá ❝❤❛♠❛❞♦ ❞❡ ❝♦♠♣♦st♦✳ P♦rt❛♥t♦✱ s❡ ✉♠ ♥ú♠❡r♦ n é ❝♦♠♣♦st♦✱ ❡①✐st✐rá ✉♠ ❞✐✈✐s♦r n1 ❞❡ n t❛❧ q✉❡ n1 6= 1 ❡ n1 6= n✳
▲♦❣♦✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ n2 t❛❧ q✉❡
n=n1n2✱ ❝♦♠ 1< n1 < n ❡1< n2 < n
✶✳✶✳ P❘❖❱❆ ✶ ✷
P♦r ❡①❡♠♣❧♦✱ ✷✱ ✸✱ ✺✱ ✼✱ ✶✶ ❡ ✶✸ sã♦ ♥ú♠❡r♦s ♣r✐♠♦s✱ ❡♥q✉❛♥t♦ q✉❡ ✹✱ ✻✱ ✽✱ ✾✱ ✶✵ ❡ ✶✷ sã♦ ❝♦♠♣♦st♦s✳
❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛ ❡str✉t✉r❛ ♠✉❧t✐♣❧✐❝❛t✐✈❛ ❞♦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♠❛ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛ q✉❡ ♠♦str❛ q✉❡ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♠❛✐♦r ❞♦ q✉❡ ✶ ♦✉ é ♣r✐♠♦ ♦✉ s❡ ❡s❝r❡✈❡ ❞❡ ♠♦❞♦ ú♥✐❝♦ ✭❛ ♠❡♥♦s ❞❛ ♦r❞❡♠ ❞♦s ❢❛t♦r❡s✮ ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✳
❚❡♦r❡♠❛✿ ❊①✐st❡♠ ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐♠♦s✳
◆❛s s❡❣✉✐♥t❡s ❞❡♠♦♥str❛çõ❡s ✐r❡♠♦s ❝♦♥s✐❞❡r❛r N ={0,1,2,3, ...} ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦
❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ Z = {...,−2,−1,0,1,2, ...} ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡ P = {2,3,5,7,11, ...} ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s✳ ❯s❛r❡♠♦s t❛♠❜é♠ ♦ ❢❛t♦ q✉❡ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s sã♦ ✐♥✜♥✐t♦s ❡ q✉❡ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ♦ ♣r♦❞✉t♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✳
✶✳✶ Pr♦✈❛ ✶
❊✉❝❧✐❞❡s ✈✐✈❡✉ ❡♠ ❆❧❡①❛♥❞r✐❛ ❝❡r❝❛ ❞❡ ✸✵✵ ❛✳ ❈✳✱ s❡♥❞♦ ✉♠ ❞♦ três ♠❛✐♦r❡s ♠❛t❡♠át✐❝♦s ❞❛ ❆♥t✐❣✉✐❞❛❞❡ ❣r❡❣❛ ❡✱ s❡♠ ❞ú✈✐❞❛✱ ❞❡ t♦❞♦s ♦s t❡♠♣♦s✳ P♦✉❝♦ s❡ s❛❜❡ ❞❛ s✉❛ ✈❡r❞❛❞❡✐r❛ ❜✐♦❣r❛✜❛✳ ❈♦♥s✐❞❡r❛❞♦ ♣❛✐ ❞❛ ●❡♦♠❡tr✐❛✱ t❡♥❞♦ ❡♠ ✈✐st❛ ♦s s❡✉s ❛✈❛♥ç♦s ❢❡✐t♦s ♥❡st❛ ár❡❛✱ ❊✉❝❧✐❞❡s✱ t❡✈❡ ✉♠❛ ♣❛rt✐❝✐♣❛çã♦ s✐❣♥✐✜❝❛t✐✈❛ ❡ ♥♦tór✐❛ ♥♦ â♠❜✐t♦ ❞❛ ❚❡♦r✐❛ ❞♦ ◆ú♠❡r♦s✳ ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛ ♣✉❜❧✐❝♦✉ ❖s ❊❧❡♠❡♥t♦s✱ ❝❡r❝❛ ❞❡ ✸✵✵ ❛✳❈✳✱ ♣r♦✈❛♥❞♦ ✈ár✐♦s r❡s✉❧t❛❞♦s s♦❜r❡ ♥ú♠❡r♦s ♣r✐♠♦s✳ ❆ ❞❡♠♦♥s✲ tr❛çã♦ q✉❡ ❤á ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐♠♦s ❛♣❛r❡❝❡ ♥♦ ❧✐✈r♦ ■❳ ❞❡ ❖s ❊❧❡♠❡♥t♦s✱ ♦♥❞❡✱ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♥❛ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛✱ ✉♠❛ ❞❡♠♦♥str❛çã♦ é ❢❡✐t❛ ❛ ♣❛rt✐r ❞♦ ✉s♦ ❞❛ r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦✳
Pr♦✈❛✿ ❙✉♣♦♥❤❛♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦s ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s P é ✜♥✐t♦✱ ✐st♦ é✱ P =
{p1, p2, ..., pr}♦♥❞❡p1 = 2< p2 = 3< p3 = 5< ...❡ ❝♦♥s✐❞❡r❡n=p1·p2·p3·. . .·pr+1.
❊♥tã♦✱ n ♣♦ss✉✐ ✉♠ ❞✐✈✐s♦r ♣r✐♠♦p✳ ▼❛s p♥ã♦ é ✉♠ ❞♦s pi ✿ ❝❛s♦ ❝♦♥trár✐♦p s❡r✐❛
✉♠ ❞✐✈✐s♦r ❞❡ n ❡ ❞♦ ♣r♦❞✉t♦ p1 ·p2 ·p3 · . . .·pr✱ ❡ ❛ss✐♠ t❛♠❜é♠ ❞❛ ❞✐❢❡r❡♥ç❛
n−p1·p2·p3·. . .·pr= 1✱ ♦ q✉❡ é ✐♠♣♦ssí✈❡❧✳ P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ P ♥ã♦ é ✜♥✐t♦✳
✶✳✷✳ P❘❖❱❆ ✷ ✸
✶✳✷ Pr♦✈❛ ✷
◆❛ s❡❣✉✐♥t❡ ❞❡♠♦♥str❛çã♦✱ ✉s❛r❡♠♦s ♦s ♥ú♠❡r♦s ❞❡ ❋❡r♠❛t ✭Fn = 22
n
+ 1✱ ❝♦♠ n = 0,1,2, ...✮✳ P✐❡rr❡ ❞❡ ❋❡r♠❛t ✭✶✻✵✶ ✲ ✶✻✻✺✮✱ ❥✉r✐st❛ ❢r❛♥❝ês ❡ ♠❛t❡♠át✐❝♦ ❛♠❛❞♦r✱ é ❝♦♥s✐❞❡r❛❞♦✱ ❛♣ós ❊✉❝❧✐❞❡s ❡ ❊r❛tóst❡♥❡s✱ ♦ ♣r✐♠❡✐r♦ ♠❛t❡♠át✐❝♦ ❛ ❝♦♥tr✐❜✉✐r ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦✳ ❋✐❧❤♦ ❞❡ ❉♦✲ ♠✐♥✐q✉❡ ❞❡ ❋❡r♠❛t ✲ r✐❝♦ ♠❡r❝❛❞♦r ❞❡ ♣❡❧❡s ✲ t❡✈❡ ✉♠❛ ❡❞✉❝❛çã♦ ♣r✐✈✐❧❡❣✐❛❞❛✱ ✐♥✐❝✐❛❧✲ ♠❡♥t❡ ♥♦ ♠♦st❡✐r♦ ❢r❛♥❝✐s❝❛♥♦ ❞❡ ●r❛♥❞s❡❧✈❡ ❡ ❞❡♣♦✐s ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❚♦✉❧♦✉s❡✳ ❋❡r♠❛t t❡✈❡ ❝♦♥tr✐❜✉✐çõ❡s s✐❣♥✐✜❝❛t✐✈❛s ♥❛ ♠❛t❡♠át✐❝❛✳ ❈♦♥❥❡❝t✉r♦✉ ❡ ❞❡♠♦♥str♦✉ ♦ ❝❤❛♠❛❞♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t q✉❡ ♣♦ss✉✐ ♦ s❡❣✉✐♥t❡ ❡♥✉♥❝✐❛❞♦✿
✏❙❡ p é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✱ ❡♥tã♦ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧ a✱ ap ≡a ✭♠♦❞ p✮✑✳
