Universidade de Brasília Instituto de Ciências Exatas Departamento de Matemática

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛

■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❖r❞❡♠ ❞❡ ❆♣❛r✐çã♦ ♥❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✿

✉♠ Pr♦❜❧❡♠❛ s♦❜r❡ ❉✐✈✐s✐❜✐❧✐❞❛❞❡

●✉st❛✈♦ ❈❛♥❞❡✐❛ ❈♦st❛

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛

■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❖r❞❡♠ ❞❡ ❆♣❛r✐çã♦ ♥❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✿

✉♠ Pr♦❜❧❡♠❛ s♦❜r❡ ❉✐✈✐s✐❜✐❧✐❞❛❞❡

♣♦r

●✉st❛✈♦ ❈❛♥❞❡✐❛ ❈♦st❛

❇r❛sí❧✐❛

Ficha catalográfica elaborada automaticamente, com os dados fornecidos pelo(a) autor(a)

C216o

Candeia, Gustavo Costa

Ordem de aparição na sequência de Fibonacci: um problema sobre divisibilidade / Gustavo Costa

Candeia; orientador Diego Marques. -- Brasília, 2015. 81 p.

Dissertação (Mestrado - Mestrado em Matemática) --Universidade de Brasília, 2015.

❆●❘❆❉❊❈■▼❊◆❚❖❙

❆ ❉❡✉s✱ ❙❡♥❤♦r ❡ ▼❡str❡ ❞❛ ♠✐♥❤❛ ✈✐❞❛✱ ♣♦r ❙❡ ♠♦str❛r ❛ ❡ss❡ ❥♦✈❡♠ ❡♠ ✉♠ ♠♦♠❡♥t♦ tã♦ ❝rít✐❝♦ ❞❡ ❜✉s❝❛ ❡ ✐♥❝r❡❞✉❧✐❞❛❞❡✳ P♦r ❝♦❧♦❝❛r ❡♠ ♠✐♥❤❛ ♠❡♥t❡ ❡ ❡♠ ♠❡✉ ❝♦r❛çã♦ ❛ ❝❡rt❡③❛ ❞❛ ❙✉❛ ♣r❡s❡♥ç❛ ✈✐✈❛✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ❡✉ ❡♥①❡r❣❛ss❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✈✐✈❡r ❡♠ ♣r♦❧ ❞❡ ✉♠ ❜❡♠ ♠❛✐♦r✳

➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♠❡✉s ♣❛✐s✳ ❊❧❡s ❢♦r❛♠ ❡ss❡♥❝✐❛✐s ♥❛ ❢♦r♠❛çã♦ ❞♦ ♠❡✉ ❝❛rát❡r✳ ❈♦♠ ❛ ♠✐♥❤❛ ♠ã❡ ❛♣r❡♥❞✐ ♦ ✈❛❧♦r ❞❛ ❡❞✉❝❛çã♦ ❡ ❝♦♠ ♦ ♠❡✉ ♣❛✐ ❛♣r❡♥❞✐ q✉❡ ♦s s♦♥❤♦s ♥ã♦ tê♠ ❧✐♠✐t❡s✦ ❙♦♠♦s ❝❛♣❛③❡s ❞❡ r❡❛❧✐③❛r ❛q✉✐❧♦ q✉❡ q✉❡r❡♠♦s✳

❆♦s ♣r♦❢❡ss♦r❡s✿ ❈❛r❧♦s✱ ❆❞❛✐❧ ❞❡ ❈❛str♦ ❈❛✈❛❧❤❡✐r♦✱ ❆❧✐♥❡ ●♦♠❡s ❞❛ ❙✐❧✈❛ P✐♥t♦✱ ❆r② ❱❛s❝♦♥❝❡❧♦s ▼❡❞✐♥♦✱ ❉❛♥✐❡❧❡ ❞❛ ❙✐❧✈❛ ❇❛r❛t❡❧❛ ▼❛rt✐♥s ◆❡t♦✱ ▲✐♥❡✉ ❞❛ ❈♦st❛ ❆r❛ú❥♦ ◆❡t♦✱ ▲✉❝❛s ❈♦♥q✉❡ ❙❡❝♦ ❋❡rr❡✐r❛✱ ▼❛✉r♦ ▲✉✐③ ❘❛❜❡❧♦✱ ❘❛q✉❡❧ ❈❛r♥❡✐r♦ ❉örr✱ ❘✐❝❛r❞♦ ❘✉✈✐❛r♦ ❡ ❘✉✐ ❙❡✐♠❡t③ ♣❡❧❛ ❝♦♥tr✐❜✉✐çã♦ ♥❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ▼❛t❡♠át✐❝❛✳

❆♦ ♣r♦❢❡ss♦r ♦r✐❡♥t❛❞♦r ❡ ❛♠✐❣♦ ✕ ❉✐❡❣♦ ▼❛rq✉❡s ✕ ♣❡❧♦ ❡♥t✉s✐❛s♠♦ ❝♦♠ q✉❡ r❡❛❧✐③❛ s✉❛s ❛✉❧❛s ❡ ♣❡sq✉✐s❛s✳ P❡❧❛s ❡①tr❛♦r❞✐♥ár✐❛s ❝♦♥tr✐❜✉✐çõ❡s à ▼❛t❡♠át✐❝❛✳ P♦r ❢❛③❡r ❞❡s♣❡rt❛r ❡♠ s❡✉s ❛❧✉♥♦s ❛ ✈♦♥t❛❞❡ ❞❡ ✐r ❛❧é♠✳ ❊ ♣❡❧❛ ✐♠❡♥s❛ ❛❥✉❞❛ ♥❛ r❡❛❧✐③❛çã♦ ❞❡ss❡ t❡①t♦✳

❆♦s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✿ ❆♥❛ P❛✉❧❛✱ ❉❛♥✐❡❧✱ ❉♦✉❣❧❛s✱ ❊♠❡rs♦♥✱ ❊♠♠❛♥✉❡❧✱ ❋r❡❞✱ ▼❛r❝♦✱ ▼❛r②♥❛✱ ❘✐❝❛r❞♦✱ ❘♦♥❛❧❞ ❡ ❯❧②ss❡s ♣♦r ❝♦♠♣❛rt✐❧❤❛r ✈ár✐❛s ❤♦r❛s ❞❡ ❡st✉❞♦ ❞✉r❛♥t❡ t♦❞♦ ♦ ♠❡str❛❞♦✳

❆♦s ❞❡♠❛✐s ❝♦❧❡❣❛s ❞❡ t✉r♠❛ ❡ ❛♦s ❡♥✈♦❧✈✐❞♦s ❝♦♠ ♦ P❘❖❋▼❆❚ ♣♦r ❡♥❣r❛♥❞❡❝❡r ❡ss❡ ♠❡str❛❞♦ ♣r♦✜ss✐♦♥❛❧✐③❛♥t❡✳

➚ ♠❡str❛♥❞❛ ❆♥♥❛ ❈❛r♦❧✐♥❛ ▲❛❢❡tá ♣♦r ❞❡❞✐❝❛r ✉♠ ♣♦✉❝♦ ❞♦ s❡✉ t❡♠♣♦ à ❧❡✐t✉r❛ ❞❡ss❡ tr❛❜❛❧❤♦ ❡ ♣❡❧❛s ❡①❝❡❧❡♥t❡s ♦❜s❡r✈❛çõ❡s ❢❡✐t❛s✳

➚ ♠✐♥❤❛ ❡s♣♦s❛ ❋❛❜✐❛♥❛ ♣♦r s❡r ❡①❡♠♣❧♦ ❞❡ ❝♦♠♣❛♥❤❡✐r✐s♠♦✳ P♦r ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦ ❡♠ ❝❛❞❛ ❞❡❝✐sã♦ t♦♠❛❞❛✳ P♦r ♠❡ ✐♥❝❡♥t✐✈❛r✳ P♦r ❢❛③❡r ❞❡ ♠✐♠ ✉♠❛ ♣❡ss♦❛ ♠❡❧❤♦r✳ P♦r ❝♦♥str✉✐r ✉♠❛ ❢❛♠í❧✐❛ ❡ ✉♠❛ ✈✐❞❛ tã♦ ♠❛r❛✈✐❧❤♦s❛ ❝♦♠✐❣♦✳ P♦r ❡♥t❡♥❞❡r ❛s ♠✐♥❤❛s ❛✉sê♥❝✐❛s ❞✉r❛♥t❡ ❡ss❡s ❞♦✐s ❛♥♦s ❡ ♠❡✐♦ ❞❡ ❡st✉❞♦s✳ ❊❧❛ q✉❡ s❡ ♠♦str♦✉ ❝♦♠♣❛♥❤❡✐r❛ ❡ ❛♠✐❣❛ ❞✉r❛♥t❡ ♦ ♥♦ss♦ t❡♠♣♦ ❥✉♥t♦s✱ ♥ã♦ ❞❡✐①♦✉ ❞❡ ❢❛③ê✲❧♦ ♥❛s ❤♦r❛s ♠❛✐s ❞✐❢í❝❡✐s ❞❡ss❛ ❝❛♠✐♥❤❛❞❛✳ ❈❛❞❛ ♠♦♠❡♥t♦ ❧♦♥❣❡ ❞♦ s❡✐♦ ❢❛♠✐❧✐❛r ❢♦✐ s✉♣♦rt❛❞♦ ❝♦♠ ❛ ❝❡rt❡③❛ ❞❡ ✉♠❛ ✉♥✐ã♦ ❡♠ ❈r✐st♦✱ ❊❧❡ q✉❡ ♥♦s ❞á ❢♦rç❛ ❡ ♥♦s ❝♦♥❞✉③ ♣❡❧♦s ❙❡✉s ❝❛♠✐♥❤♦s✳

❋❛❜✐❛♥❛✱ ❡✉ t❡ ❛♠♦✦ ◆ã♦ ❝♦♥s❡❣✉✐r✐❛ ❝❤❡❣❛r ♦♥❞❡ ❝❤❡❣✉❡✐ s❡♠ ✈♦❝ê✦

✏◗✉❡♠ ♣r♦❝✉r❛ ❛ ✈❡r❞❛❞❡ ♣r♦❝✉r❛ ❛ ❉❡✉s✱ ❛✐♥❞❛ q✉❡ ♥ã♦ ♦ s❛✐❜❛✳✑ ✭❊❞✐t❤ ❙t❡✐♥✮

❘❊❙❯▼❖

❙❡❥❛(Fn)n≥0 ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ❡z(n) ❛ ♦r❞❡♠ ❞❡ ❛♣❛r✐çã♦ ♥❡ss❛ s❡q✉ê♥❝✐❛

❞❡✜♥✐❞❛ ❝♦♠♦ ♦ ♠❡♥♦rk_∈Nt❛❧ q✉❡n❞✐✈✐❞❡Fk✳ ◆❡ss❡ tr❛❜❛❧❤♦✱ ❞✐s❝✉t✐r❡♠♦s ❛❧❣✉♠❛s

♣r♦♣r✐❡❞❛❞❡s ❞❡ss❛ ❢✉♥çã♦✳ ❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ é ♣r♦✈❛r q✉❡ ❡①✐st❡♠ ✐♥✜♥✐t❛s s♦❧✉çõ❡s ♣❛r❛ ❛ ❡q✉❛çã♦z(n) = z(n+ 2)❡ ❡①✐❜✐r ❢ór♠✉❧❛s ❢❡❝❤❛❞❛s ♣❛r❛z(Fm±1)✳ ▼❛s✱ ❛♥t❡s