❆t✉❛❧♠❡♥t❡ ❡st❡ t❡♦r❡♠❛ é ❛ ❜❛s❡ ❞❡ ♠✉✐t♦s r❡s✉❧t❛❞♦s ❞❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ❡ ❞❡ ♠ét♦❞♦s ♣❛r❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✱ ✉t❧✐③❛❞♦s ❡♠ ❧❛r❣❛ ❡s❝❛❧❛ ♥❛ ❝♦♠♣✉t❛çã♦ ❡ ♥❛ ❝r✐♣t♦❣r❛✜❛✳ ❊ss❛ ❝♦♥❥❡❝t✉r❛ ❢♦✐ ♣r♦✈❛❞❛ ♣♦r ❊✉❧❡r ✭✈❡❥❛ ❇ ❞♦ ❛♣ê♥❞✐❝❡✮✱ ❡♠ ✶✼✸✻✳
❖s r❡s✉❧t❛❞♦s ❞❡ ❋❡r♠❛t ❢♦r❛♠ ❞✐✈✉❧❣❛❞♦s ♣♦r ♠❡✐♦ ❞❡ s✉❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛✱ ♣r✐♥✲ ❝✐♣❛❧♠❡♥t❡ ❝♦♠ ♦ ♣❛❞r❡ ▼❛r✐♥ ▼❡rs❡♥♥❡✱ q✉❡ ❞❡s❡♠♣❡♥❤❛✈❛ ♦ ♣❛♣❡❧ ❞❡ ❞✐✈✉❧❣❛❞♦r ❞❛ ▼❛t❡♠át✐❝❛✳ ◆✉♠❛ ❞❡ s✉❛s ❝❛rt❛s ❞❡ ✶✻✹✵✱ ❋❡r♠❛t ❡♥✉♥❝✐♦✉ ♦ s❡✉ P❡q✉❡♥♦ ❚❡♦r❡♠❛✱ ❞✐③❡♥❞♦ q✉❡ ♥ã♦ ❡s❝r❡✈❡r✐❛ ❛ ❞❡♠♦♥str❛çã♦ ♣♦r s❡r ❧♦♥❣❛ ❞❡♠❛✐s✳ ❆ s✉❛ ❝♦♥tr✐❜✉✐çã♦ ♠❛✐s ♠❛r❝❛♥t❡ ❢♦✐ ❛ ❛♥♦t❛çã♦ ❞❡✐①❛❞❛ ♥❛ ♠❛r❣❡♠ ❞♦ Pr♦❜❧❡♠❛ ✽✱ ❞♦ ▲✐✈r♦ ✷✱ ❞❡ s✉❛ ❝ó♣✐❛ ❞❡ ❇❛❝❤❡t ❞❛ ❆r✐t♠ét✐❝❛ ❞❡ ❉✐♦❢❛♥t♦✱ ♦♥❞❡ s❡ ❡♥❝♦♥tr❛✈❛♠ ❞❡s❝r✐t❛s ❛s ✐♥✜♥✐t❛s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ♣✐t❛❣ór✐❝❛ x2+y2 =z2✱ ❞✐③✐❛✿
✏P♦r ♦✉tr♦ ❧❛❞♦✱ é ✐♠♣♦ssí✈❡❧ s❡♣❛r❛r ✉♠ ❝✉❜♦ ❡♠ ❞♦✐s ❝✉❜♦s✱ ♦✉ ✉♠❛ ❜✐q✉❛❞r❛❞❛ ❡♠ ❞✉❛s ❜✐q✉❛❞r❛❞❛s✱ ♦✉✱ ❡♠ ❣❡r❛❧✱ ✉♠❛ ♣♦tê♥❝✐❛ q✉❛❧q✉❡r✱ ❡①❝❡t♦ ✉♠ q✉❛❞r❛❞♦ ❡♠
❞✉❛s ♣♦tê♥❝✐❛s s❡♠❡❧❤❛♥t❡s✳ ❊✉ ❞❡s❝♦❜r✐ ✉♠❛ ❞❡♠♦♥str❛çã♦ ✈❡r❞❛❞❡✐r❛♠❡♥t❡ ♠❛r❛✈✐❧❤♦s❛ ❞✐st♦✱ q✉❡ t♦❞❛✈✐❛ ❡st❛ ♠❛r❣❡♠ ♥ã♦ é s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ♣❛r❛
❝❛❜ê✲❧❛✳✑
❆♣❡s❛r ❞❡ ♥ã♦ ❞❡♠♦♥str❛❞❛ ♣♦r ❡❧❡✱ ❛❝❛❜♦✉ s❡♥❞♦ ❝❤❛♠❛❞❛ ❞❡ Ú❧t✐♠♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t ❡ ❢♦✐ ❞❡♠♦♥str❛❞❛ ❡♠ ✶✾✾✺✱ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ✐♥❣❧ês✱ ❆♥❞r❡✇ ❲✐❧❡s✳
❍á ♠✉✐t♦ ❡r❛ ♣r♦❝✉r❛❞♦ ♣♦r ♠❛t❡♠át✐❝♦s ❢ór♠✉❧❛s q✉❡ ❣❡r❛ss❡♠ ♥ú♠❡r♦s ♣r✐♠♦s✳ ❋❡r♠❛t ♣r♦♣ôs q✉❡ ❛ ❢ór♠✉❧❛ 22n
+ 1 ❝♦♠ n ∈ N ♣r♦❞✉③✐❛ ♥ú♠❡r♦s ♣r✐♠♦s✳ P❛r❛ n = 0,1,2,3 ❡ 4✱ t❡♠♦s✿
n = 0 →F0 = 3
n = 1 →F1 = 5
n = 2 →F2 = 17
n = 3 →F3 = 257
✶✳✸✳ P❘❖❱❆ ✸ ✹
sã♦ ♥ú♠❡r♦s ♣r✐♠♦s✳ ❆♣❡s❛r ❞❡ ♥ã♦ t❡r ✉♠ ♣r♦✈❛ ❝♦♥✈✐❝t❛ ❞♦ s❡✉ r❡s✉❧t❛❞♦✱ s✉❛ ❝r❡♥ç❛ ❢♦✐ ♣♦st❡r✐♦r♠❡♥t❡ ❞❡♠♦♥str❛❞❛ ❝♦♠♦ ❢❛❧s❛ ❝♦♠ ❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ ❢❛✲ t♦r❛çã♦ ❞❡ 225
+ 1 ♣r♦♣♦st❛ ♣♦r ▲❡♦♥❛r❞ ❊✉❧❡r✳ ❖s ♥ú♠❡r♦s ❞❡ ❋❡r♠❛t ♣r✐♠♦s sã♦ ❝❤❛♠❛❞♦s ❞❡ ♣r✐♠♦s ❞❡ ❋❡r♠❛t✳ ❆té ❤♦❥❡✱ ♥ã♦ s❡ s❛❜❡ s❡ ❡①✐st❡♠ ♦✉tr♦s ♣r✐♠♦s ❞❡ ❋❡r♠❛t ❛❧é♠ ❞♦s ❝✐♥❝♦ ♣r✐♠❡✐r♦s✳ ❯♠ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦ ❛❝❡r❝❛ ❞❡ss❡s ♥ú♠❡r♦s✱ ❛✜r♠❛ q✉❡ q✉❛✐sq✉❡r ❞♦✐s ♣r✐♠♦s ❞❡ ❋❡r♠❛t sã♦ r❡❧❛t✐✈❛♠❡♥t❡s ♣r✐♠♦s✱ ♦ q✉❡ ♥♦s ❧❡✈❛ ❛ ♠❛✐s ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞❡ q✉❡ ❤á ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐♠♦s✱ ♣♦✐s ❝❛❞❛ ♥ú♠❡r♦ ❞❡ ❋❡r♠❛t t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞✐✈✐s♦r ♣r✐♠♦ ❡ ❡ss❡s ❞✐✈✐s♦r❡s ♣r✐♠♦s sã♦ t♦❞♦s ❞✐st✐♥t♦s✳ Pr♦✈❛✿ ❈♦♥s✐❞❡r❡♠♦s ♦s ♥ú♠❡r♦s ❞❡ ❋❡r♠❛t✱ Fn = 22
n
+ 1 ♣❛r❛ n = 0,1,2,3,· · ·✳ ❱❛♠♦s ♠♦str❛r ❛ s❡❣✉✐♥t❡ r❡❝♦rrê♥❝✐❛✿
n−1 Y
k=0
Fk =Fn−2 (n≥1),
❯s❡♠♦s ✐♥❞✉çã♦ s♦❜r❡ ♥✳ P❛r❛ ♥ ❂ ✶✱ t❡♠♦s F0 = 3 ❡ F1 −2 = 3✳ P♦r ✐♥❞✉çã♦
❝♦♥❝❧✉í♠♦s q✉❡✿
n
Y
k=0
Fk= ( n−1 Y
k=0
Fk)Fn= (Fn−2).Fn= (22
n
−1)(22n
+ 1) = 22n+1
−1 =Fn+1−2
❙❡❣✉❡✲s❡ ❞❛ ❢ór♠✉❧❛ ❛♥t❡r✐♦r✱ q✉❡ ❞♦✐s ♥ú♠❡r♦s ❞❡ ❋❡r♠❛t ❞✐❢❡r❡♥t❡s sã♦ r❡❧❛t✐✈❛✲ ♠❡♥t❡s ♣r✐♠♦s✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ ❞ é ✉♠ ❞✐✈✐s♦r ❞❡ Fk ❡ Fn (k < n)✱ ❡♥tã♦ ❞ ❞✐✈✐❞❡ ✷
❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❞ ❂ ✶ ♦✉ ❞ ❂ ✷✳ ▼❛s✱ ❞ ❂ ✷ é ✐♠♣♦ssí✈❡❧✱ ❥á q✉❡ t♦❞♦ ♥ú♠❡r♦ ❞❡ ❋❡r♠❛t é í♠♣❛r✳ P♦rt❛♥t♦✱ ❝♦♠♦ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♥ú♠❡r♦s ❞❡ ❋❡r♠❛t✱ t♦❞♦s ❝♦♣r✐♠♦s ❞♦✐s ❛ ❞♦✐s✱ ❝♦♥❧✉í♠♦s q✉❡ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐♠♦s✳
✶✳✸ Pr♦✈❛ ✸
P❛r❛ ❛ s❡❣✉✐♥t❡ ♣r♦✈❛✱ ✉s❛r❡♠♦s ♦s ♥ú♠❡r♦s ❞❡ ▼❡rs❡♥♥❡ ✭Mp = 2p −1✱ ❝♦♠
✶✳✹✳ P❘❖❱❆ ✹ ✺
❆ss✐♠ ❝♦♠♦ ❋❡r♠❛t✱ ▼❡rs❡♥♥❡ ♣r♦♣ôs ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ s❡ ♦❜t❡r ♥ú♠❡r♦s ♣r✐♠♦s✳ ❯s❛♥❞♦ ❛ r❡❧❛çã♦Mp = 2p−1✱ ♦♥❞❡pé ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❡2≤p≤5000♦s ♥ú♠❡r♦s
❞❡ ▼❡rs❡♥♥❡ q✉❡ sã♦ ♣r✐♠♦s✱ ❝❤❛♠❛❞♦s ❞❡ ♣r✐♠♦s ❞❡ ▼❡rs❡♥♥❡✱ ❝♦rr❡s♣♦♥❞❡♠ ❛♦s s❡❣✉✐♥t❡s ✈❛❧♦r❡s ❞❡ p✿ ✸✱ ✺✱ ✼✱ ✶✸✱ ✶✾✱ ✸✶✱ ✻✶✱ ✽✾✱ ✶✵✼✱ ✶✷✼✱ ✺✷✶✱ ✶✷✼✾✱ ✷✷✵✸✱ ✷✷✽✶✱ ✸✷✶✼✱ ✹✷✺✸ ❡ ✹✹✷✸✳ ▼❡rs❡♥♥❡ s❛❜✐❛ q✉❡ s❡ n é ❝♦♠♣♦st♦✱ ❡♥tã♦ Mn t❛♠❜é♠ ♦ é✳ ❏á
s❡ n é ♣r✐♠♦✱ ♥❡♠ s❡♠♣r❡ Mn é ♣r✐♠♦ ✭211−1 = 2047 = 23·89é ❝♦♠♣♦st♦✮✳ ❆té
❞❡③❡♠❜r♦ ❞❡ ✷✵✵✶✱ ♦ ♠❛✐♦r ♣r✐♠♦ ❞❡ ▼❡rs❡♥♥❡ ❝♦♥❤❡❝✐❞♦ ❡r❛M13466917✱ q✉❡ ♣♦ss✉✐ ♥♦
s✐st❡♠❛ ❞❡❝✐♠❛❧ ✹✵✺✸✾✹✻ ❞í❣✐t♦s✱ ❡ é ♦ tr✐❣és✐♠♦ ♥♦♥♦ ♣r✐♠♦ ❞❡ ▼❡rs❡♥♥❡ ❝♦♥❤❡❝✐❞♦✳ Pr♦✈❛✿ ❙✉♣♦♥❤❛♠♦s q✉❡ P é ✜♥✐t♦✳ ❙❡❥❛ p ♦ ♠❛✐♦r ♥ú♠❡r♦ ♣r✐♠♦✳ ❈♦♥s✐❞❡r❡✲ ♠♦s ♦ ♥ú♠❡r♦ ❞❡ ▼❡rs❡♥♥❡ 2p −1✳ ▼♦str❛r❡♠♦s q✉❡ q✉❛❧q✉❡r ❢❛t♦r ♣r✐♠♦ q ❞❡
2p−1s❡❥❛ ♠❛✐♦r ❞♦ q✉❡p✱ ♦ q✉❡ ♥♦s ❧❡✈❛rá ❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱
❝♦♥❝❧✉✐r❡♠♦s q✉❡ P é ✐♥✜♥✐t♦✳ ❙❡❥❛ q ✉♠ ♣r✐♠♦ q✉❡ ❞✐✈✐❞❡ 2p −1✱ ❞❡ ❢♦r♠❛ q✉❡
2p ≡1 ♠♦❞ q✳ ❯♠❛ ✈❡③ q✉❡ p é ♣r✐♠♦✱ s✐❣♥✐✜❝❛ q✉❡ ♦ ❡❧❡♠❡♥t♦ ✷ t❡♠ ♦r❞❡♠ p ♥♦
❣r✉♣♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ Zq\{0} ❞♦ ❝♦r♣♦ Zq✳ ❊ss❡ ❣r✉♣♦ t❡♠ q −1 ❡❧❡♠❡♥t♦s✳ P❡❧♦
t❡♦r❡♠❛ ❞❡ ▲❛❣r❛♥❣❡ ✭✈❡❥❛ ❈✳✶ ❞♦ ❛♣ê♥❞✐❝❡✮✱ s❛❜❡♠♦s q✉❡ ❛ ♦r❞❡♠ ❞❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞✐✈✐❞❡ ❛ ♦r❞❡♠ ❞♦ ❣r✉♣♦✱ ♦✉ s❡❥❛✱ p|q−1✱ ❡ ❞❛í✱ p < q✳
✶✳✹ Pr♦✈❛ ✹
●❛✉ss✱ ❛ss✐♠ ❝♦♠♦ ♦✉tr♦s ♠❛t❡♠át✐❝♦s✱ s❡♠♣r❡ ❜✉s❝♦✉ r❡s♣♦♥❞❡r ♣❡r❣✉♥t❛s r❡❧❛✲ ❝✐♦♥❛❞❛s ❛ ❞✐str✐❜✉✐çã♦ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ ◗✉❛♥t♦s ♥ú♠❡r♦s ♣r✐♠♦s ❡①✐st❡♠ ❡♥tr❡ ✶ ❡ ✉♠ ♥ú♠❡r♦xq✉❛❧q✉❡r❄ ●❛✉ss tr❛❜❛❧❤♦✉ ❝♦♠ ✉♠❛ ❢✉♥çã♦✱ q✉❡ ♣♦st❡r✐♦r♠❡♥t❡ ❢♦✐ ❞❡♥♦t❛❞❛ ♣♦r π(x)✱ ❞❡✜♥✐❞❛ ❝♦♠♦ ♦ ♥ú♠❡r♦ ❞❡ ♣r✐♠♦s q✉❡ sã♦ ♠❡♥♦r❡s q✉❡ ♦✉ ✐❣✉❛✐s ❛♦ ♥ú♠❡r♦ r❡❛❧ x✱ π(x) := ★ {p ≤x : p∈ P}✱ ❝❤❛♠❛❞❛
❞❡ ❢✉♥çã♦ ❞❡ ❝♦♥t❛❣❡♠ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s✳ ❆ss✐♠ t❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✿ π(1) = 0✱ π(2) = 1✱ π(3) = 2✱ π(5) = 3✱ π(10) = 4✱ π(100) = 25✱ π(1000) = 168✱ π(√2) = 0✱ π(e) = 1✱ ❡t❝✳ ❆ss✐♠✱ ❛ ♣r♦♣♦rçã♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ❡♥tr❡ ✶ ❡ x é ❞❛❞❛ ♣♦r π(x)
x ✳ ▼❛t❡♠át✐❝♦s ❜✉s❝❛r❛♠ ❛❝❤❛r ❜♦❛s ❛♣r♦①✐♠❛çõ❡s ♣❛r❛ π(x) ♣♦r ❢✉♥çõ❡s ❝♦♥tí♥✉❛s✳ ❊♠ ✶✾✼✷✱ ●❛✉ss ❝♦♥❥❡❝❝t✉r♦✉ q✉❡ π(x) ❡r❛ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❛❞❡r❡♥t❡ ❛ ❢✉♥çã♦ ✐♥✲ t❡❣r❛❧ ❧♦❣❛rít♠✐❝❛ ✭f(x) = Rx
2
dt
lnt✮✱ s❡♥❞♦ ♣r♦✈❛❞❛ ❡♠ ✶✽✾✻ ♣♦r ❍❛❞❛♠❛r❞ ❡ ❉❡
▲❛ ❱❛❧❧é❡ P♦✉ss✐♥✳ ❈♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ ❞♦ ◆ú♠❡r♦ Pr✐♠♦ q✉❡ ❛✜r♠❛ q✉❡ ♦ lim
x→∞ π(x)
x/lnx = 1 ✳
●❛✉ss q✉❛♥❞♦ ❝♦♥❥❡❝t✉r♦✉ ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ π(x) ♣❡❧❛ ❢✉♥çã♦ ✐♥t❡❣r❛❧ ❧♦❣❛rít♠✐❝❛✱ ❝♦♥t❛✈❛ ❛♣❡♥❛s ❝♦♠ ✶✺ ❛♥♦s ❞❡ ✐❞❛❞❡✳ ❊♠ ✶✼✾✽✱ ✐♥s♣✐r❛❞♦ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ◆ú♠❡r♦ Pr✐♠♦✱ ▲❡❣❡♥❞r❡ ❝♦♥❥❡❝t✉r♦✉ q✉❡ π(x)∼ x
✶✳✹✳ P❘❖❱❆ ✹ ✻
❡①✐st❡ ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐♠♦s ✉s❛♥❞♦ ❈á❝✉❧♦ ❊❧❡♠❡♥t❛r✳
Pr♦✈❛✿ ❈♦♥s✐❞❡r❡♠♦s π(x) := ★ {p ≤ x : p ∈ P}✳ ❊♥✉♠❡r❡♠♦s ♦s ♣r✐♠♦s
P = {p1, p2, ...} ❡♠ ♦r❞❡♠ ❝r❡s❝❡♥t❡✳ ❈♦♥s✐❞❡r❡ ♦ ❧♦❣❛r✐t♠♦ ♥❛t✉r❛❧ logx✱ ❞❡✜♥✐❞♦
❝♦♠♦ logx = Rx 1
1
tdt✳ ■r❡♠♦s ❝♦♠♣❛r❛r ❛ ár❡❛ ❛❜❛✐①♦ ❞♦ ❣rá✜❝♦ ❞❡ f(t) =
1
t ❝♦♠
✉♠❛ ❢✉♥çã♦ ❡s❝❛❞❛ s✉♣❡r✐♦r ✭♣❛r❛ ❡ss❡ ♠ét♦❞♦✱ ✈❡❥❛ ❉ ❞♦ ❛♣ê♥❞✐❝❡✮✳ ❆ss✐♠✱ ♣❛r❛ n ≤x≤n+ 1 ♥ós t❡♠♦s✿
logx≤1 + 1 2 +
1
3 +...+ 1 n−1+
1 n ≤
P 1
m
♦♥❞❡ ❛ s♦♠❛ s❡ ❡st❡♥❞❡ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧ m q✉❡ t❡♥❤❛ ❛♣❡♥❛s ❞✐✈✐s♦r❡s ♣r✐♠♦s p ≤x✳ ❯♠❛ ✈❡③ q✉❡ ❝❛❞❛ m ♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❡ ❢♦r♠❛ ú♥✐❝❛ ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ❞❛ ❢♦r♠❛ Y
p6x
pkp✱ ✈❡♠♦s q✉❡ ❛ ú❧t✐♠❛ s♦♠❛ é ✐❣✉❛❧ ❛
Y
p∈P, p6x
X
k≥0
1 pk
.
❖ s♦♠❛tór✐♦ é ✉♠❛ sér✐❡ ❣❡♦♠étr✐❝❛ ❝♦♠ r❛③ã♦ 1
p✱ ❞❡ ♦♥❞❡
logx≤ Y
p∈P, p6x
1 1−1p =
Y
p∈P, p6x
p p−1 =
π(x) Y
k=1
pk
pk−1
.