❞✐ss♦✱ ❞❡t❛❧❤❛r❡♠♦s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❡ ♥ú♠❡r♦s ❞❡ ▲✉❝❛s✳

P❛❧❛✈r❛s✲❝❤❛✈❡

◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✱ ♥ú♠❡r♦s ❞❡ ▲✉❝❛s ❡ ♦r❞❡♠ ❞❡ ❛♣❛r✐çã♦✳

❆❇❙❚❘❆❈❚

▲❡t (Fn)n≥0 ❜❡ t❤❡ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡ ❛♥❞ ❧❡t z(n) ❜❡ t❤❡ ♦r❞❡r ♦❢ ❛♣♣❡❛r❛♥❝❡ ✐♥

t❤✐s s❡q✉❡♥❝❡ ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❛s t❤❡ s♠❛❧❧❡st k _∈ N s✉❝❤ t❤❛t n ❞✐✈✐❞❡s Fk✳ ■♥ t❤✐s

✇♦r❦✱ ✇❡ s❤❛❧❧ ❞✐s❝✉ss s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤✐s ❢✉♥❝t✐♦♥✳ ❚❤❡ ♠❛✐♥ ❣♦❛❧ ✐s t♦ ♣r♦✈❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ✐♥✜♥✐t❡❧② ♠❛♥② s♦❧✉t✐♦♥s t♦ t❤❡ ❡q✉❛t✐♦♥ z(n) = z(n+ 2) ❛s ✇❡❧❧ ❛s

t♦ ❡①❤✐❜✐t ❝❧♦s❡❞ ❢♦r♠✉❧❛s ❢♦r z(Fm ±1)✳ ❆t ✜rst✱ ✇❡ s❤❛❧❧ ❞❡s❝r✐❜❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢

❋✐❜♦♥❛❝❝✐ ❛♥❞ ▲✉❝❛s ♥✉♠❜❡rs✳

❑❡②✇♦r❞s

◆✉♠❜❡rs ♦❢ ❋✐❜♦♥❛❝❝✐✱ ♥✉♠❜❡rs ♦❢ ▲✉❝❛s ❛♥❞ ♦r❞❡r ♦❢ ❛♣♣❡❛r❛♥❝❡✳

▲■❙❚❆ ❉❊ ❋■●❯❘❆❙

✶✳✶ ❘❡tâ♥❣✉❧♦s1_×4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✶✳✷ ❊s♣✐r❛❧ ❞❡ ❋✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸ ❙♦♠❛ ❞❛s ❞✐❛❣♦♥❛✐s ❞♦ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✹ ■♥t❡r✈❛❧♦s ❡♥❝❛✐①❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✹✳✶ ❚✐❥♦❧♦ ❞❡ ❋✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

❆✳✶ ❊s❢❡r❛ ❝✐r❝✉♥s❝r✐t❛ ❛♦ t✐❥♦❧♦ ❞❡ ❋✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹

❙❯▼➪❘■❖

■♥tr♦❞✉çã♦ ✶

✶ ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❡ ◆ú♠❡r♦s ❞❡ ▲✉❝❛s ✸

✶✳✶ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❙♦♠❛s ❞❡ ♥ú♠❡r♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❋✐❜♦♥❛❝❝✐ ❡ ❛❧❣✉♠❛s r❡❧❛çõ❡s ✐♥t❡r❡ss❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✹ ◆ú♠❡r♦s ❞❡ ▲✉❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✷ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ❡ ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❡ ❞❡ ▲✉❝❛s ✷✸

✷✳✶ ❘❡s✉❧t❛❞♦s ❝❧áss✐❝♦s s♦❜r❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷ ❙í♠❜♦❧♦ ❞❡ ▲❡❣❡♥❞r❡ ❡ r❡s✉❧t❛❞♦s ✉s❛♥❞♦ ❝♦♥❣r✉ê♥❝✐❛s ♠ó❞✉❧♦ p ♣r✐♠♦✳ ✷✾ ✷✳✸ ❚❡♦r❡♠❛s ❛✉①✐❧✐❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✸ ❖r❞❡♠ ❞❡ ❆♣❛r✐çã♦ ♥❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✸✼

✸✳✶ ❖r❞❡♠ ❞❡ ❛♣❛r✐çã♦ ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✷ ■♥✜♥✐t❛s s♦❧✉çõ❡s ♣❛r❛z(n) =z(n+ 2)❡ ❢ór♠✉❧❛s ❢❡❝❤❛❞❛s ♣❛r❛z(Fm±1) ✹✹

✹ ❆♣❧✐❝❛çõ❡s ❛♦ ❊♥s✐♥♦ ▼é❞✐♦ ✺✸

✹✳✶ P❡q✉❡♥♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✷ ❙✉❣❡stã♦ ❞❡ ❛t✐✈✐❞❛❞❡s ❡ ♣r♦❜❧❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻

❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✻✵

❆ Pr♦❜❧❡♠❛s ❆♣❧✐❝á✈❡✐s ❛♦ ❊♥s✐♥♦ ▼é❞✐♦ ✻✷

❘❡❢❡rê♥❝✐❛s ✻✻

■◆❚❘❖❉❯➬➹❖

❊st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛rá ✉♠ ❡st✉❞♦ s♦❜r❡ ❛ ♦r❞❡♠ ❞❡ ❛♣❛r✐çã♦ ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✳ P❛r❛ ❛t✐♥❣✐r ❡ss❡ ♦❜❥❡t✐✈♦✱ s❡rá ♥❡❝❡ssár✐♦ ❝♦♥❤❡❝❡r ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❡ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳

❉❛❞❛ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐(Fn)n≥0 ❞❡✜♥✐❞❛ ♣♦rFn+2 =Fn+1+Fn✱ ♣❛r❛ n≥0✱

♦♥❞❡ F0 = 0 ❡ F1 = 1✱ ❞❡♠♦♥str❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❝❧áss✐❝♦s ❞❛ ❧✐t❡r❛t✉r❛ ❡

♦✉tr♦s q✉❡ s❡r✈✐rã♦ ❞❡ s✉♣♦rt❡ ♣❛r❛ ❛ ♣❛rt❡ ❝❡♥tr❛❧ ❞♦ t❡①t♦✱ q✉❡ é ❛ ♦r❞❡♠ ❞❡ ❛♣❛r✐çã♦ ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✳

❙❡❥❛Fn ♦ n✲és✐♠♦ ♥ú♠❡r♦ ❞❡ ❋✐❜♦♥❛❝❝✐✳ ❆ ♦r❞❡♠ ❞❡ ❛♣❛r✐çã♦ z(n) ❞❡ ✉♠ ♥ú♠❡r♦

♥❛t✉r❛❧n ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ é ❞❡✜♥✐❞❛ ❝♦♠♦ ♦ ♠❡♥♦r ♥ú♠❡r♦ ♥❛t✉r❛❧kt❛❧ q✉❡ n ❞✐✈✐❞❡ Fk✳

❈♦♠ ✐ss♦ ❛❧❣✉♠❛s ♣❡r❣✉♥t❛s s✉r❣❡♠ ♥❛t✉r❛❧♠❡♥t❡✳ P♦r ❡①❡♠♣❧♦✱z(n)❡stá s❡♠♣r❡

❞❡✜♥✐❞❛❄ ❊①✐st❡♠ ❢ór♠✉❧❛s ❢❡❝❤❛❞❛s ♣❛r❛z(n)❄ ◗✉❛✐s ❛s ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡z(n)s❡❥❛

✐❣✉❛❧ ❛z(n+ 1) ❡ ♣❛r❛ z(n) =z(n+ 2)❄ ◗✉❛♥❞♦ z(n) = 2n❄ ❙❡rá q✉❡ z(Fn) ❝♦✐♥❝✐❞❡

❝♦♠ ❛ ♣♦s✐çã♦ ❞❡ Fn❄ ❉❡ss❡s q✉❡st✐♦♥❛♠❡♥t♦s✱ ♦ ú❧t✐♠♦ é ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞❛

❞❡✜♥✐çã♦ ❞❡ ♦r❞❡♠ ❞❡ ❛♣❛r✐çã♦ ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✳

❉❡♠♦♥str❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s s♦❜r❡ ♦r❞❡♠ ❞❡ ❛♣❛r✐çã♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ s❡ m_|Fn✱ ❡♥tã♦z(m)|n✱ ❡ z(Fm±1)> m=z(Fm)✱ ♣❛r❛ t♦❞♦ m≥5✳ ▼♦str❛r❡♠♦s q✉❡

❡①✐st❡♠ ✐♥✜♥✐t❛s s♦❧✉çõ❡s ♣❛r❛z(n) = z(n+ 2)✳

❋♦r♥❡❝❡r❡♠♦s ❢ór♠✉❧❛s ❡①♣❧í❝✐t❛s ♣❛r❛z(Fm±1)❞❡♣❡♥❞❡♥❞♦ ❞❛ ❝❧❛ss❡ ❞❡ r❡st♦s ❞❡

m ♠ó❞✉❧♦ 4✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ z(Fm±1)≥(m2/2)−2✱ ♣❛r❛ m≡0 (mod 4)✳

■♥tr♦❞✉çã♦ ✷

❆❧é♠ ❞❡ r❡❧❡♠❜r❛r♠♦s ♦ ❢❛♠♦s♦ ♣r♦❜❧❡♠❛s ❞♦s ❝♦❡❧❤♦s✱ ✈❛♠♦s ❞❛r ❡①❡♠♣❧♦s ❛❜str❛t♦s ♦♥❞❡ ❛ r❡s♣♦st❛ é ❞❛❞❛ ♣♦r ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳ ❚❛♠❜é♠ ♥♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ✈❛♠♦s ❞❡♠♦♥str❛r✱ ✉s❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s s♦❜r❡ s♦♠❛s ❞❡ ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳

❊♥❝❡rr❛♥❞♦ ❡ss❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞♦ tr❛❜❛❧❤♦✱ ✈❛♠♦s r❡❧❛❝✐♦♥❛r ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦✲ ♥❛❝❝✐ ❛ ❛❧❣✉♥s tó♣✐❝♦s ✐♥t❡r❡ss❛♥t❡s ❧✐❣❛❞♦s ❛ ❡❧❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ tr✐♣❧❛s ♣✐t❛❣ór✐❝❛s✱ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧ ❡ r❛③ã♦ á✉r❡❛✳

❖ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛rá r❡s✉❧t❛❞♦s ❡ ♣r♦♣r✐❡❞❛❞❡s ❝❧áss✐❝❛s ❡♥✈♦❧✈❡♥❞♦ ❝♦♥❝❡✐t♦s ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ r❡❧❛❝✐♦♥❛❞♦s ❛♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❡ ❞❡ ▲✉❝❛s✳ ❊ss❡s r❡s✉❧t❛❞♦s ❞❛rã♦ s✉♣♦rt❡ ♣❛r❛ ❛s ❞❡♠♦♥str❛çõ❡s q✉❡ s❡rã♦ ❢❡✐t❛s ♥♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡✳ ◆♦ ✜♥❛❧ ❞♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❝♦♠❡♥t❛r❡♠♦s s♦❜r❡ ❛s s♦♠❛s ❞❡ ♣♦tê♥❝✐❛s ❞❡ ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❝♦♥s❡❝✉t✐✈♦s✳

◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ tr❛t❛r❡♠♦s ❞❛ ♦r❞❡♠ ❞❡ ❛♣❛r✐çã♦ ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✳ ❆♣r❡s❡♥t❛r❡♠♦s ✉♠❛ t❛❜❡❧❛ ❝♦♠ ❛ ♦r❞❡♠ ❞❡ ❛♣❛r✐çã♦ ❞♦s 100 ♣r✐♠❡✐r♦s ♥ú♠❡r♦s

♥❛t✉r❛✐s✳ ❊ss❛ t❛❜❡❧❛ ♣♦❞❡ s❡r ✉s❛❞❛ ♣❛r❛ ❛t✐✈✐❞❛❞❡s ♦♥❞❡ ♦ ♦❜❥❡t✐✈♦ é ❢❛③❡r ✐♥❢❡rê♥❝✐❛s ❡ ❝♦♥❥❡❝t✉r❛s s♦❜r❡ ❞❡t❡r♠✐♥❛❞♦s ♣❛❞rõ❡s ♥✉♠ér✐❝♦s✳ ❱❛♠♦s ♠♦str❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡♥✈♦❧✈❡♥❞♦ ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✱ ♦s ♥ú♠❡r♦s ❞❡ ▲✉❝❛s ❡ ❛ ♦r❞❡♠ ❞❡ ❛♣❛r✐çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ◆♦ á♣✐❝❡ ❞♦ t❡①t♦✱ ❞❡♠♦♥str❛r❡♠♦s q✉❡ ❤á ✐♥✜♥✐t❛s s♦❧✉çõ❡s ♣❛r❛z(n) = z(n+ 2) ❡ ❝❛r❛❝t❡r✐③❛r❡♠♦s z(Fm±1)✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦ r❡st♦ ❞❛

❞✐✈✐sã♦ ❞❡m ♣♦r 4✳

CAP´

ITULO 1

◆Ú▼❊❘❖❙ ❉❊ ❋■❇❖◆❆❈❈■ ❊

◆Ú▼❊❘❖❙ ❉❊ ▲❯❈❆❙

✶✳✶ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐

▲❡♦♥❛r❞♦ ❞❡ P✐s❛ ♥❛s❝❡✉ ♥❛ ■tá❧✐❛ ♣♦r ✈♦❧t❛ ❞❡ 1175 ❡ ✜❝♦✉ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ▲❡♦✲

♥❛r❞♦ ❋✐❜♦♥❛❝❝✐✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❋✐❜♦♥❛❝❝✐ ✭✜❧❤♦ ❞❡ ❇♦♥❛❝❝✐✮✱ ✉♠❛ ✈❡③ q✉❡ ♦ ♥♦♠❡ ❞♦ s❡✉ ♣❛✐ ❡r❛ ●✉✐❧✐❡❧♠♦ ❇♦♥❛❝❝✐✳ ◆❛ s✉❛ ♦❜r❛ ▲✐❜❡r ❛❜❛❝❝✐✱ ♦✉ ❧✐✈r♦ ❞♦ á❜❛❝♦✱ ❤á ♦ r❡❣✐str♦ ❞♦ ♣r♦❜❧❡♠❛ ❞♦s ❝♦❡❧❤♦s✱ ♦ q✉❛❧ ❣❡r❛r✐❛ ✉♠❛ ❞❛s s❡q✉ê♥❝✐❛s ♥✉♠ér✐❝❛s ♠❛✐s ❢❛♠♦s❛s ❞❛ ❤✉♠❛♥✐❞❛❞❡✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❬✺❪✱ ❋✐❜♦♥❛❝❝✐ ❢♦✐ ✉♠ ❞♦s ♠❡❧❤♦r❡s ♠❛t❡✲ ♠át✐❝♦s ❞♦ ♣❡rí♦❞♦ ♠❡❞✐❡✈❛❧✱ ♣✉❜❧✐❝❛♥❞♦✱ ❛❧é♠ ❞♦ ▲✐❜❡r ❛❜❛❝❝✐✱ ♦s tr❛❜❛❧❤♦s Pr❛❝t✐❝❛ ●❡♦♠❡tr✐❛❡✱ ❡♠ ✶✷✷✵✱ s♦❜r❡ ❣❡♦♠❡tr✐❛ ❡ tr✐❣♦♥♦♠❡tr✐❛ ❡ ▲✐❜❡r q✉❛❞r❛t♦r✉♠✱ ❡♠ ✶✷✷✺✱ s♦❜r❡ ❛♥á❧✐s❡ ✐♥❞❡t❡r♠✐♥❛❞❛✳

❖ ♣r♦❜❧❡♠❛ ❞♦s ❝♦❡❧❤♦s ❡r❛ ♣r❛t✐❝❛♠❡♥t❡ ♦ s❡❣✉✐♥t❡✿ ✏❯♠ ❤♦♠❡♠ ♣ôs ✉♠ ❝❛s❛❧ ❞❡ ❝♦❡❧❤♦s ❡♠ ✉♠ ❧✉❣❛r ❝❡r❝❛❞♦ ♣♦r t♦❞♦s ♦s ❧❛❞♦s ♣♦r ✉♠ ♠✉r♦✳ ◗✉❛♥t♦s ♣❛r❡s ❞❡ ❝♦❡❧❤♦s ♣♦❞❡♠ s❡r ❣❡r❛❞♦s ❛ ♣❛rt✐r ❞❡ss❡ ♣❛r ❡♠ ✉♠ ❛♥♦ s❡✱ s✉♣♦st❛♠❡♥t❡✱ t♦❞♦ ♠ês ❝❛❞❛ ❝❛s❛❧ ❞❡ ❝♦❡❧❤♦s ❣❡r❛ ✉♠ ♥♦✈♦ ❝❛s❛❧✱ q✉❡ é ❢ért✐❧ ❛ ♣❛rt✐r ❞♦ s❡❣✉♥❞♦ ♠ês❄✑

✶✳✶ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✹

❝♦❡❧❤♦s✳ P❡❧♦ ❝♦♥trár✐♦✱ ❡❧❡ ♣r♦♣ôs ✉♠❛ q✉❡stã♦ ♠❛t❡♠át✐❝❛ s✉♣♦st❛♠❡♥t❡ s❡♠ ❛ ♣r❡t❡♥sã♦ ❞❡ ❧✐❣á✲❧❛ ❛ ♦✉tr♦s ❝❛♠♣♦s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳

❆♦ ❛♥❛❧✐s❛r ♦ ♣r♦❜❧❡♠❛✱ ♣❡r❝❡❜❡✲s❡ q✉❡ ♥♦ ♠♦♠❡♥t♦ ✐♥✐❝✐❛❧ ❤á ✉♠ ♣❛r ❞❡ ❝♦❡❧❤♦s ❥♦✈❡♥s ❡ ✐♥❢ért❡✐s✳ ❆♣ós ♦ ♣r✐♠❡✐r♦ ♠ês✱ q✉❛♥❞♦ ♦ ❝❛s❛❧ s❡ t♦r♥❛ ❢ért✐❧✱ ❡❧❡ ♣♦❞❡ r❡♣r♦❞✉③✐r✱ ♠❛s ❛✐♥❞❛ só ❡①✐st❡ ✉♠ ❝❛s❛❧✳

◆♦ s❡❣✉♥❞♦ ♠ês ❤❛✈❡rá ❞♦✐s ❝❛s❛✐s✱ ✉♠ ❛❞✉❧t♦ ❡ ♦ ♦✉tr♦ ❥♦✈❡♠✳ ◆♦ t❡r❝❡✐r♦ ♠ês✱ ♦ ❝❛s❛❧ ✐♥✐❝✐❛❧ t❡rá ❣❡r❛❞♦ ♠❛✐s ✉♠ ❝❛s❛❧ t♦t❛❧✐③❛♥❞♦ três ❝❛s❛✐s✳

◆♦ q✉❛rt♦ ♠ês✱ ♦ ❝❛s❛❧ ♠❛tr✐③ ❣❡r❛rá ♦✉tr♦ ❝❛s❛❧✳ ❖ ♣r✐♠❡✐r♦ ❝❛s❛❧ ❞❡ ❝♦❡❧❤♦s ❣❡r❛❞♦ t❛♠❜é♠ ❝♦♥tr✐❜✉✐rá ❝♦♠ ♦✉tr♦ ❝❛s❛❧ ❞❡ ❝♦❡❧❤♦s✱ ❞❡ ♠♦❞♦ q✉❡ ❥á sã♦ ❝✐♥❝♦ ❝❛s❛✐s✳

❈♦♥t✐♥✉❛♥❞♦ ❡ss❡ ♣r♦❝❡ss♦✱ ♦♥❞❡ ❛ ♣❛rt✐r ❞♦ ✐♥í❝✐♦ ❞♦ t❡r❝❡✐r♦ ♠ês ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛s❛✐s ❞❡ ❝♦❡❧❤♦s é ✐❣✉❛❧ ❛ s♦♠❛ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛s❛✐s ❞♦s ❞♦✐s ♠❡s❡s ✐♠❡❞✐❛t❛♠❡♥t❡ ❛♥t❡r✐♦r❡s✱ s✉r❣❡ ❛ s❡q✉ê♥❝✐❛ ♥✉♠ér✐❝❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✶_✳

❆ss✐♠ ♦s ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ sã♦✿

1,1, 2, 3,5, 8, 13, 21,34, 55, 89, 144, 233, 377, 610,987, 1597, 2584, . . .