❈♦♠♦ pk≥k+ 1✱ t❡♠♦s✿
pk
pk−1
= 1 + 1 pk−1 ≤
1 + 1 k =
k+ 1 k ❡ ♣♦rt❛♥t♦
logx≤
π(x) Y
k=1
k+ 1
✶✳✺✳ P❘❖❱❆ ✺ ✼
❙❛❜❡♠♦s q✉❡ logx ♥ã♦ é ❧✐♠✐t❛❞♦✱ ❞❛í ❝♦♥❝❧✉í♠♦s q✉❡ π(x) é ✐❧✐♠✐t❛❞♦ ❡ ❛ss✐♠ ❝♦♥❝❧✉✐✲s❡ q✉❡ ❤á ✉♠ ♥ú♠❡r♦ ✐♥✜♥✐t♦ ❞❡ ♣r✐♠♦s✳
✶✳✺ Pr♦✈❛ ✺
❯♠❛ ♦✉tr❛ ❞❡♠♦♥str❛çã♦ ❞❡ q✉❡ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐♠♦s ❡stá r❡❧❛✲ ❝✐♦♥❛❞❛ à ❚♦♣♦❧♦❣✐❛ ✲ r❛♠♦ ❞❛ ▼❛t❡♠át✐❝❛ ♥♦ q✉❛❧ sã♦ ❡st✉❞❛❞❛s✱ ❝♦♠ ❣r❛♥❞❡ ❣❡✲ ♥❡r❛❧✐❞❛❞❡✱ ❛s ♥♦çõ❡s ❞❡ ❧✐♠✐t❡s✱ ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡ ❡ ❛s ✐❞❡✐❛s ❝♦♠ ❡❧❛s r❡❧❛❝✐♦♥❛❞❛s✳ ❚❛❧ ❞❡♠♦♥str❛çã♦ ❢♦✐ ♣r♦♣♦st❛ ♣♦r ❋r✉st❡♥❜❡r❞✱ ♠❛t❡♠át✐❝♦ ✐sr❛❡❧❡♥s❡✱ ♠✉✐t♦ r❡s✲ ♣❡✐t❛❞♦ ♣❡❧♦s r❡s✉❧t❛❞♦s q✉❡ ♦❜t❡✈❡ ♥❛ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡ t❡♦r✐❛ ❡r❣ó❞✐❝❛✱ q✉❡ ✜❝♦✉ ❢❛♠♦s♦ ❧♦❣♦ ♥♦ ❝♦♠❡ç♦ ❞❛ ❝❛rr❡✐r❛ ❛♦ ♣✉❜❧✐❝❛r✱ ❡♠ ✶✾✺✺✱ q✉❛♥❞♦ ❡❧❡ ❡r❛ ❛♣❡♥❛s ✉♠ ❛❧✉♥♦ ❞❡ ❣r❛❞✉❛çã♦✱ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞❛ ✐♥✜♥✐t✉❞❡ ❞♦s ♣r✐♠♦s ✉s❛♥❞♦ ❛♣❡♥❛s ❛ ❞❡✜♥✐çã♦ ❞❡ t♦♣♦❧♦❣✐❛ ❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ s❡q✉ê♥❝✐❛s ✐♥✜♥✐t❛s ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❊ss❛ ❞❡♠♦♥str❛çã♦ ✜❝♦✉ ❢❛♠♦s❛✱ ♣♦✐s ❡❧❛ ❡stá ❜❡♠ ❧♦♥❣❡ ❞❛s ❞❡♠♦♥str❛çõ❡s r♦t✐♥❡✐r❛s ❞❛ ár❡❛ ❞❡ t♦♣♦❧♦❣✐❛✿ tr❛t❛✲s❡ ❞❡ ✉♠❛ té❝♥✐❝❛ t♦♣♦❧ó❣✐❝❛ ❛♣❧✐❝❛❞❛ à t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✦ P♦r ✐ss♦ ♠❡s♠♦ é ❝♦♥s✐❞❡r❛❞❛ ♠✉✐t♦ ❡str❛♥❤❛ ❡ ❢❛s❝✐♥❛♥t❡✳ P❛r❛ t❛❧ ❡♥t❡♥❞✐♠❡♥t♦✱ é ✐♠♣♦rt❛♥t❡ ❝♦♥❤❡❝❡r♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ✉t✐❧✐③❛❞♦s ❡♠ t❛❧ ♣r♦✈❛ ✭✈❡❥❛ ❊ ❞♦ ❛♣ê♥❞✐❝❡✮✳
Pr♦✈❛✿ ❱❛♠♦s ✐♥tr♦❞✉③✐r ✉♠❛ t♦♣♦❧♦❣✐❛ ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s Z✳ P❛r❛ a, b∈Z, b >0✱ ❢❛ç❛♠♦s
Na,b={a+nb:n ∈Z}✳
❉✐③❡♠♦s q✉❡ ✉♠ ❝♦♥❥✉♥t♦O ⊆Zé ❛❜❡rt♦ s❡O é ✈❛③✐♦ ♦✉ s❡✱ ♣❛r❛ ❝❛❞❛a∈O✱ ❡①✐st❡ ❛❧❣✉♠b > 0❝♦♠Na,b ⊆O✳ ❈❧❛r❛♠❡♥t❡ ❛ ✉♥✐ã♦ ❞❡ ❝♦♥❥✉♥t♦s ❛❜❡rt♦s é t❛♠❜é♠ ✉♠
❝♦♥❥✉♥t♦ ❛❜❡rt♦✳ ❙❡ O1, O2 sã♦ ❛❜❡rt♦s ❡ a∈O1∩O2 ❝♦♠ Na,b1 ⊆O1 ❡Na,b2 ⊆O2✱ ❡♥tã♦ a ∈ Na,b1b2 ⊆ O1 ∩O2✳ ❊♥tã♦ ❝♦♥❝❧✉í♠♦s q✉❡ q✉❛q✉❡r ✐♥t❡rs❡❝çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❝♦♥❥✉♥t♦s ❛❜❡rt♦s é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✳ ❆ss✐♠✱ ❡ss❛ ❢❛♠í❧✐❛ ❞❡ ❝♦♥❥✉♥t♦s ❛❜❡rt♦s ✐♥❞✉③ ✉♠❛ t♦♣♦❧♦❣✐❛ ❡♠ Z✳ ❈♦♥✈é♠ ♦❜s❡r✈❛r ❞♦✐s ❢❛t♦s✿
❛✮ ◗✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ♥ã♦ ✈❛③✐♦ é ✐♥✜♥✐t♦✳ ❜✮ ◗✉❛❧q✉❡r ❝♦♥❥✉♥t♦ Na,b t❛♠❜é♠ é ❢❡❝❤❛❞♦✳
❖ ♣r✐♠❡✐r♦ ❢❛t♦ ❞❡❝♦rr❡ ❞❛ ❞❡✜♥✐çã♦✳ ◗✉❛♥t♦ ❞♦ s❡❣✉♥❞♦✱ ♦❜s❡r✈❡ q✉❡✿
Na,b=Z\ b−1 [
i=1
Na+i,b
♦ q✉❡ ♣r♦✈❛ q✉❡ Na,b é ♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡✱ ♣♦rt❛♥t♦✱ ❢❡❝❤❛❞♦✳
✶✳✻✳ P❘❖❱❆ ✻ ✽
q✉❡ q✉❛❧q✉❡r ♥ú♠❡r♦ n 6= 1,−1 t❡♠ ✉♠ ❞✐✈✐s♦r ♣r✐♠♦ p ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❡stá ❝♦♥t✐❞♦ ❡♠ N0,p✱ ❝♦♥❝❧✉í♠♦s ❡♥tã♦ q✉❡
Z\{1,−1}= [
p∈P
N0,p
❆❣♦r❛ s❡ P ❢♦ss❡ ✜♥✐t♦✱ ❡♥tã♦ [
p∈P
N0,p s❡r✐❛ ✉♠❛ ✉♥✐ã♦ ✜♥✐t❛ ❞❡ ❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦s
✭♣♦r ❜✮✱ ❡ ♣♦rt❛♥t♦ ❢❡❝❤❛❞♦✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ {−1,1} s❡r✐❛ ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✱ ❡♠ ❝♦♥tr❛❞✐çã♦ ❝♦♠ ✭❛✮✳ P♦rt❛♥t♦✱ P é ✐♥✜♥✐t♦✳
✶✳✻ Pr♦✈❛ ✻
◆♦ss❛ ♣r♦✈❛ ✜♥❛❧ ❞á ✉♠ ❝♦♥s✐❞❡rá✈❡❧ ♣❛ss♦ ❛❞✐❛♥t❡ ❡ ❞❡♠♦♥str❛ ♥ã♦ s♦♠❡♥t❡ q✉❡ ❤á ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐♠♦s✱ ♠❛s t❛♠❜é♠ q✉❡ ❛ sér✐❡X
p∈P
1
p ❞✐✈❡r❣❡✳ ❆ ♣r✐♠❡✐r❛ ♣r♦✈❛ ❞❡ss❡ r❡s✉❧t❛❞♦ ✐♠♣♦rt❛♥t❡ ❢♦✐ ❞❛❞❛ ♣♦r ❊✉❧❡r ✭❡ é ✐♥t❡r❡ss❛♥t❡ ❡♠ s✐ ♠❡s♠❛✮✱ ♠❛s ♥♦ss❛ ♣r♦✈❛✱ ❝♦♥❝❡❜✐❞❛ ♣♦r ❊r❞➤s✱ é ❞❡ ✉♠❛ ❜❡❧❡③❛ ✐rr❡s✐stí✈❡❧✳
✶✳✻✳ P❘❖❱❆ ✻ ✾