❉❡ss❛ ❢♦r♠❛✱ ❛♣ós ✉♠ ❛♥♦✱ sã♦ ❣❡r❛❞♦s ✷✸✷ ♣❛r❡s ❞❡ ❝♦❡❧❤♦s ❛ ♣❛rt✐r ❞♦ ❝❛s❛❧ ❞❡ ❝♦❡❧❤♦s ✐♥✐❝✐❛❧✳

❙❡♠ ♦ ❛s♣❡❝t♦ ❤✐stór✐❝♦ ♦✉ ❛ ❝♦♥t❡①t✉❛❧✐③❛çã♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳

❉❡✜♥✐çã♦ ✶✳✶✳ ❆ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ (Fn)n≥0 é ❞❡✜♥✐❞❛ ♣♦r Fn+2 = Fn+1 +Fn✱

♣❛r❛n _≥0✱ ♦♥❞❡ F0 = 0 ❡ F1 = 1✳

❊ss❛ s❡q✉ê♥❝✐❛ ♥✉♠ér✐❝❛ t❡♠ ♠✉✐t❛s ♣r♦♣r✐❡❞❛❞❡s ✐♥t❡r❡ss❛♥t❡s ❡ ❛❧❣✉♠❛s ❞❡❧❛s s❡rã♦ ❛❜♦r❞❛❞❛s ♥❡ss❡ t❡①t♦✳ P❛r❛ ✐♥❢♦r♠❛çõ❡s ♠❛✐s ❛✈❛♥ç❛❞❛s✱ ♦ ❧❡✐t♦r ♣♦❞❡ ❝♦♥s✉❧t❛r ❬✶❪ ♦✉ ❬✷✸❪✳ ❙❡q✉ê♥❝✐❛s ❝♦♠♦ ❛ ❞❡ ❋✐❜♦♥❛❝❝✐ sã♦ ❝❤❛♠❛❞❛s ❞❡ r❡❝♦rr❡♥t❡s ♦✉ r❡❝✉rs✐✈❛s✳ ❆♥❛❧✐s❛♥❞♦ ❛ r❡❝♦rrê♥❝✐❛ ❧✐♥❡❛r ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❤♦♠♦❣ê♥❡❛ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡s ✭Fn+2 =Fn+1+Fn✱ ♣❛r❛n ≥0✱ ♦♥❞❡F0 = 0❡F1 = 1✮✱ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r

✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ♦ t❡r♠♦ ❣❡r❛❧ ❞❡ss❛ r❡❝♦rrê♥❝✐❛✳ ❊ss❛ ❡①♣r❡ssã♦ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❢ór♠✉❧❛ ❞❡ ❇✐♥❡t✳

❖❜s❡r✈❡ q✉❡ ❛ ❡q✉❛çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ r❡❝♦rrê♥❝✐❛ ❛❝✐♠❛ é ϕ2_{−ϕ−1 = 0 ✭✈❡r}

❬✶✶❪✮✱ ❝✉❥❛s r❛í③❡s sã♦α= (1 +√5)/2♦✉ β= (1₋√5)/2✳

❙❛❜❡♠♦s q✉❡ s❡ r1 ❡ r2 sã♦ r❛í③❡s ❞✐st✐♥t❛s ❞❡ r2 +pr+q = 0✱ p, q ∈ R, ❡♥tã♦

Xn = c1rn1 +c2r2n é s♦❧✉çã♦ ❞❛ r❡❝♦rrê♥❝✐❛ Xn+2 +pXn+1 +qXn = 0✱ q✉❛✐sq✉❡r q✉❡

s❡❥❛♠ ♦s ✈❛❧♦r❡s ❞❛s ❝♦♥st❛♥t❡sc1 ❡ c2✳

✶✳✶ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✺

❉❡ ❢❛t♦✱ s❡❥❛♠ Yn=rn1, Zn=rn2 ❡Xn=c1Yn+c2Zn✳ ❆ss✐♠

Xn+2+pXn+1+qXn = (c1Yn+2+c2Zn+2) +p(c1Yn+1+c2Zn+1) +q(c1Yn+c2Zn)

= c1(Yn+2+pYn+1+qYn) +c2(Zn+2+pZn+1+qZn)

= c1(0) +c2(0) = 0.

❉❡ss❛ ❢♦r♠❛✱ ♦ t❡r♠♦ ❣❡r❛❧ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ é ❞❛❞♦ ♣♦rFn =c1αn+c2βn,

❝♦♠ F0 = 0, F1 = 1✱ c1, c2 ∈R ❡α ❡β r❛í③❡s ❞❛ ❡q✉❛çã♦ ϕ2−ϕ−1 = 0✳

▲♦❣♦✱       

F0 = 0 =c1α0+c2β0 =c1+c2

F1 = 1 =c1α1+c2β1 =c1α+c2β

⇒       

c1+c2 = 0

c1α+c2β = 1

.

❖ q✉❡ ✐♠♣❧✐❝❛

c1 =−c2 ⇒ −c2α+c2β = 1⇒c2(−α+β) = 1 ⇒c2 =−1/

√

5 ❡ c1 = 1/

√

5. P♦rt❛♥t♦✱

Fn=

αn_−βn

α₋β . ✭✶✳✶✮

➱✱ ♥♦ ♠í♥✐♠♦✱ ❜❛st❛♥t❡ ❝✉r✐♦s♦ ♦ ❢❛t♦ ❞❡ q✉❡ ♣❛r❛ q✉❛❧q✉❡r ✐♥t❡✐r♦ n ♥ã♦ ♥❡✲ ❣❛t✐✈♦ Fn = (αn − βn)/

√

5 t❛♠❜é♠ s❡❥❛ ✐♥t❡✐r♦✳ P♦r ❡①❡♠♣❧♦ F7 = 13✱ F67 =

44.945.570.212.853 ❡F127 = 155.576.970.220.531.065.681.649.693✳

❊ss❡s três ♥ú♠❡r♦s ❡①❡♠♣❧✐✜❝❛♠ ❛ ♣❡r✐♦❞✐❝✐❞❛❞❡ ❞♦s ❞í❣✐t♦s ❞❛s ✉♥✐❞❛❞❡s ❞♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✱ ♦✉ s❡❥❛✱ Fk+60 ≡ Fk (mod 10) ♣❛r❛ k ≥ 0✳ ❊ss❛ ❞❡♠♦♥s✲

tr❛çã♦ ♣♦❞❡ s❡r ❢❡✐t❛ ♣♦r ✐♥❞✉çã♦✳ ❱❛♠♦s ♠♦str❛r ❡ss❡ ❢❛t♦ ♥♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ♥❛ Pr♦♣♦s✐çã♦ ✷✳✶✶✳

❖✉tr♦ ❢❛t♦ ✐♥t❡r❡ss❛♥t❡ é q✉❡✱ ♣❛r❛ p ♣r✐♠♦ (p < 300), Fp é ♣r✐♠♦ ♦✉ ✉♠ ♣r♦❞✉t♦

❞❡ ♣r✐♠♦s ❞✐st✐♥t♦s✷_{✱ ♦✉ s❡❥❛✱F}

p é ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s✳ ❖s10♣r✐♠❡✐r♦s ♥ú♠❡r♦s ♣r✐♠♦s

Fp sã♦ ♦s s❡❣✉✐♥t❡s✿ F3, F5, F7, F11, F13, F17, F23, F29, F43, F47✳

❖❜s❡r✈❡ q✉❡ F19 = 4181 = 37 × 113, F31 = 1346269 = 557 × 2417✱ ❡ F59 =

956722026041 = 353_×2710260697 sã♦ ♥ú♠❡r♦s ❝♦♠♣♦st♦s✳ ❚❛♠❜é♠ sã♦ ❝♦♠♣♦st♦s

♦s ♥ú♠❡r♦sF37, F41, F53✱ ❛♣❡s❛r ❞❡19,31,37,41,53 ❡59 s❡r❡♠ ♣r✐♠♦s✳

◆❡ss❡ s❡♥t✐❞♦ é ♣♦ssí✈❡❧ ♣r♦✈❛r q✉❡ s❡k =m_×n,(k >4),1< m, n < k✱ é ❝♦♠♣♦st♦✱ ❡♥tã♦ Fk t❛♠❜é♠ é ❝♦♠♣♦st♦✳ ❆ Pr♦♣♦s✐çã♦ ✷✳✻ ♠♦str❛rá ❡ss❡ ❢❛t♦✳ ❉❡ss❛ ❢♦r♠❛✱

✷_{❊ss❡s ❢❛t♦s ♣♦❞❡♠ s❡r ✈✐st♦s ❡♠ ❤tt♣✿✴✴✇✇✇✳♠❛t❤s✳s✉rr❡②✳❛❝✳✉❦✴❤♦st❡❞✲}

✶✳✶ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✻

p ♣r✐♠♦ é ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛✱ ♣♦ré♠ ♥ã♦ s✉✜❝✐❡♥t❡✱ ♣❛r❛ Fp s❡r ♣r✐♠♦✳ ❆té ♦

♠♦♠❡♥t♦ ❞❛ r❡❛❧✐③❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✱ ❛✐♥❞❛ ❡st❛✈❛ ❡♠ ❛❜❡rt♦ ❛ q✉❡stã♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✐♥✜♥✐t♦s ♣r✐♠♦s ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✳

❊♥tr❡ ♦s ♣r✐♠❡✐r♦s300 ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ♦ ♠❛✐♦r ♣r✐♠♦ ❡♥❝♦♥tr❛❞♦ é ♦ F137 =

19.134.702.400.093.278.081.449.423.917✳ ❚❛♠❜é♠ sã♦ ♣r✐♠♦s ♦s s❡❣✉✐♥t❡s ♥ú♠❡r♦s ❞❡

❋✐❜♦♥❛❝❝✐✿ F359, F431, F433 ❡F449✳

❊♠ ♠✉✐t♦s ❧✐✈r♦s✱ ♣♦r ❡①❡♠♣❧♦ ❡♠ ❬✶❪ ❡ ❬✺❪✱ ♦ t❡r♠♦ ❣❡r❛❧ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ Fn= (αn−βn)/

√

5✱ é ❛♣r❡s❡♥t❛❞♦ ❝♦♠♦ ❛ ❢ór♠✉❧❛ ❞❡ ❇✐♥❡t✸✳ ◆❡ss❡s ❧✐✈r♦s✱ ♥♦ ❧✉❣❛r ❞❛

❝♦♥str✉çã♦ ❞❛ ❡①♣r❡ssã♦ ✭✶✳✶✮✱ ✉s❛♥❞♦ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ s♦❜r❡ r❡❝♦rrê♥❝✐❛s✱ é ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ❞❡♠♦♥str❛çã♦ ♣♦r ✐♥❞✉çã♦ ❝♦♠♦ ❛ s❡❣✉✐♥t❡✳

Pr♦♣♦s✐çã♦ ✶✳✷✳ ❖ n✕és✐♠♦ ♥ú♠❡r♦ ❞❡ ❋✐❜♦♥❛❝❝✐ é ❞❛❞♦ ♣♦r Fn = (αn−βn)/

√

5 = (αn_−βn_{)/(α−β)✱ ♦♥❞❡ α = (1 +}√_{5)/2 ❡ β = (1−}√_{5)/2 sã♦ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦}

ϕ2_{−ϕ−1 = 0✳}

❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❡ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ Fn = (αn −βn)/

√

5 é ✈á❧✐❞❛ ♣❛r❛ n = 0 ❡

n= 1✱ ♣♦✐s F0 = (α0−β0)/

√

5 = 0 ❡F1 = (α1 −β1)/

√

5 = 1✳

❆❣♦r❛✱ ✈❛♠♦s s✉♣♦r q✉❡✱ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ 1 < k _≤ n + 1✱ ❛ ❡①♣r❡ssã♦ Fk =

(αk_−βk_)/√_{5 s❡❥❛ ✈á❧✐❞❛✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ ✭✶✳✶✮ s❡ ✈❡r✐✜❝❛ ♣❛r❛ k =n+ 2✱ ♦✉}

s❡❥❛✱Fn+2 = (αn+2−βn+2)/

√

5✳

P❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦s✿ Fn= (αn−βn)/

√

5 ❡ Fn+1 = (αn+1−βn+1)/

√

5✳

❙✉❜st✐t✉✐♥❞♦ Fn ❡ Fn+1 ♥❛ ❡①♣r❡ssã♦ ♣❛r❛ Fn+2✱ ❞❛❞❛ ♥❛ ❞❡✜♥✐çã♦ ❞❛ s❡q✉ê♥❝✐❛ ❞❡

❋✐❜♦♥❛❝❝✐✱ t❡♠♦s✿

Fn+2 = Fn+1+Fn

= √1

5(α

n+1

−βn+1) + √1

5(α

n

−βn) = _√1

5(α

n_{)(α+ 1)}

−√1

5(β

n_{)(β+ 1)}

= _√1 5(α

n₎ 3 +

√

5 2

!