❞❡ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❡①tr❛♦r❞✐♥❛r✐❛♠❡♥t❡ ❞✐❢í❝❡✐s✳ ❖ s❡✉ ❡st✐❧♦ ❝❛r❛❝t❡ríst✐❝♦ ❝♦♥✲ s✐st✐❛ ❡♠ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❡ ✉♠❛ ❢♦r♠❛ ❡❧❡❣❛♥t❡ ❡ ✈✐s✐♦♥ár✐❛✳ ❘❡❝❡❜❡✉ ♦ Pré♠✐♦ ❈♦❧❡ ❞❛ ❙♦❝✐❡❞❛❞❡ ❆♠❡r✐❝❛♥❛ ❞❡ ▼❛t❡♠át✐❝❛ ❡♠ ✶✾✺✶ ♣❡❧♦s s❡✉s ♠✉✐t♦s ❛rt✐❣♦s ❡♠ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ❡ ❡♠ ♣❛rt✐❝✉❧❛r ♣❡❧♦ ❛rt✐❣♦ ✏❖♥ ❛ ♥❡✇ ♠❡t❤♦❞ ✐♥ ❡❧❡♠❡♥t❛r② ♥✉♠❜❡r t❤❡♦r② ✇❤✐❝❤ ❧❡❛❞s t♦ ❛♥ ❡❧❡♠❡♥t❛r② ♣r♦♦❢ ♦❢ t❤❡ ♣r✐♠❡ ♥✉♠❜❡r t❤❡♦r❡♠✑✱ ♣✉❜❧✐❝❛❞♦ ♥♦s Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ◆❛t✐♦♥❛❧ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ❡♠ ✶✾✹✾✳ ❊r❞➤s ♦❝✉♣♦✉ ♦✜❝✐❛❧♠❡♥t❡ ♣♦s✐çõ❡s ❡♠ ✈ár✐❛s ✉♥✐✈❡rs✐❞❛❞❡s ❞❡ ■sr❛❡❧✱ ❊st❛❞♦s ❯♥✐❞♦s ❡ ❘❡✐♥♦ ❯♥✐❞♦✳ ❊ss❛s ♣♦s✐çõ❡s ❡r❛♠ ❛♣❡♥❛s ❢♦r♠❛✐s✳ ◆❛ r❡❛❧✐❞❛❞❡ ❡❧❡ ❡r❛ ✉♠ ♥ó♠❛❞❛ s❡♠ ♦❜❥❡t✐✈♦s ❞❡✜♥✐❞♦s✱ ✈✐❛❥❛♥❞♦ ♣❡❧❛s ✉♥✐✈❡rs✐❞❛❞❡s ♠❛✐s ♣r❡st✐❣✐❛❞❛s✳ ❚r❛❜❛❧❤❛✈❛ ♦❜s❡ss✐✈❛♠❡♥t❡✱ ❞♦r♠✐❛ ✹ ❛ ✺ ❤♦r❛s ♣♦r ❞✐❛ ❡ t♦♠❛✈❛ ❛♥❢❡t❛♠✐♥❛s ♣❛r❛ ♠❛♥t❡r ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ tr❛❜❛❧❤♦✳ ❆ ❞❛❞❛ ❛❧t✉r❛✱ ✉♠ ❛♠✐❣♦ ❞❡s❛✜♦✉✲♦ ❛ ♥ã♦ t♦♠❛r ❛ ❞r♦❣❛ ❞✉r❛♥t❡ ✉♠ ♠ês❀ ❡❧❡ q✉❡✐①♦✉✲s❡ ♠❛✐s t❛r❞❡ q✉❡ ❞✉r❛♥t❡ ❡ss❡ ♠ês ❛ s✉❛ ♣r♦❞✉t✐✈✐❞❛❞❡ ❜❛✐①❛r❛ ✐♠❡♥s❛♠❡♥t❡✳ ❊r❞➤s r❡❝❡❜❡✉ ♠✉✐t♦s ♣rê♠✐♦s✱ ✐♥❝❧✉✐♥❞♦ ♦ Prê♠✐♦ ❲♦❧❢ ❞❡ ▼❛t❡♠át✐❝❛ ❞❡ ✶✾✽✸✳ ◆♦ ❡♥t❛♥t♦✱ ❞❡✈✐❞♦ ❛♦ s❡✉ ❡st✐❧♦ ❞❡ ✈✐❞❛✱ ♣r❡❝✐s❛✈❛ ❞❡ ♣♦✉❝♦ ❞✐♥❤❡✐r♦✳ P♦r ✐ss♦ ❛❥✉❞♦✉ ❡st✉❞❛♥t❡s t❛❧❡♥t♦s♦s ❡ ♦❢❡r❡❝❡✉ ♣ré♠✐♦s ♣❡❧❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ♣r♦♣♦st♦s ♣♦r ❡❧❡ ✳ ▼♦rr❡✉ ❡♠ ❱❛rsó✈✐❛✱ P♦❧ó♥✐❛ ❛ ✷✵ ❞❡ s❡t❡♠❜r♦ ❞❡ ✶✾✾✻✳
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ p1, p2, p3, ... ❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ❡♠ ♦r❞❡♠ ❝r❡s✲
❝❡♥t❡ ❡ ❛ss✉♠✐r❡♠♦s q✉❡ X
p∈P
1
p ❝♦♥✈❡r❣❡✳ ❊♥tã♦ ❞❡✈❡ ❡①✐st✐r ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ k t❛❧ q✉❡ X
i>k+1
1 pi
< 1
2✳ ❈❤❛♠❛r❡♠♦s p1, ..., pk ❞❡ ♣r✐♠♦s ♣❡q✉❡♥♦s ❡ pk+1, pk+2, ... ❞❡ ♣r✐♠♦s ❣r❛♥❞❡s✳ P❛r❛ ✉♠ ♥❛t✉r❛❧ ❛r❜✐trár✐♦ N ♥ós ❡♥❝♦♥tr❛r❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡
X
i≥k+1
N pi
< N
2. ✭✶✳✶✮
❙❡❥❛ Nb ♦ ♥ú♠❡r♦ ❞❡ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦sn ≤N q✉❡ sã♦ ❞✐✈✐sí✈❡✐s ♣♦r ♣❡❧♦ ♠❡♥♦s ✉♠
♣r✐♠♦ ❣r❛♥❞❡ ❡ Ns ♦ ♥ú♠❡r♦ ❞❡ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s n≤N q✉❡ t❡♠ s♦♠❡♥t❡ ❞✐✈✐s♦r❡s
♣r✐♠♦s ♣❡q✉❡♥♦s✳ ▼♦str❡♠♦s q✉❡ ♣❛r❛ ✉♠ N ❛❞❡q✉❛❞♦✱ Nb +Ns < N✱ q✉❡ s❡rá
♥♦ss❛ ❝♦♥tr❛❞✐çã♦✱ ✉♠❛ ✈❡③ q✉❡✱ ♣♦r ❞❡✜♥✐çã♦✱ Nb+Nst❡r✐❛ q✉❡ s❡r ✐❣✉❛❧ ❛N✳ P❛r❛
❡st✐♠❛r Nb ♥♦t❡ q✉❡⌊Npi⌋ ❝♦♥t❛ ♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦sn 6N q✉❡ sã♦ ♠ú❧t✐♣❧♦s ❞❡pi✳
❉❛í ♣♦r ✭✶✳✶✮ ♥ós ♦❜t❡♠♦s
Nb ≤
X
i≥k+1 jN
pi
k
< N
2. ✭✶✳✷✮
❱❛♠♦s ♦❧❤❛r ♣❛r❛ Ns✳ ❊s❝r❡✈❡♠♦s ❝❛❞❛ n ≤ N q✉❡ t❡♠ s♦♠❡♥t❡ ❞✐✈✐s♦r❡s ♣r✐♠♦s
♣❡q✉❡♥♦s ❞❛ ❢♦r♠❛ n = anb2n✱ ♦♥❞❡ an é ❛ ♣❛rt❡ ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s✳ ❚♦❞♦ an
é ♣♦rt❛♥t♦✱ ✉♠ ♣r♦❞✉t♦ ❞❡ ❞✐❢❡r❡♥t❡s ♣r✐♠♦s ♣❡q✉❡♥♦s ❡ ♥ós ❝♦♥❝❧✉í♠♦s q✉❡ ❤á ♣r❡❝✐s❛♠❡♥t❡2k ❞✐❢❡r❡♥t❡s ♣❛rt❡s ❧✐✈r❡s ❞❡ q✉❛❞r❛❞♦s✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦b
✶✳✻✳ P❘❖❱❆ ✻ ✶✵
√
N✱ ❞❡s❝♦❜r✐♠♦s q✉❡ sã♦ ♥♦ ♠á①✐♠♦ √N ♣❛rt❡s q✉❛❞r❛❞❛s ❡ ❛ss✐♠ Ns ≤ 2k √
N✳ ❈♦♠♦ ✭✶✳✷✮ ✈❛❧❡ ♣❛r❛ q✉❛❧q✉❡r ◆✱ r❡st❛ ❡♥❝♦♥tr❛r ✉♠ ♥ú♠❡r♦ ◆ ❝♦♠2k√N ≤ N
2 ♦✉
2k+1 ≤√N✱ ❛ss✐♠N = 22k+2✳
❈❛♣ít✉❧♦ ✷
P♦st✉❧❛❞♦ ❞❡ ❇❡rtr❛♥❞
❏á ✈✐♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ✷✱ ✸✱ ✺✱ ✼✱ ✳✳✳ é ✐♥✜♥✐t❛✳ P❛r❛ ✈❡r q✉❡ ♦ t❛♠❛♥❤♦ ❞❛s ❧❛❝✉♥❛s ❡♥tr❡ ✉♠ ♣r✐♠♦ ❡ ♦✉tr♦ ♥ã♦ é ❧✐♠✐t❛❞♦✱ ✈❛♠♦s ✉s❛r N := 2·3·5·. . .·p ♣❛r❛ ❞❡♥♦t❛r ♦ ♣r♦❞✉t♦ ❞❡ t♦❞♦s ♦s ♣r✐♠♦s q✉❡ sã♦ ♠❡♥♦r❡s q✉❡ k+ 2✱ ❡ ✈❡r✐✜❝❛r q✉❡ ♥❡♥❤✉♠ ❞♦s k ♥ú♠❡r♦s
N + 2, N + 3, N + 4,· · · , N + (k+ 1)
é ♣r✐♠♦✱ ✉♠❛ ✈❡③ q✉❡✱ ♣❛r❛ 2≤i≤k+ 1✱ s❛❜❡♠♦s q✉❡ i t❡♠ ✉♠ ❢❛t♦r ♣r✐♠♦ q✉❡ é ♠❡♥♦r q✉❡ k+ 2✱ ❡ ❡ss❡ ❢❛t♦r t❛♠❜é♠ ❞✐✈✐❞❡N ❡✱ ❝♦s❡q✉❡♥t❡♠❡♥t❡✱ t❛♠❜é♠ ❞✐✈✐❞❡ N +i✳ ❈♦♠ ❡ss❛ r❡❝❡✐t❛✱ ❡♥❝♦♥tr❛♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❛r❛ k = 10✱ q✉❡ ♥❡♥❤✉♠ ❞♦s ❞❡③ ♥ú♠❡r♦s
2.312,2.313,2.314, . . . ,2.321 é ♣r✐♠♦✳