− √1

5(β

n₎ 3−

√

5 2

!

= _√1 5(α

n_)α2_−√1

5(β

n_)β2

= α

n+2_−βn+2

α₋β .

❖❜s❡r✈❡ q✉❡ ✉s❛♠♦s ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s✿ (3 +√5)/2 = α2 _{❡ (3−}√_{5)/2 = β}2_✳

✶✳✶ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✼

P♦rt❛♥t♦✱ ❛ ❡①♣r❡ssã♦ ✭✶✳✶✮ s❡ ✈❡r✐✜❝❛ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦n✳

➱ ❜❛st❛♥t❡ ❝♦♠✉♠✱ ♥♦s ❧✐✈r♦s q✉❡ ❛❜♦r❞❛♠ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ ❡①❡♠♣❧♦s ❞❛ ✈✐❞❛ r❡❛❧ ♦♥❞❡ s❡ ❡♥❝♦♥tr❛♠ t❡r♠♦s ❞❛ r❡❢❡r✐❞❛ s❡q✉ê♥❝✐❛ ❧✐❣❛❞♦s ❛ ♦❜s❡r✈❛çõ❡s ❞❛ ♥❛t✉r❡③❛✳ P♦r ❡①❡♠♣❧♦✱ ❡♠ ❬✺❪ ❡ ❬✷❪ ❡①✐st❡♠ ❡①❝❡❧❡♥t❡s ✐♥❢♦r♠❛çõ❡s✳ ❚❛♠❜é♠ é ❜❛st❛♥t❡ ❝♦♠✉♠ ❡①♣❧♦r❛r ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ s✐♠✉❧t❛♥❡❛♠❡♥t❡ à r❛③ã♦ á✉r❡❛✳ ❚♦❞❛✈✐❛✱ ♥❡ss❡ t❡①t♦✱ ✈❛♠♦s ❡♠ ✉♠ s❡♥t✐❞♦ ❞✐❢❡r❡♥t❡✳ ❆ s❡❣✉✐r✱ ❝✐t❛r❡♠♦s ❞♦✐s ❡①❡♠♣❧♦s ❛❜str❛t♦s ♦♥❞❡ s✉r❣❡♠ ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳

❊①❡♠♣❧♦ ✶✳✸✳ ❈♦♥s✐❞❡r❡ ✉♠ r❡tâ♥❣✉❧♦1_×n✱ ♦ q✉❛❧ ♣♦❞❡ s❡r ♣r❡❡♥❝❤✐❞♦ ♣♦r ❞♦✐s t✐♣♦s ❞❡ r❡tâ♥❣✉❧♦s ♠❡♥♦r❡s✱ 1_×1 ❡ 1_×2✳ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ♣♦❞❡♠♦s ❢❛③❡r ✐ss♦❄

❙♦❧✉çã♦✳ ❙❡ n = 1✱ só ❤á ✉♠❛ ♠❛♥❡✐r❛ ❞❡ ❝♦❜r✐r ♦ r❡tâ♥❣✉❧♦✳ ❙❡ n = 2✱ ❤á ❞✉❛s

♠❛♥❡✐r❛s✳ ❙❡ n = 3✱ ❡♥tã♦ ❡①✐st❡♠ três ♠❛♥❡✐r❛s ❞✐st✐♥t❛s ❞❡ ♣r❡❡♥❝❤❡r ♦ r❡tâ♥❣✉❧♦✳

❙❡n= 4✱ ❤á 5 ♠♦❞♦s ❞✐st✐♥t♦s✱ ❛ s❛❜❡r✿

❋✐❣✉r❛ ✶✳✶✿ ❘❡tâ♥❣✉❧♦s 1_×4

❉❡ ♠❛♥❡✐r❛ ❣❡r❛❧✱ s❡❥❛Xn♦ ♥ú♠❡r♦ ❞❡ ♠❛♥❡✐r❛s ❞✐st✐♥t❛s ❞❡ ♣r❡❡♥❝❤❡r ♦ r❡tâ♥❣✉❧♦

1_×n✳ ❆ss✐♠✱ X1 = 1, X2 = 2, X3 = 3, X4 = 5, X5 = 8✱ ✳ ✳ ✳

❖❜s❡r✈❡ q✉❡ ♣❛r❛ ❝♦❜r✐r ♦ r❡tâ♥❣✉❧♦1_×n✱ ♦✉ ❝♦♠❡ç❛♠♦s ❝♦♠ ✉♠ r❡tâ♥❣✉❧♦ 1_×1✱

❢❛❧t❛♥❞♦(n₋1)❝❛s❛s ♣❛r❛ s❡r❡♠ ♣r❡❡♥❝❤✐❞❛s✱ ♦ q✉❡ ♣♦❞❡ s❡r ❢❡✐t♦ ❞❡Xn−1 ♠❛♥❡✐r❛s✱

♦✉ ❝♦♠❡ç❛♠♦s ❝♦♠ ✉♠ r❡tâ♥❣✉❧♦1_×2✱ r❡st❛♥❞♦(n₋2)❝❛s❛s ♣❛r❛ s❡r❡♠ ♣r❡❡♥❝❤✐❞❛s✱

♦ q✉❡ ♣♦❞❡ s❡r ❢❡✐t♦ ❞❡Xn−2 ♠♦❞♦s✳

P♦rt❛♥t♦✱ ♦ ♥ú♠❡r♦ ❞❡ ♠❛♥❡✐r❛s ❞✐st✐♥t❛s ❞❡ ❝♦❜r✐r ♦ r❡tâ♥❣✉❧♦ 1 _×n é Xn =

Xn−1+Xn−2✱ ❝♦♠X1 = 1 ❡ X2 = 2✳

❆ s❡q✉ê♥❝✐❛ ♥✉♠ér✐❝❛✱ q✉❡ é s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❝♦♠ n= 1,2,3, . . . ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ ♣♦ré♠ ❝♦♠ ✉♠ ❞❡s❧♦❝❛♠❡♥t♦✳ ❉❡ss❡ ♠♦❞♦✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♠❛♥❡✐r❛s ❞♦ r❡tâ♥❣✉❧♦ 1_×n s❡r ♣r❡❡♥❝❤✐❞♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s ❝♦♥❞✐çõ❡s ❞❛❞❛s✱ é ✐❣✉❛❧ ❛

Xn=

1+√5

2 n+1

−1−√5 2

n+1

√

5 =

αn+1_−βn+1

✶✳✷ ❙♦♠❛s ❞❡ ♥ú♠❡r♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✽

❊①❡♠♣❧♦ ✶✳✹✳ ❍á n ❧â♠♣❛❞❛s ❡♥✜❧❡✐r❛❞❛s ❡♠ ✉♠❛ s❛❧❛✳ ◗✉❛♥t❛s ❝♦♥✜❣✉r❛çõ❡s ❡①✐s✲ t❡♠ s❡ ♥ã♦ ♣✉❞❡r ❤❛✈❡r ❞✉❛s ❧â♠♣❛❞❛s ❛❞❥❛❝❡♥t❡s ❧✐❣❛❞❛s s✐♠✉❧t❛♥❡❛♠❡♥t❡❄

❙♦❧✉çã♦✳ ❙❡❥❛An ♦ ♥ú♠❡r♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ♣❛r❛ n ❧â♠♣❛❞❛s✳

❱❛♠♦s ❝♦♥t❛r s❡♣❛r❛❞❛♠❡♥t❡ ♦s ❝❛s♦s ♦♥❞❡ ❛ ♣r✐♠❡✐r❛ ❧â♠♣❛❞❛ ❡stá ❞❡s❧✐❣❛❞❛ ❡ ♣♦st❡r✐♦r♠❡♥t❡ s♦♠❛r à q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛s♦s ♦♥❞❡ ❛ ♣r✐♠❡✐r❛ ❧â♠♣❛❞❛ ❡stá ❧✐❣❛❞❛ ❡ ❛ss✐♠ ♦❜t❡r ♦ t♦t❛❧An✳

❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s✿

An = An−1 | {z }

Pr✐♠❡✐r❛ ❧â♠♣❛❞❛ ❞❡s❧✐❣❛❞❛

+ An−2 | {z }

Pr✐♠❡✐r❛ ❧â♠♣❛❞❛ ❧✐❣❛❞❛

✳

❖❜s❡r✈❡ q✉❡ s❡ ❛ ♣r✐♠❡✐r❛ ❧â♠♣❛❞❛ ❡stá ❧✐❣❛❞❛✱ ❡♥tã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛ ❧â♠♣❛❞❛ ❛❞❥❛❝❡♥t❡ ❞❡✈❡ ❡st❛r ❞❡s❧✐❣❛❞❛ ❡ ❝♦♠ ✐ss♦ ❤á An−2 ❝♦♥✜❣✉r❛çõ❡s ❞✐st✐♥t❛s ♣❛r❛ ♦ ❝❛s♦

❡♠ q✉❡ ❛ ♣r✐♠❡✐r❛ ❧â♠♣❛❞❛ ❡stá ❧✐❣❛❞❛✳

❉❡ss❛ ❢♦r♠❛✱ A1 = 2, A2 = 3, A3 = 5 ❡✱ ❛ ❝❛❞❛ ❧â♠♣❛❞❛ ❛❝r❡s❝❡♥t❛❞❛ ♥❛ s❛❧❛✱ ❛

♣❛rt✐r ❞❛ t❡r❝❡✐r❛✱ ♦ t♦t❛❧ ❞❡ ❝♦♥✜❣✉r❛çõ❡s é ❞❛❞♦ ♣❡❧❛ s♦♠❛ ❞❛s ❞✉❛s q✉❛♥t✐❞❛❞❡s ✐♠❡❞✐❛t❛♠❡♥t❡ ❛♥t❡r✐♦r❡s✳

❆♥❛❧♦❣❛♠❡♥t❡ ❛♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ s✉r❣❡ ♥❛s r❡s♣♦st❛s ♣❛r❛ ❛s ❝♦♥✜❣✉r❛çõ❡s ❝♦♠ n = 1,2,3, . . . ❧â♠♣❛❞❛s ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ❝♦♠ ✉♠ ❞❡s❧♦❝❛♠❡♥t♦ ❞❡ ❞✉❛s ♣♦s✐çõ❡s✱ ♦✉ s❡❥❛✱ An = Fn+2✳ P♦r ❡①❡♠♣❧♦✱ A10 = F12 = 144✳ ❉❡ss❡ ♠♦❞♦✱ ♦ ♥ú♠❡r♦

❞❡ ❝♦♥✜❣✉r❛çõ❡s ♣❛r❛ n ❧â♠♣❛❞❛s é ✐❣✉❛❧ ❛

An=

1

√

5



 1 +

√

5 2

!n+2

− 1− √

5 2

!n+2 = α

n+2_−βn+2

α₋β ✳

✶✳✷ ❙♦♠❛s ❞❡ ♥ú♠❡r♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐

❆♦ ♦❜s❡r✈❛r ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ ❛❧❣✉♥s ♣❛❞rõ❡s ♣❛r❡❝❡♠ s❡ r❡♣❡t✐r✳ ❖❜s❡r✲ ✈❛♥❞♦ ❡ss❡s ♣❛❞rõ❡s✱ ✈❛♠♦s ❞❡st❛❝❛r ❛❧❣✉♠❛s r❡❧❛çõ❡s ❡♥tr❡ ❛ s♦♠❛ ❞❡ ❞❡t❡r♠✐♥❛❞♦s ♥ú♠❡r♦s ❡ ♣♦st❡r✐♦r♠❡♥t❡ ❞❡♠♦♥strá✲❧❛s✳

P♦r ❡①❡♠♣❧♦✱ ❛ s♦♠❛ ❞♦sn ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ é Fn+2−1✱ ✐st♦ é✱

F1 +F2+F3+· · ·+Fn =Fn+2−1✳

❆ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❞❡ í♥❞✐❝❡ ♣❛r é ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❋✐❜♦♥❛❝❝✐ s❡❣✉✐♥t❡ ♠❡♥♦s ✉♠❛ ✉♥✐❞❛❞❡✱ ♦✉ s❡❥❛✱

✶✳✷ ❙♦♠❛s ❞❡ ♥ú♠❡r♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✾

❆ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❞❡ í♥❞✐❝❡ í♠♣❛r é ❛ s❡❣✉✐♥t❡✿

F1 +F3+F5+· · ·+F2n−1 =F2n✳

❆ s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❞♦s n ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ é

F2

1 +F22+F32+· · ·+Fn2 =FnFn+1✳

❆❧é♠ ❞✐ss♦✱ ♦ ♣❛❞rã♦ ❞♦s ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ s✉❣❡r❡ q✉❡ ❞♦✐s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❝♦♥s❡❝✉t✐✈♦s sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✱ ♦✉ s❡❥❛✱ ♠❞❝(Fn, Fn+1) = 1✱

♣❛r❛ t♦❞♦n_∈N✳

◗✉❛♥❞♦ ♥ã♦ ❤♦✉✈❡r ♣r❡❥✉í③♦ ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ✈❛♠♦s ❞❡♥♦t❛r ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ ❞♦✐s ♥ú♠❡r♦sa ❡ b ♣♦r (a, b)✳

❱❛♠♦s ♠♦str❛r ❡ss❡s r❡s✉❧t❛❞♦s ✉s❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦✳

Pr♦♣♦s✐çã♦ ✶✳✺✳ ◗✉❛✐sq✉❡r ❞♦✐s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❝♦♥s❡❝✉t✐✈♦s sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✱ ♦✉ s❡❥❛✱ (Fn, Fn+1) = 1,∀n ∈N✳

❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❡ q✉❡ (F1, F2) = 1❡ (F2, F3) = 1✳ ❙✉♣♦♥❤❛ q✉❡ (Fn, Fn+1) = 1✳

◗✉❡r❡♠♦s ♠♦str❛r q✉❡ (Fn+1, Fn+2) = 1✳ ❙❛❜❡♠♦s q✉❡ Fn+2 = Fn+1 +Fn ❡ q✉❡ s❡

(a, b₋na)❡stá ❞❡✜♥✐❞♦✱ a, b, n_∈Z✱ ❡♥tã♦ (a, b) = (a, b₋na)✱ ✈❡r ❬✼❪✳ ❆ss✐♠✱ (Fn+1, Fn+2) = (Fn+1, Fn+1+Fn) = (Fn+1, Fn+1+Fn−Fn+1) = (Fn+1, Fn)✳

❈♦♠♦✱ ♣♦r ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ (Fn+1, Fn) = 1✱ s❡❣✉❡ q✉❡ (Fn+1, Fn+2) = 1✳

P♦rt❛♥t♦✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ✜♥✐t❛✱(Fn, Fn+1) = 1,∀n∈N✳

Pr♦♣♦s✐çã♦ ✶✳✻✳ F1+F2+F3+· · ·+Fn=Fn+2−1 ♣❛r❛ t♦❞♦ n∈N✳

❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❡ q✉❡ F1 = F1+2−1✱ ✐st♦ é✱ 1 = F3 −1 = 1 ❡ q✉❡ F1 +F2 =

F2+2−1 =F4 −1 = 2✳ ❆❣♦r❛✱ s✉♣♦♥❞♦ q✉❡ F1+F2+F3+· · ·+Fn =Fn+2 −1 s❡❥❛

✈❡r❞❛❞❡✐r❛✱ q✉❡r❡♠♦s ♣r♦✈❛r q✉❡F1+F2+F3+· · ·+Fn+Fn+1 =Fn+3−1 t❛♠❜é♠ é

✈❡r❞❛❞❡✐r❛✳

P♦r ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦sF1+F2+F3+· · ·+Fn =Fn+2−1✳ ❆ss✐♠✱ s♦♠❛♥❞♦

Fn+1 ❛ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ❛♥t❡r✐♦r✱ t❡♠♦s✿

F1+F2+· · ·+Fn+Fn+1 =Fn+1+Fn+2−1✳

P♦ré♠✱ Fn+1+Fn+2−1 = Fn+3−1✳ ❉❡ss❛ ❢♦r♠❛✱

✶✳✷ ❙♦♠❛s ❞❡ ♥ú♠❡r♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✶✵

P♦rt❛♥t♦✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦✱ F1 +F2 +F3 +F4 +· · ·+Fn = Fn+2 −1 é

✈á❧✐❞❛ ♣❛r❛ t♦❞♦n _∈N✳

Pr♦♣♦s✐çã♦ ✶✳✼✳ F2+F4+F6+· · ·+F2n=F2n+1−1✱ ♣❛r❛ t♦❞♦ n∈N✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ ❛ ❜❛s❡ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦sF2 =F2+1−1 = 1❡F2+F4 =F5−1 = 4✳

❙✉♣♦♥❤❛ q✉❡F2+F4+F6+· · ·+F2n=F2n+1−1 ♣❛r❛ ✉♠ ❝❡rt♦ n ♥❛t✉r❛❧✳

◗✉❡r❡♠♦s ♣r♦✈❛r q✉❡

F2+F4+F6+· · ·+F2n+F2(n+1) =F2(n+1)+1−1 = F2n+3−1✳

❙♦♠❛♥❞♦ F2(n+1) à ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦s✿

F2+F4+F6+· · ·+F2n+F2(n+1) =F2n+1+F2(n+1)−1✳

❖❜s❡r✈❡ q✉❡ F2n+1+F2(n+1) =F2n+3✳ ▲♦❣♦✱

F2+F4+F6+· · ·+F2n+F2(n+1) =F2(n+1)+1−1 = F2n+3−1✳

❖ q✉❡ ♣r♦✈❛ ♦ r❡s✉❧t❛❞♦ ♣❛r❛ t♦❞♦ n ♥❛t✉r❛❧✳

Pr♦♣♦s✐çã♦ ✶✳✽✳ F1+F3+F5+· · ·+F2n−1 =F2n✱ ♣❛r❛ t♦❞♦ n ♥❛t✉r❛❧✳

❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❡ q✉❡ F1 = F2·1 = 1 ❡ q✉❡ F1 +F3 = F2·2 = F4 = 3✳ ❆❣♦r❛

s✉♣♦♥❤❛ q✉❡ ♣❛r❛ ✉♠ ❝❡rt♦n ♥❛t✉r❛❧

F1+F3+F5+· · ·+F2n−1 =F2n s❡❥❛ ✈❡r❞❛❞❡✐r❛✳

◗✉❡r❡♠♦s ♣r♦✈❛r q✉❡

F1+F3 +F5+· · ·+F2n−1+F2n+1 =F2n+2✳

❙♦♠❛♥❞♦ F2n+1 à ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦s✿

F1 +F3+F5+· · ·+F2n−1+F2n+1 =F2n+F2n+1 =F2n+2✳

P♦rt❛♥t♦✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ✜♥✐t❛✱ F1 +F3+F5 +· · ·+F2n−1 =F2n✱ ♣❛r❛

t♦❞♦n ♥❛t✉r❛❧✳

Pr♦♣♦s✐çã♦ ✶✳✾✳ (F1)2+ (F2)2 + (F3)2+· · ·+ (Fn)2 =FnFn+1✱ ♣❛r❛ t♦❞♦ n ♥❛t✉r❛❧✳

❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ (F1)2 = F1F2 = 1 ❡ (F1)2+ (F2)2 = F2F3 = 2✱ ♠♦str❛♥❞♦ ❛

✶✳✷ ❙♦♠❛s ❞❡ ♥ú♠❡r♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✶✶

(F1)2+ (F2)2+ (F3)2+· · ·+ (Fn)2 =FnFn+1✱

♣❛r❛ ✉♠ ❝❡rt♦ ✈❛❧♦rn ♥❛t✉r❛❧✳ ◗✉❡r❡♠♦s ♣r♦✈❛r q✉❡

(F1)2+ (F2)2+ (F3)2+· · ·+ (Fn)2+ (Fn+1)2 =Fn+1Fn+2✳

❆❞✐❝✐♦♥❛♥❞♦(Fn+1)2 à ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦s✿

(F1)2+ (F2)2+ (F3)2+· · ·+ (Fn)2+ (Fn+1)2 =FnFn+1+ (Fn+1)2✳

❈♦❧♦❝❛♥❞♦ Fn+1 ❡♠ ❡✈✐❞ê♥❝✐❛✱ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡✱ ❡ ✉t✐❧✐③❛♥❞♦ ❛

❞❡✜♥✐çã♦ r❡❝✉rs✐✈❛ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ ♦❜t❡♠♦s✿

(F1)2+ (F2)2+ (F3)2+· · ·+ (Fn)2+ (Fn+1)2 =Fn+1(Fn+Fn+1) = Fn+1Fn+2✳

❆ss✐♠✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦✱ ♦ r❡s✉❧t❛❞♦ é ✈❡r❞❛❞❡✐r♦ ♣❛r❛ t♦❞♦n ♥❛t✉r❛❧✳

❆ Pr♦♣♦s✐çã♦ ✶✳✾ ♠♦str❛ ✉♠ r❡s✉❧t❛❞♦ ❞❡ r❡❧❡✈❛♥t❡ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛✱ ❛ s❛❜❡r✿ ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s ♣r✐♠❡✐r♦s n q✉❛❞r❛❞♦s ❝✉❥♦s ❧❛❞♦s sã♦ ♦s ♣r✐♠❡✐r♦s n ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ r❡s♣❡❝t✐✈❛♠❡♥t❡ é ❡q✉✐✈❛❧❡♥t❡ à ár❡❛ ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❝✉❥♦s ❧❛❞♦s sã♦ Fn ❡ Fn+1✳