▼❛s t❛♠❜é♠ ❡①✐st❡♠ ❧✐♠✐t❛♥t❡s s✉♣❡r✐♦r❡s ♣❛r❛ ❛s ❧❛❝✉♥❛s ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ✉♠❛ ❢❛♠♦s❛ ❡st✐♠❛t✐✈❛ ❡st❛❜❡❧❡❝❡ q✉❡ ✏❛ ❧❛❝✉♥❛ ❛té ♦ ♣ró①✐♠♦ ♣r✐♠♦ ♥ã♦ ♣♦❞❡ s❡r ♠❛✐♦r q✉❡ ♦ ♥ú♠❡r♦ ❝♦♠ ♦ q✉❛❧ ✐♥✐❝✐❛♠♦s ♥♦ss❛ ❜✉s❝❛✑✳❵❊❧❛ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ♣♦st✉❧❛❞♦ ❞❡ ❇❡tr❛♥❞✱ ✉♠❛ ✈❡③ q✉❡ ❢♦✐ ❝♦♥❥❡❝t✉r❛❞❛ ❡ ✈❡r✐✜❝❛❞❛ ❡♠♣✐r✐❝❛♠❡♥t❡ ♣❛r❛ n < 3.000.000 ♣♦r ❇❡rtr❛♥❞✳ ❏♦s❡♣❤ ▲♦✉✐s ❇❡rtr❛♥❞ ✭✶✽✷✷✲✶✾✵✵✮ ❢♦✐ ✉♠ ❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ♠❛t❡♠át✐❝♦s ❡ ❣❡ô♠❡tr❛s ❞❛ ❋r❛♥ç❛✱ ❝♦♠ ✐♠♣♦rt❛♥t❡s tr❛❜❛❧❤♦s ♣✉❜❧✐❝❛❞♦s ❡♠ ❣❡♦♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧ ❡ t❡♦r✐❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s✳ ❋♦✐ ❝♦♥s✐❞❡r❛❞♦ ✉♠ ❣ê♥✐♦ ♣r❡❝♦❝❡✱ ♣♦✐s ❛♦s ♥♦✈❡ ❛♥♦s ❞❡ ✐❞❛❞❡ ❡❧❡ ❡♥t❡♥❞❡✉ á❧❣❡❜r❛ ❡ ❣❡♦♠❡tr✐❛ ❡❧❡✲ ♠❡♥t❛r✱ ❜❡♠ ❝♦♠♦ ❢♦✐ ❝❛♣❛③ ❞❡ ❢❛❧❛r ❧❛t✐♠ ✢✉❡♥t❡♠❡♥t❡✳ ❉♦✐s ❛♥♦s ❞❡♣♦✐s✱ q✉❛♥❞♦ t✐♥❤❛ ♦♥③❡ ❛♥♦s✱ ❡❧❡ r❡❝❡❜❡✉ ♣❡r♠✐ssã♦ ♣❛r❛ ❛ss✐st✐r ❛ ♣❛❧❡str❛s ♥❛ ❊s❝♦❧❛ P♦❧✐té❝✲ ♥✐❝❛✳ ❉❡❢❡♥❞❡✉ s✉❛ t❡s❡ ❞❡ ❞♦✉t♦r❛❞♦ ❡♠ t❡r♠♦❞✐♥â♠✐❝❛ ❝♦♠ ❞❡③❡ss❡t❡ ❛♥♦s ❞❡ ✐❞❛❞❡✱ s❡♥❞♦ ❛❞♠✐t✐❞♦ ❡♠ s❡❣✉✐❞❛ ❝♦♠♦ ♣r♦❢❡ss♦r ❞❛ ❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛✳ ◆♦ ❈♦❧✲ ❧è❣❡ ❞❡ ❋r❛♥❝❡✱ ✐♥st✐t✉✐çã♦ ♠❛✐s ♣r❡st✐❣✐♦s❛ ❞♦ ♣❛ís✱ ❡♥s✐♥♦✉ ❢ís✐❝❛ ❡ ♠❛t❡♠át✐❝❛ ❞✉r❛♥t❡ q✉❛s❡ ❝✐♥q✉❡♥t❛ ❛♥♦s✳ ❚❛♠❜é♠ ❧❡❝✐♦♥♦✉ ♥❛ ❊s❝♦❧❛ ❞❡ ▼✐♥❛s ❡ ♥❛ ❊s❝♦❧❛ ◆♦r♠❛❧ ❙✉♣❡r✐♦r✳ ❊♥tr♦✉ ♣❛r❛ ❛ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛ ❛♦ ❢♦r♠✉❧❛r ❡ r❡s♦❧✈❡r ♦
❈❆P❮❚❯▲❖ ✷✳ P❖❙❚❯▲❆❉❖ ❉❊ ❇❊❘❚❘❆◆❉ ✶✷
❝❤❛♠❛❞♦ ✏♣r♦❜❧❡♠❛ ❞❡ ❇❡rtr❛♥❞✑ ❡ ❛♦ ❞❡s❝r❡✈❡r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛q✉❡❧❛s q✉❡ ♣❛s✲ s❛r❛♠ ❛ s❡r ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ✏❝✉r✈❛s ❞❡ ❇❡rtr❛♥❞✑✳ ❊♠ ✶✽✺✻ ❢♦✐ ❡❧❡✐t♦ ♠❡♠❜r♦ ❞❛ ❆❝❛❞❡♠✐❛ ❞❡ ❈✐ê♥❝✐❛s✱ t♦r♥❛♥❞♦✲s❡ s❡✉ s❡❝r❡tár✐♦ ♣❡r♣ét✉♦ ❛ ♣❛rt✐r ❞❡ ✶✽✼✹✳ ❊♠ ✶✽✽✹ t♦r♥♦✉✲s❡ ♠❡♠❜r♦ t❛♠❜é♠ ❞❛ ❆❝❛❞❡♠✐❛ ❋r❛♥❝❡s❛ ♥♦ ❧✉❣❛r ❞❡ ❏❡❛♥✲❇❛♣t✐st❡ ❉✉♠❛s✳ ❊ss❛s ❛❧t❛s ♣♦s✐çõ❡s ❛❝❛❞ê♠✐❝❛s✱ ❝♦♠❜✐♥❛❞❛s ❝♦♠ s✉❛ ❡r✉❞✐çã♦✱ s✉❛ ❡❧♦✲ q✉ê♥❝✐❛ ❡ s❡✉ ❝❤❛r♠❡✱ ❝♦❧♦❝❛r❛♠✲♥♦ ❡♠ ✉♠❛ ♣♦s✐çã♦ ❞❡ ❣r❛♥❞❡ ♣r♦❡♠✐♥ê♥❝✐❛ ♥♦ ❝❡♥ár✐♦ ❢r❛♥❝ês ♥❛ s❡❣✉♥❞❛ ♠❡t❛❞❡ ❞♦ sé❝✉❧♦ ❳■❳✳ ❘❡✉♥✐✉ ❡♠ t♦r♥♦ ❞❡ s✐ ✉♠ ❝ír✲ ❝✉❧♦ ❝✉❧t✉r❛❧ ❞❡ ❣r❛♥❞❡ ♣r❡stí❣✐♦✳ ❋♦✐ ♠❡♠❜r♦ ❞❛ ▲❡❣✐ã♦ ❞❡ ❍♦♥r❛✳ ❉❡ ✶✽✻✺ ❛té s✉❛ ♠♦rt❡✱ ❇❡rtr❛♥❞ ❡❞✐t♦✉ ♦ ✏❏♦✉r♥❛❧ ❞❡s ❙❛✈❛♥ts✑✳ ❊s❝r❡✈❡✉ ✐♥ú♠❡r♦s ❛rt✐❣♦s ❞❡ ❞✐✈✉❧❣❛çã♦ ❡ ❞❡ ❤✐stór✐❛ ❞❛ ❝✐ê♥❝✐❛✱ ❜❡♠ ❝♦♠♦ s♦❜r❡ ✈✐❞❛ ❡ ♦❜r❛ ❞❡ ❝✐❡♥t✐st❛s ❝♦♠♦ ▲❛✈♦✐s✐❡r✱ ❈♦♠t❡✱ ❉✬❆❧❡♠❜❡rt✱ P❛s❝❛❧✱ ❡♥tr❡ ♦✉tr♦s✳ ❇❡rtr❛♥❞ ♣✉❜❧✐❝♦✉ ♠✉✐t♦s tr❛✲ ❜❛❧❤♦s s♦❜r❡ ❣❡♦♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧ ❡ t❡♦r✐❛ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ❊❧❡ ❡s❝r❡✈❡✉ ✉♠❛ sér✐❡ ❞❡ ♥♦t❛s s♦❜r❡ ❛ t❡♦r✐❛ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♥❛ r❡❞✉çã♦ ❞♦s ❞❛❞♦s ❞❡ ♦❜s❡r✈❛çõ❡s✳ ❊❧❡ ♣✉❜❧✐❝♦✉ ❡st❛s ♥♦t❛s ❝♦♠❡ç❛♥❞♦ ♣♦r ✈♦❧t❛ ❞❡ ✶✽✼✺ ❡✱ ❛♣ós ✉♠❛ ❜r❡✈❡ ♣❛✉s❛ ❞❡ três ❛♥♦s ❛ ♣❛rt✐r ❞❡ ✶✽✽✹✱ ❡❧❡ ❝♦♠❡ç♦✉ ❛ ♣✉❜❧✐❝❛r ♥♦t❛s ❛❞✐❝✐♦♥❛✐s ❡♠ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ❖ ♣♦st✉❧❛❞♦ ❢♦✐ ♣r♦✈❛❞♦ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♣❛r❛ t♦❞♦ n ♣♦r P❛❢♥✉t② ❈❤❡❜②s❤❡✈ ❡♠ ✶✽✺✵✳ ❯♠❛ ♣r♦✈❛ ♠✉✐t♦ ♠❛✐s s✐♠♣❧❡s ❢♦✐ ❞❛❞❛ ♣❡❧♦ ❣ê♥✐♦ ✐♥❞✐❛♥♦ ❘❛♠❛♥✉❥❛♠ ✭✈❡❥❛ ❋ ❞♦ ❛♣ê♥❞✐❝❡✮✳ ◆♦ss❛ ♣r♦✈❛ s❡ ❞❡✈❡ ❛ P❛✉❧ ❊r❞➤s ❡ ❢♦✐ ❡①tr❛í❞❛ ❞♦ ♣r✐♠❡✐r♦ ❛rt✐❣♦ ♣♦r ❡❧❡ ♣✉❜❧✐❝❛❞♦✱ q✉❡ ❛♣❛r❡❝❡✉ ❡♠ ✶✾✸✷✱ q✉❛♥t♦ t✐♥❤❛ ✶✾ ❛♥♦s✳
P♦st✉❧❛❞♦ ❞❡ ❇❡tr❛♥❞
P❛r❛ ❝❛❞❛ n ≥1✱ ❡①✐st❡ ❛❧❣✉♠ ♥ú♠❡r♦ ♣r✐♠♦ p ❝♦♠ n < p≤2n✳
Pr♦✈❛✳ ❱❛♠♦s ❢❛③❡r ✉♠❛ ❡st✐♠❛t✐✈❛ ❞♦ t❛♠❛♥❤♦ ❞♦ ❝♦❡✜❝✐❡♥t❡ ❜✐♥♦♠✐❛❧ 2n n
❞❡ ♠❛♥❡✐r❛ ❝✉✐❞❛❞♦s❛ ♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ✈❡r q✉❡✱ s❡ ❡❧❡ ♥ã♦ t✐✈❡ss❡ ♥❡♥❤✉♠ ❢❛t♦r ♣r✐♠♦ ♥♦ ✐♥t❡r✈❛❧♦ n < p ≤ 2n✱ ❡♥tã♦ ❡❧❡ s❡r✐❛ ✏♠✉✐t♦ ♣❡q✉❡♥♦✑✳ ◆♦ss♦ ❛r❣✉♠❡♥t♦ s❡rá
❢❡✐t♦ ❡♠ ❝✐♥❝♦ ❡t❛♣❛s✳
✭✶✮ Pr♦✈❡♠♦s ♣r✐♠❡✐r♦ ♦ ♣♦st✉❧❛❞♦ ❞❡ ❇❡rtr❛♥❞ ♣❛r❛ n <4.000✳ P❛r❛ ✐ss♦✱ ♥ã♦ é ♣r❡❝✐s♦ ❝❤❡❝❛r ✹✳✵✵✵ ❝❛s♦s✿ é s✉✜❝✐❡♥t❡ ✭❡st❡ é ♦ ✏tr✉q✉❡ ❞❡ ▲❛♥❞❛✉✑✮ ❝❤❡❝❛r q✉❡
2,3,5,7,13,23,43,83,163,317,631,1.259,2.503,4.001
é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ♦♥❞❡ ❝❛❞❛ ✉♠ é ♠❡♥♦r q✉❡ ❞✉❛s ✈❡③❡s ♦ ❛♥t❡r✐♦r✳ ❊♠ ❝♦♥s❡q✉ê♥❝✐❛✱ ❝❛❞❛ ✐♥t❡r✈❛❧♦ {y : n < y ≤2n}✱ ❝♦♠n ≤4.000✱
❝♦♥té♠ ✉♠ ❞❡ss❡s ✶✹ ♣r✐♠♦s✳ ✭✷✮ ❊♠ s❡❣✉✐❞❛ ♣r♦✈❡♠♦s q✉❡
Y
p≤x
p≤4x−1 ✭✷✳✶✮
♣❛r❛ t♦❞♦ r❡❛❧x≥2✱ ♦♥❞❡ ♥♦ss❛ ♥♦t❛çã♦ ✕ ❛q✉✐ ❡ ♥♦ q✉❡ s❡❣✉❡ ✕ é ❡♥t❡♥❞✐❞❛
❈❆P❮❚❯▲❖ ✷✳ P❖❙❚❯▲❆❉❖ ❉❊ ❇❊❘❚❘❆◆❉ ✶✸
p≤x✳ ❆ ♣r♦✈❛ q✉❡ ❛♣r❡s❡♥t❛♠♦s ♣❛r❛ ❡ss❛ ❢❛t♦ ♥ã♦ ✈❡♠ ❞♦ ❛rt✐❣♦ ♦r✐❣✐♥❛❧ ❞❡ ❊r❞➤s✱ ♠❛s t❛♠❜é♠ é ❞❡❧❡✱ ❡ é ✉♠❛ ♣r♦✈❛ ❞✐❣♥❛ ❞✬❖❧✐✈r♦✶✳ Pr✐♠❡✐r♦ ♥♦t❛♠♦s
q✉❡✱ s❡ q é ♦ ♠❛✐♦r ♥ú♠❡r♦ ♣r✐♠♦ ❝♦♠q ≤x✱ ❡♥tã♦
Y
p≤x
p=Y
p≤q
p❡ 4q−1 ≤4x−1.
❆ss✐♠ ❜❛st❛ ❝❤❡❣❛r q✉❡ ✭✷✳✶✮ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ x= q é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳ P❛r❛ q = 2✱ t❡♠♦s“2≤ 4”✱ ❞❡ ❢♦r♠❛ q✉❡ ♥♦ss♦ ♣r♦❝❡❞✐♠❡♥t♦ s❡rá ❝♦♥s✐❞❡r❛r ♥ú♠❡r♦s ♣r✐♠♦s q= 2m+ 1✳ P❛r❛ ❡ss❡s ♥ú♠❡r♦s✱ ✈❛♠♦s ❞❡❝♦♠♣♦r ♦ ♣r♦❞✉t♦ ❡ ❝♦♠♣✉t❛r
Y
p≤2m+1
p= Y
p≤m+1
p· Y
m+1<p<2m+1
p≤4m
2m+ 1 m
≤4m22m = 42m.
❚♦❞❛s ❛s ♣❛rt❡s ❞❡ss❛ ✏❝♦♠♣✉t❛çã♦ ❞❡ ✉♠❛ ❧✐♥❤❛✑ sã♦ ❢á❝❡✐s ❞❡ ✈❡r✳ ❉❡ ❢❛t♦✱
Y
p≤m+1
p≤4m
♣❡❧❛ ✐♥❞✉çã♦✳ ❆ ❞❡s✐❣✉❛❧❞❛❞❡
Y
m+1<p≤2m+1
p≤
2m+ 1 m
s❡❣✉❡ ❞❛ ♦❜s❡r✈❛çã♦ ❞❡ q✉❡ 2m+1
m
= m(2!(mm+1)!+1)! é ✉♠ ✐♥t❡✐r♦✱ ♦♥❞❡ ♦s ♣r✐♠♦s q✉❡ ❝♦♥s✐❞❡r❛♠♦s sã♦ t♦❞♦s ❢❛t♦r❡s ❞♦ ♥✉♠❡r❛❞♦r (2m+ 1)!✱ ♠❛s ♥ã♦ ❞♦ ❞❡✲ ♥♦♠✐♥❛❞♦r m!(m+ 1)!✳ ❋✐♥❛❧♠❡♥t❡✱
2m+ 1 m
≤22m
✈❛❧❡✱ ♣♦✐s
2m+ 1 m
❡
2m+ 1 m+ 1
sã♦ ❞✉❛s ♣❛r❝❡❧❛s ✭✐❣✉❛✐s✦✮ q✉❡ ❛♣❛r❡❝❡♠ ❡♠
2m+1 X
k=0
2m+ 1 k
= 22m+1.
✶❖ ▲✐✈r♦ ✭❡♠ ✐♥❣❧ês✱ ❚❤❡ ❇♦♦❦✮ r❡❢❡r❡✲s❡ ❛ ✉♠ ❧✐✈r♦ ✐♠❛❣✐♥ár✐♦ ♥♦ q✉❛❧ ❉❡✉s t❡r✐❛ ❡s❝r✐t♦
❈❆P❮❚❯▲❖ ✷✳ P❖❙❚❯▲❆❉❖ ❉❊ ❇❊❘❚❘❆◆❉ ✶✹
✭✸✮ ❉♦ t❡♦r❡♠❛ ❞❡ ▲❡❣❡♥❞r❡ ✭✈❡❥❛ ❈ ❞♦ ❛♣ê♥❞✐❝❡✮✱ ♦❜t❡♠♦s q✉❡ 2n n
= (2n!nn)!! ❝♦♥té♠ ♦ ❢❛t♦r ♣r✐♠♦ p ❡①❛t❛♠❡♥t❡
X
k≥1 j2n
pk
k
−2j n pk
k
✈❡③❡s✳ ❆q✉✐✱ ❝❛❞❛ ♣❛r❝❡❧❛ é ♥♦ ♠á①✐♠♦ ✶✱ ❥á q✉❡ ❡❧❛ s❛t✐s❢❛③
j2n
pk
k
−2jn pk
k
< 2n pk −2
n
pk −1
= 2
é ✐♥t❡✐r❛✳ ❆❧é♠ ❞✐ss♦✱ ❛s ♣❛r❝❡❧❛s ❛♥✉❧❛♠✲s❡ s❡♠♣r❡ q✉❡ pk >2n✳ ❆ss✐♠✱ 2n n
❝♦♥té♠ p ❡①❛t❛♠❡♥t❡
X
k≥1 j2n
pk
k
−2jn pk
k
≤max{r :pr ≤2n}
✈❡③❡s✳ ❉❛í s❡❣✉❡ q✉❡ ❛ ♠❛✐♦r ♣♦tê♥❝✐❛ ❞❡ p q✉❡ ❞✐✈✐❞❡ 2nn
♥ã♦ é ♠❛✐♦r q✉❡ 2n✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦s ♥ú♠❡r♦s ♣r✐♠♦s p > √2n ❛♣❛r❡❝❡♠ ♥♦ ♠á①✐♠♦ ✉♠❛ ✈❡③ ❡♠ 2n
n
✳ ❆❧é♠ ❞♦ ♠❛✐s ✕ ❡ ✐ss♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❊r❞➤s✱ é ♦ ❢❛t♦✲❝❤❛✈❡ ♣❛r❛ s✉❛ ♣r♦✈❛ ✕✱ ♥ú♠❡r♦s ♣r✐♠♦s p q✉❡ s❛t✐s❢❛③❡♠ 2
3n < p ≤ n ♥ã♦ ❞✐✈✐❞❡♠ 2n
n
❞❡ ❢♦r♠❛ ❛❧❣✉♠❛✦ ❉❡ ❢❛t♦✱ 3p > 2n ✐♠♣❧✐❝❛ ✭♣❛r❛ n≥3 ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ p≥3✮ q✉❡ p ❡2p sã♦ ♦s ú♥✐❝♦s ♠ú❧t✐♣❧♦s ❞❡p q✉❡ ❛♣❛r❡❝❡♠ ❝♦♠♦ ❢❛t♦r❡s ♥♦ ♥✉♠❡r❛❞♦r ❞❡ (2n)!
n!n!✱ ❛♦ ♣❛ss♦ q✉❡ t❡♠♦s ❞♦✐s p✲ ❢❛t♦r❡s ♥♦ ❞❡♥♦♠✐♥❛❞♦r✳
✭✹✮ ❆❣♦r❛ ❡st❛♠♦s ♣r♦♥t♦s ♣❛r❛ ❡st✐♠❛r 2n n
✳ P❛r❛ n≥3✱ ✉s❛♥❞♦ ✉♠❛ ❡st✐♠❛t✐✈❛
✭✈❡❥❛ ● ❞♦ ❛♣ê♥❞✐❝❡✮ ♣❛r❛ ❛ ❝♦t❛ ✐♥❢❡r✐♦r✱ ♦❜t❡♠♦s 4n 2n ≤ 2n n ≤ Y
p≤√2n
2n· Y √
2n<p≤2 3n
p· Y
n<p≤2n
p
❡ ❛ss✐♠✱ ✉♠❛ ✈❡③ q✉❡ ♥ã♦ ❤á ♠❛✐s q✉❡ √2n ♣r✐♠♦s p≤√2n✱ 4n ≤(2n)1+√2n· Y
√
2n<p≤2 3n
p· Y
n<p<2n
p ♣❛r❛ n≥3 ✭✷✳✷✮
✭✺✮ ❱❛♠♦s ❛ss✉♠✐r ❛❣♦r❛ q✉❡ ♥ã♦ ❡①✐st❡ ♣r✐♠♦p❝♦♠ n < p≤2n✱ ❞❡ ❢♦r♠❛ q✉❡ ♦ s❡❣✉♥❞♦ ♣r♦❞✉t♦ ❡♠ ✭✷✳✷✮ é ✶✳ ❙✉❜st✐t✉✐♥❞♦ ✭✷✳✶✮ ❡♠ ✭✷✳✷✮ ♦❜t❡♠♦s
4n≤(2n)1+√2n423n
♦✉
❈❆P❮❚❯▲❖ ✷✳ P❖❙❚❯▲❆❉❖ ❉❊ ❇❊❘❚❘❆◆❉ ✶✺
q✉❡ é ❢❛❧s♦ ♣❛r❛ n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✦ ❉❡ ❢❛t♦✱ ✉s❛♥❞♦ a+ 1 < 2a ✭q✉❡
✈❛❧❡ ♣❛r❛ t♦❞♦ a≥2 ♣♦r ✐♥❞✉çã♦✮✱ ♦❜t❡♠♦s
2n = (√6
2n)6 <j√6
2nk+ 16 <26⌊√62n⌋≤26√62n, ✭✷✳✹✮ ❡ ❛ss✐♠✱ ♣❛r❛ n ≥ 50 ✭ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ 18< 2√2n✮✱ ♦❜t❡♠♦s✱ ❞❡ ✭✷✳✸✮ ❡ ✭✷✳✹✮✱
22n≤(2n)3(1+√2n) <2√62n(18+18√2n)<220√62n√2n = 220(2n)2/3.