❖s q✉❛❞r❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝❛✐①❛❞♦s ♣❛r❛ ❢♦r♠❛r ♦ r❡tâ♥❣✉❧♦ ❛❜❛✐①♦✳ ❊♠ ❝❛❞❛ q✉❛✲ ❞r❛❞♦ ❢♦✐ tr❛ç❛❞♦ ✉♠ q✉❛rt♦ ❞❡ ❝ír❝✉❧♦ ❢♦r♠❛♥❞♦ ❛ s✉r♣r❡❡♥❞❡♥t❡ ❡s♣✐r❛❧ ❞❡ ❋✐❜♦♥❛❝❝✐✱✹

✐❧✉str❛❞❛ ♥❛ ✜❣✉r❛ s❡❣✉✐♥t❡✱ ❛ q✉❛❧ tr❛❞✉③ ✈✐s✉❛❧♠❡♥t❡ ❛ ✐❞❡✐❛ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✾✳

1 1

2

3 5 8

13 21

34

55

89 144

❋✐❣✉r❛ ✶✳✷✿ ❊s♣✐r❛❧ ❞❡ ❋✐❜♦♥❛❝❝✐

❊①❝❡❧❡♥t❡s tr❛❜❛❧❤♦s ❣❡♦♠étr✐❝♦s ❥á ❢♦r❛♠ ❢❡✐t♦s ❝♦♠ ❡ss❡ t❡♠❛ ❡ ❡①✐st❡♠ ❜♦❛s ❢♦♥t❡s ❞❡ ♣❡sq✉✐s❛ ❝♦♥❤❡❝✐❞❛s✱ ❛s q✉❛✐s ♣♦❞❡♠ s❡r ❡①♣❧♦r❛❞❛s ♣♦r ❛q✉❡❧❡s q✉❡ t❡♥❤❛♠ ✐♥t❡r❡ss❡✳ P♦r ❡①❡♠♣❧♦✱ ❡♠ ❬✺❪ ❡ ❬✷❪✱ ✈ár✐❛s ✐♥❢♦r♠❛çõ❡s sã♦ ❡♥❝♦♥tr❛❞❛s✳

✶✳✸ ❋✐❜♦♥❛❝❝✐ ❡ ❛❧❣✉♠❛s r❡❧❛çõ❡s ✐♥t❡r❡ss❛♥t❡s ✶✷

▼✉✐t♦s ❛✉t♦r❡s ❛❜♦r❞❛♠ ❛ r❡❧❛çã♦ ❡♥tr❡ ♦ r❡tâ♥❣✉❧♦ ❝♦♥str✉í❞♦ ♥❛ ❋✐❣✉r❛ ✶✳✷ ❡ ❛ r❛③ã♦ á✉r❡❛✳ ▼❛s ❛♣❡s❛r ❞❛ ✈❛st❛ ❧✐t❡r❛t✉r❛ s♦❜r❡ r❛③ã♦ á✉r❡❛✱ ♥ã♦ ❡♥tr❛r❡♠♦s ♣r♦❢✉♥❞❛♠❡♥t❡ ♥❡ss❛ s❡❛r❛✳ ❊♠ r❡❧❛çã♦ ❛ ❡❧❛✱ ✈❛♠♦s ♥♦s ❧✐♠✐t❛r ❛ ❝♦♠❡♥t❛r s♦❜r❡ ♦ ❧✐♠✐t❡ ❛❜❛✐①♦✳

lim

n→∞ Fn+1

Fn

= 1 +

√

5 2 .

✶✳✸ ❋✐❜♦♥❛❝❝✐ ❡ ❛❧❣✉♠❛s r❡❧❛çõ❡s ✐♥t❡r❡ss❛♥t❡s

◆❛ t❡r❝❡✐r❛ s❡çã♦ ❞❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉tór✐♦✱ ✈❛♠♦s ❛❜♦r❞❛r ♦✉tr♦s tó♣✐❝♦s ♠❛✲ t❡♠át✐❝♦s ♥♦s q✉❛✐s é ❡♥❝♦♥tr❛❞❛ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ❡ t❛♠❜é♠ ❞❡♠♦♥str❛r ❛ ❧✐❣❛çã♦ ❞♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❛♦ ♥ú♠❡r♦ ❞❡ ♦✉r♦(1 +√5)/2_≈1,618✳

❖ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦ é s♦❜r❡ ♦ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧ ✱ ♦ q✉❛❧ t❡♠ ♠✉✐t❛s ♣r♦♣r✐❡❞❛❞❡s ✐♥t❡r❡ss❛♥t❡s ❝♦♥❤❡❝✐❞❛s✳

❖s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ t❛♠❜é♠ sã♦ ❡♥❝♦♥tr❛❞♦s ♥♦ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧ q✉❛♥❞♦ sã♦ s♦♠❛❞♦s ♦s ♥ú♠❡r♦s ♥❛s ❞✐❛❣♦♥❛✐s ♣❛r❛❧❡❧❛s ❝♦♥❢♦r♠❡ ❛ ❋✐❣✉r❛1.3✳

❊①❡♠♣❧♦ ✶✳✶✵ ✭❋✐❜♦♥❛❝❝✐ ❡ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✮✳ ❆ s♦♠❛ ❞♦s ♥ú♠❡r♦s ♥❛s ❞✐❛❣♦♥❛✐s ♣❛r❛❧❡❧❛s✱ ❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ♣r♦❞✉③ ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳

❋✐❣✉r❛ ✶✳✸✿ ❙♦♠❛ ❞❛s ❞✐❛❣♦♥❛✐s ❞♦ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧

❆ ❥✉st✐✜❝❛t✐✈❛ ❞❡ss❡ ❢❛t♦ é ❞❡✈✐❞❛ ❛♦ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês ➱❞♦✉❛r❞ ▲✉❝❛s✱ q✉❡✱ ❡♠

✶✳✸ ❋✐❜♦♥❛❝❝✐ ❡ ❛❧❣✉♠❛s r❡❧❛çõ❡s ✐♥t❡r❡ss❛♥t❡s ✶✸

❚❡♦r❡♠❛ ✶✳✶✶✳ Fn+1 = n₀

+ n−₁1+ n−₂2+_{· · ·}+ n−_jj✱ ♦♥❞❡ j é ♦ ♠❛✐♦r ✐♥t❡✐r♦ ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ n/2✳

❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ❞❡♠♦♥str❛r ✉t✐❧✐③❛♥❞♦ ✐♥❞✉çã♦ s♦❜r❡ n✳ ❖❜s❡r✈❡ q✉❡ ♣❛r❛ n = 0,1 ❡ 2♦ r❡s✉❧t❛❞♦ é ✈á❧✐❞♦✳ ❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ k,0_≤k < n✱ ❛ ❛✜r♠❛çã♦ s❡❥❛ ✈❡r❞❛❞❡✐r❛✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ ❡❧❛ ✈❛❧❡ ♣❛r❛k+ 1 =n✳

❙❛❜❡♠♦s q✉❡Fk+1 =Fk+Fk−1✳ P❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦s✿

Fk+1 = Fk+Fk−1

=

k₋1 0

+

k₋2 1

+

k₋3 2

+_{· · ·}+

k₋j₋1

j

+

k₋2 0

+

k₋3 1

+

k₋4 2

+_{· · ·}+

k₋j₋1

j₋1

=

k₋1 0

+

k₋2 0

+

k₋2 1

+

k₋3 1

+

k₋3 2

+_{· · ·}+

k₋j₋1

j ₋1

+

k₋j₋1

j

=

k₋1 0

+

k₋1 1

+

k₋2 2

+_{· · ·}+

k₋j j

.

■st♦ é✱Fk+1 = k−₀1

+ k−₁1+ k−₂2+_{· · ·}+ k−_jj✳ ❖♥❞❡ ✉s❛♠♦s ❛ ❝♦♥❤❡❝✐❞❛ r❡❧❛çã♦

❞❡ ❙t✐❢❡❧ ✺_✳

❖❜s❡r✈❛♥❞♦ q✉❡ k−1 0

= k₀✱ t❡♠♦s Fk+1 = k₀

+ k−₁1+ k−₂2+_{· · ·}+ k−_jj✳

P♦rt❛♥t♦✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦✱ ❛ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ n ✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦✳

❖ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧ é ✉♠❛ ❢❡rr❛♠❡♥t❛ r✐q✉íss✐♠❛ ♣❛r❛ ❡①♣❧♦r❛r ❝♦♥❝❡✐t♦s ✐♥t❡r✲ ❞✐s❝✐♣❧✐♥❛r❡s ♥❛ ▼❛t❡♠át✐❝❛✳ ❆❧é♠ ❞❛s s✉❛s ✐♥ú♠❡r❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❞❛ r❡❧❛çã♦ ❝♦♠ ♦ ❜✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥✱ ♦ ♣r♦❢❡ss♦r ♣♦❞❡ ✐♥tr♦❞✉③✐r ♦s ❝♦♥❝❡✐t♦s ❞❡ s❡q✉ê♥❝✐❛s ♥✉♠ér✐❝❛s ❡ ♣r♦❣r❡ssõ❡s✱ tr❛❜❛❧❤❛r ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s ❞❡ ♦r❞❡♠ n✱ ❛❝❤❛♥❞♦ ❞❡t❡r♠✐♥❛❞♦s t❡r♠♦s ❝♦♠ ♦ ❛✉①í❧✐♦ ❞♦s s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❡✱ ❛❧é♠ ❞✐ss♦✱ ❞❡✐①❛r q✉❡ ♦s ❛❧✉♥♦s ❡①♣❧♦r❡♠ ❡ss❡ tr✐â♥❣✉❧♦ ♥♦ ✐♥t✉✐t♦ ❞❡ ♣❡r❝❡❜❡r ♣❛❞rõ❡s✳ P❛r❛ ♠❛✐s ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧ ❝♦♥s✉❧t❡ ❬✶✶❪✳

◆♦ ❈❛♣ít✉❧♦ 3 ❞❡ss❡ tr❛❜❛❧❤♦✱ ❡♥❝♦♥tr❛✲s❡ ❛ ❚❛❜❡❧❛ 3.2✳ ❉❛ ♠❡s♠❛ ♠❛♥❡✐r❛ q✉❡

é ♣♦ssí✈❡❧ ❡①♣❧♦r❛r ♣❛❞rõ❡s ♥✉♠ér✐❝♦s ♥♦ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✱ é ♣♦ssí✈❡❧ ✉t✐❧✐③❛r ❛

✺ n p + n p+1

= n+1

p+1

✶✳✸ ❋✐❜♦♥❛❝❝✐ ❡ ❛❧❣✉♠❛s r❡❧❛çõ❡s ✐♥t❡r❡ss❛♥t❡s ✶✹

r❡❢❡r✐❞❛ t❛❜❡❧❛ ♣❛r❛ tr❛❜❛❧❤❛r ❡♠ s❛❧❛ ❛ ❤❛❜✐❧✐❞❛❞❡ ❞♦s ❛❧✉♥♦s ❡♠ ❡①♣❧♦r❛r ♣❛❞rõ❡s ❡ ❢❛③❡r ✐♥❢❡rê♥❝✐❛s✳ ◆♦ ❞❡❝♦rr❡r ❞♦ t❡①t♦✱ s❡rã♦ ❛❜♦r❞❛❞♦s ♦s ❝♦♥❝❡✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ♣r♦❢❡ss♦r ♣♦❞❡r ❝♦♥❞✉③✐r ❡ss❡ t✐♣♦ ❞❡ ❛t✐✈✐❞❛❞❡✳