■ss♦ ✐♠♣❧✐❝❛ q✉❡ (2n)1/3 <20❡✱ ❛ss✐♠✱n <4.000✳ P♦❞❡♠♦s ❡①tr❛✐r ❛✐♥❞❛ ♠❛✐s
❞❡ss❛ ♣r♦✈❛✿ ❞❡ ✭✷✳✷✮✱ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❡st✐♠❛t✐✈❛ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ✉s❛r ♣r♦✈❛ q✉❡
Y
n<p≤2n
p≥2301n ♣❛r❛ n ≥4.000
❡✱ ❛ss✐♠✱ q✉❡ ❡①✐st❡♠ ♥♦ ♠í♥✐♠♦
log2n2301n
= 1 30
n
log2n+ 1 > 1 30
n log2n
♣r✐♠♦s ❡♥tr❡ n ❡ 2n✳ ❆té q✉❡ ❡ss❛ ❡st✐♠❛t✐✈❛ ♥ã♦ é tã♦ ♠á ❛ss✐♠✿ ♦ ♥ú♠❡r♦ ✏✈❡r❞❛❞❡✐r♦✑ ❞❡ ♣r✐♠♦s ♥❡ss❡ ✐♥t❡r✈❛❧♦ é ❛♣r♦①✐♠❛❞❛♠❡♥t❡n/logn✳ ■st♦ ❞❡❝♦rr❡ ❞♦ ❢❛♠♦s♦ ✏t❡♦r❡♠❛ ❞♦ ♥ú♠❡r♦ ♣r✐♠♦✑✱ q✉❡ ❡st❛❜❡❧❡❝❡ q✉❡ ♦ ❧✐♠✐t❡
lim
n→∞
#{p≤n :♣ é ♣r✐♠♦}
n/logn
❡①✐st❡ ❡ é ✐❣✉❛❧ ❛ ✶✳ ■ss♦ ❢♦✐ ♣r♦✈❛❞♦ ♣r✐♠❡✐r♦ ♣♦r ❍❛❞❛♠❛r❞ ❡ ❞❡ ❧❛ ❱❛❧❧é❡✲ P♦✉ss✐♥ ❡♠ ✶✾✽✻❀ ❙❡❧❜❡r❣ ❡ ❊r❞➤s ❡♥❝♦♥tr❛r❛♠ ✉♠❛ ♣r♦✈❛ ❡❧❡♠❡♥t❛r ✭s❡♠ ❛s ❢❡rr❛♠❡♥t❛s ❞❡ ❛♥á❧✐s❡ ❝♦♠♣❧❡①❛✱ ♠❛s ❛✐♥❞❛ ❛ss✐♠ ❧♦♥❣❛ ❡ ✐♥tr✐♥❝❛❞❛✮ ❡♠ ✶✾✹✽✳ ❆ r❡s♣❡✐t♦ ❞♦ t❡♦r❡♠❛ ❞♦ ♥ú♠❡r♦ ♣r✐♠♦ ♣r♦♣r✐❛♠❡♥t❡✱ ♣❛r❡❝❡ q✉❡ ❛ ♣❛❧❛✈r❛ ✜♥❛❧ ❛✐♥❞❛ ♥ã♦ ❢♦✐ ❞❛❞❛✿ ♣♦r ❡①❡♠♣❧♦✱ ✉♠❛ ♣r♦✈❛ ❞❛ ❤✐♣ót❡s❡ ❞❡ ❘✐❡♠❛♥♥ ✭✈❡❥❛ ❍ ❞♦ ❛♣ê♥❞✐❝❡✮✱ ✉♠ ❞♦s ♠❛✐♦r❡s ♣r♦❜❧❡♠❛s ❛❜❡rt♦s ❡♠ ♠❛t❡♠át✐❝❛✱ t❛♠❜é♠ ❞❛r✐❛ ✉♠❛ ♠❡❧❤♦r❛ s✉❜st❛♥❝✐❛❧ ♥❛s ❡st✐♠❛t✐✈❛s ❞♦ t❡♦r❡♠❛ ❞♦ ♥ú♠❡r♦ ♣r✐♠♦✳ ❚❛♠❜é♠ ♣❛r❛ ♦ ♣♦st✉❧❛❞♦ ❞❡ ❇❡rtr❛♥❞ s❡ ♣♦❞❡r✐❛♠ ❡s♣❡r❛r ❛♣❡r❢❡✐ç♦❛♠❡♥t♦s ❞r❛♠át✐❝♦s✳ ❉❡ ❢❛t♦✱ ♦ s❡❣✉✐♥t❡ é ✉♠ ♣r♦❜❧❡♠❛ ♥ã♦ r❡s♦❧✈✐❞♦✿
❈❛♣ít✉❧♦ ✸
❈♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s ✭q✉❛s❡✮
♥✉♥❝❛ sã♦ ♣♦tê♥❝✐❛s
❏❛♠❡s ❏♦s❡♣❤ ❙②❧✈❡st❡r ✭✶✽✶✹✲✶✽✾✼✮ ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ✐♥❣❧ês✳ ❈♦♥tr✐❜✉✐✉ ❢✉♥✲ ❞❛♠❡♥t❛❧♠❡♥t❡ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ♠❛tr✐❝✐❛❧✱ t❡♦r✐❛ ❞♦s ✐♥✈❛r✐❛♥t❡s✱ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❡ ❛♥á❧✐s❡ ❝♦♠❜✐♥❛tór✐❛✳ ❉❡s❡♠♣❡♥❤♦✉ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♠❛t❡♠át✐❝❛ ♥♦s ❊st❛❞♦s ❯♥✐❞♦s ♥❛ s❡❣✉♥❞❛ ♠❡t❛❞❡ ❞♦ sé❝✉❧♦ ❳■❳✱ q✉❛♥❞♦ ♣r♦❢❡ss♦r ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❏♦❤♥s ❍♦♣❦✐♥s ❡ ❢✉♥❞❛❞♦r ❞♦ ❆♠❡r✐❝❛♥ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✳ ❙②❧✈❡st❡r ❢r❡q✉❡♥t♦✉ ❞✉❛s ❡s❝♦❧❛s ❡♠ ▲♦♥❞r❡s✱ ❛ ♣r✐♠❡✐r❛ s❡♥❞♦ ✉♠ ✐♥t❡r♥❛t♦ ❡♠ ❍✐❣❤❣❛t❡ q✉❡ ❡❧❡ ❢r❡q✉❡♥t♦✉ ❛té ✶✽✷✼✱ ❞❡♣♦✐s ❡❧❡ r❡❛❧✐③♦✉ ✉♠ ❡st✉❞♦ ♠❛✐s ❛♣r♦❢✉♥❞❛❞♦ ❡♠ ✶✽ ♠❡s❡s ❡♠ ✉♠❛ ❡s❝♦❧❛ ❡♠ ■s❧✐♥❣t♦♥✳ ❊♠ ✶✽✷✽✱ ❝♦♠ ❛ ✐❞❛❞❡ ❞❡ ✶✹ ❛♥♦s✱ ❡❧❡ ❡♥tr♦✉ ♥♦ ❯♥✐✈❡r✲ s✐t② ❈♦❧❧❡❣❡ ❞❡ ▲♦♥❞r❡s ❡ ❝♦♠❡ç♦✉ s❡✉s ❡st✉❞♦s ♥♦ ♣r✐♠❡✐r♦ ❛♥♦ q✉❡ ♦ ❈♦❧é❣✐♦ r❡❝❡❜❡✉ ❛❧✉♥♦s✳ ❊❧❡ t❛♠❜é♠ t✐♥❤❛ ♦ t❛✲❧❡♥t♦s♦ ❉❡ ▼♦r❣❛♥ ❝♦♠♦ s❡✉ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛✳ ❋♦✐ ❡❧❡ q✉❡♠ ✐♥✈❡♥t♦✉ ❛ ♣❛❧❛✈r❛ t♦t✐❡♥t❡✱ ♣❡❧❛ q✉❛❧ é r❡❝♦♥✲ ❤❡❝✐❞❛ ❛ ❋✉♥çã♦ t♦t✐❡♥t❡ ❞❡ ❊✉❧❡r✱ ✉s❛❞❛ ❡♠ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ❡ ❝r✐♣t♦❣r❛✜❛ ❘❙❆ ✭✈❡❥❛ ■ ❞♦ ❛♣ê♥❞✐❝❡✮✱ ❛ q✉❛❧ ❢♦✐ ✉s❛❞❛ ♣♦r ▲❡♦♥❤❛r❞ ❊✉❧❡r ♣❛r❛ ♣r♦✈❛r ♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✳
❊①✐st❡ ✉♠ ❡♣í❧♦❣♦ ♣❛r❛ ♦ ♣♦st✉❧❛❞♦ ❞❡ ❇❡rtr❛♥❞ q✉❡ ❧❡✈❛ ❛ ✉♠ ❜❡❧♦ r❡s✉❧t❛❞♦ s♦❜r❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s✳ ❊♠ ✶✽✾✷✱ ❙②❧✈❡st❡r r❡❢♦rç♦✉ ♦ ♣♦st✉❧❛❞♦ ❞❡ ❇❡rtr❛♥❞ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿
❙❡ n≥2k✱ ❡♥tã♦ ♥♦ ♠í♥✐♠♦ ✉♠ ❞♦s ♥ú♠❡r♦s n, n−1, ..., n−k+ 1 t❡♠ ✉♠ ❞✐✈✐s♦r ♣r✐♠♦ ♠❛✐♦r ❞♦ q✉❡ k✳
◆♦t❡ q✉❡ ♣❛r❛ n = 2k✱ ♦❜t❡♠♦s ♣r❡❝✐s❛♠❡♥t❡ ♦ ♣♦st✉❧❛❞♦ ❞❡ ❇❡rtr❛♥❞✳ ❊♠ ✶✾✸✹✱ ❊r❞➤s ❞❡✉ ✉♠❛ ❝✉rt❛ ❡ ❡❧❡♠❡♥t❛r ♣r♦✈❛ ❞✬❖ ▲✐✈r♦ ❞♦ r❡s✉❧t❛❞♦ ❞❡ ❙②❧✈❡st❡r✱ ♥❛ ♠❡s♠❛ ❧✐♥❤❛ ❞❡ s✉❛ ♣r♦✈❛ ❞♦ ♣♦st✉❧❛❞♦ ❞❡ ❇❡rtr❛♥❞✳ ❊①✐st❡