❖✉tr♦ ❢❛t♦ ✐♥t❡r❡ss❛♥t❡ s♦❜r❡ ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ é q✉❡ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ ❛ s♦♠❛ ❞❡ ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❞✐st✐♥t♦s ❡ ♥ã♦ ❝♦♥s❡❝✉✲ t✐✈♦s✳ ❊ss❡ ❢❛t♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❡ ❩❡❝❦❡♥❞ör✛✳

P❛r❛ ❡①❡♠♣❧✐✜❝❛r✱ ♦❜s❡r✈❡ q✉❡ 1,2,3,5 ❡ 8 sã♦ ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❡ 4 = F4 +

F1,6 = F5+F1,7 =F5+F3,9 = F6 +F1,10 = F6+F3 ❡ 11 = F6+F4 sã♦ s♦♠❛s ❞❡

♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❞✐❢❡r❡♥t❡s ❡ ♥ã♦ ❝♦♥s❡❝✉t✐✈♦s✳ ❆ ♣r♦✈❛ s❡rá ❢❡✐t❛ ♣♦r ✐♥❞✉çã♦✳

❚❡♦r❡♠❛ ✶✳✶✷ ✭❚❡♦r❡♠❛ ❞❡ ❩❡❝❦❡♥❞ör✛✮✳ ❚♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞❡ ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❞✐st✐♥t♦s ❡ ♥ã♦ ❝♦♥s❡❝✉t✐✈♦s✳

❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❡ q✉❡ ❛ ❜❛s❡ ❞❡ ✐♥❞✉çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s n = 1 = F1✳

❙✉♣♦♥❤❛ q✉❡ ♦ r❡s✉❧t❛❞♦ s❡❥❛ ✈❡r❞❛❞❡✐r♦ ♣❛r❛ ✉♠ ❝❡rt♦ ♥❛t✉r❛❧n✱ ♦✉ s❡❥❛✱

n =Fm1 +Fm2 +· · ·+Fmk✱ ❝♦♠mi+1−mi ≥2✱ ♣❛r❛ i∈ {1,2, . . . , k−1}✳

◗✉❡r❡♠♦s ♠♦str❛r q✉❡ n+ 1 ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞❡ ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐

❞✐st✐♥t♦s ❡ ♥ã♦ ❝♦♥s❡❝✉t✐✈♦s✱ ♦✉ s❡❥❛✱

n+ 1 = 1 +Fm1 +Fm2 +· · ·+Fmk✳

◆♦t❡ q✉❡ s❡ m1 ≥3✱ ♦ r❡s✉❧t❛❞♦ ❡stá ♣r♦✈❛❞♦ ♣♦✐s1 = F1 ❡ ♣♦rt❛♥t♦

n+ 1 = 1 +Fm1 +Fm2 +· · ·+Fm_k ❡m1−1≥2✳

❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ✈❛♠♦s s✉♣♦r q✉❡m1 = 2✱ ♣♦✐sF1 =F2 = 1✳ ❆ss✐♠

n+ 1 =F1+F2 | {z }

F3

+Fm2 +· · ·+Fmk =F3+Fm2 +· · ·+Fmk✳

❆❣♦r❛ ♦❜s❡r✈❡ q✉❡ s❡ m2 ≥5 ♦ r❡s✉❧t❛❞♦ ❡stá ♣r♦✈❛❞♦✳ ❙✉♣♦♥❞♦ m2 = 4✱ t❡♠♦s✿

n+ 1 =F3+F4 | {z }

F5

+Fm3+· · ·+Fm_k ❡ ❧♦❣♦n+ 1 =F5+Fm3 +· · ·+Fm_k✳

✶✳✸ ❋✐❜♦♥❛❝❝✐ ❡ ❛❧❣✉♠❛s r❡❧❛çõ❡s ✐♥t❡r❡ss❛♥t❡s ✶✺

P❛r❛ ❡①❡♠♣❧✐✜❝❛r ♦ ❚❡♦r❡♠❛ ✶✳✶✷✱ ♦❜s❡r✈❡ q✉❡2015 =F17+F14+F9 +F5+F3 =

1597 + 377 + 34 + 5 + 2✳ ❊ss❛ r❡♣r❡s❡♥t❛çã♦ é ú♥✐❝❛✱ ❛ ♠❡♥♦s ❞❛ ♦r❞❡♠ ❞❛s ♣❛r❝❡❧❛s

❞❛ s♦♠❛✱ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❞✐st✐♥t♦s ❡ ♥ã♦ ❝♦♥s❡❝✉t✐✈♦s✳ ❆ s❡❣✉✐r ✈❛♠♦s r❡❧❛❝✐♦♥❛r ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ à r❛③ã♦ á✉r❡❛ ❡ ♣♦st❡r✐♦r♠❡♥t❡✱ ❡♥❝❡rr❛♥❞♦ ❡ss❛ s❡çã♦✱ ❛ss♦❝✐á✲❧❛ às tr✐♣❧❛s ♣✐t❛❣ór✐❝❛s✳

P❛r❛ ♦ q✉❡ s❡❣✉❡✱ ❛ ❞❡✜♥✐çã♦ ❛❧❣é❜r✐❝❛ ❞❡ r❛③ã♦ á✉r❡❛ é ❛ s❡❣✉✐♥t❡✳

❉❡✜♥✐çã♦ ✶✳✶✸ ✭❘❛③ã♦ ➪✉r❡❛✮✳ ❆ r❛③ã♦ á✉r❡❛ é ✉♠❛ r❡❧❛çã♦ ♠❛t❡♠át✐❝❛ ❞❡✜♥✐❞❛ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ♣❡❧❛ ❡①♣r❡ssã♦(a+b)/a=a/b=α✱ ❡♠ q✉❡ a ❡ b r❡♣r❡s❡♥t❛♠ ♥ú♠❡r♦s r❡❛✐s✱ ❡ α r❡♣r❡s❡♥t❛ ✉♠❛ ❝♦♥st❛♥t❡ ❞❡ ✈❛❧♦r ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✐❣✉❛❧ ❛ 1,618✳

❆ ♣❛rt✐r ❞❛ ❞❡✜♥✐çã♦ ❛❧❣é❜r✐❝❛ (a+b)/a = a/b = α✱ ✈❡r✐✜❝❛✲s❡ q✉❡ 1 +b/a = α✱ ✐st♦ é✱1 +α−1 _{=α✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ♣♦r α✱ ♦❜t❡♠♦s✿ α}2_{−α−1 = 0✱}

❝♦♠ r❛í③❡sα = (1 +√5)/2 ❡β = (1₋√5)/2✳

➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡

lim

n→∞ Fn+1

Fn

= 1 +

√

5 2 .

P❛r❛ ♠♦str❛r♠♦s ❡ss❡ ❢❛t♦✱ ✈❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ❞❡♠♦♥str❛r✱ ✉t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ❇✐♥❡t✱ ❛s ❞✉❛s ✐❞❡♥t✐❞❛❞❡s s❡❣✉✐♥t❡s✳

Pr♦♣♦s✐çã♦ ✶✳✶✹✳ F2n+2F2n+1−F2nF2n+3 = 1✳

❉❡♠♦♥str❛çã♦✳ ❯t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ❇✐♥❡t✱ t❡♠♦s q✉❡✿

F2n+2F2n+1−F2nF2n+3 =

1 5

(α2n+2₋β2n+2)(α2n+1₋β2n+1)

−1₅(α2n₋β2n)(α2n+3₋β2n+3) = 1

5(α

4n+3_+β4n+3

−α2n+2β2n+1₋β2n+2α2n+1)

−1₅(α4n+3+β4n+3₋α2nβ2n+3₋β2nα2n+3) = 1

5(−α

2n+2_β2n+1

−β2n+2α2n+1+α2nβ2n+3+β2nα2n+3) = 1

5

−(αβ)2n+1α₋(βα)2n+1β+ (αβ)2nβ3 + (βα)2nα3

= 1

5(α+β+β

3_+α3₎

= 1

5(1 + 4) = 1,

✶✳✸ ❋✐❜♦♥❛❝❝✐ ❡ ❛❧❣✉♠❛s r❡❧❛çõ❡s ✐♥t❡r❡ss❛♥t❡s ✶✻

Pr♦♣♦s✐çã♦ ✶✳✶✺✳ F2

n =Fn−1Fn+1−(−1)n✳

❉❡♠♦♥str❛çã♦✳ ❯t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ❇✐♥❡t✱ t❡♠♦s q✉❡✿

Fn−1Fn+1−(−1)n =

(αn−1_−βn−1_{) (α}n+1_−βn+1₎

(α₋β)2 −(−1)

n

= α

2n_−(αβ)n_α−1_β−(αβ)n_β−1_α+β2n

(α₋β)2 −(αβ)

n

= α

2n_−(αβ)n_(α−1_β+β−1_{α) +β}2n

(α₋β)2 −(αβ)

n

= α

2n_−(αβ)n_{(−ββ−αα) +β}2n

(α₋β)2 −(αβ)

n

= α

2n_{+ (αβ)}n_(β2_+α2_{) +β}2n

(α₋β)2 −(αβ)

n

= α

2n_{+ (αβ)}n_{(β+ 1 +α+ 1) +β}2n

(α₋β)2 −(αβ)

n

= α

2n_{+ 3(αβ)}n_+β2n

(α₋β)2 −(αβ)

n

= α

2n_{+ 3(αβ)}n_+β2n_−5(αβ)n

(α₋β)2 = α

2n_−2(αβ)n_+β2n

(α₋β)2 = (α

n_−βn₎2

(α₋β)2 =

αn_−βn

α₋β

2

= Fn2.

❖❜s❡r✈❡ q✉❡✱ ♥❛s ♠❛♥✐♣✉❧❛çõ❡s ❛❧❣é❜r✐❝❛s ❛❝✐♠❛✱ ✉s❛♠♦s ♦s s❡❣✉✐♥t❡s ❢❛t♦s ❝♦♥❤❡❝✐❞♦s✿ α+β = 1, αβ =₋1, α−1 _{=−β, β}−1 _{=−α, α}2 _{=α+ 1 ❡ β}2 _{=β+ 1✳}

❱♦❧t❛♥❞♦ à ❛♥á❧✐s❡ ❞❡

lim

n→∞ Fn+1

Fn

= 1 +

√

5 2 ,

s❡❥❛♠ rn = Fn+1/Fn ❛ r❛③ã♦ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❝♦♥s❡❝✉t✐✈♦s ❡

In = [r2n−1, r2n], n = 1,2,3, . . . ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✐♥t❡r✈❛❧♦s ❢❡❝❤❛❞♦s✱ t❛✐s q✉❡

✶✳✸ ❋✐❜♦♥❛❝❝✐ ❡ ❛❧❣✉♠❛s r❡❧❛çõ❡s ✐♥t❡r❡ss❛♥t❡s ✶✼

✈❡r ❬✶✵❪✱ ❛✜r♠❛ q✉❡ s❡I1, I2, I3, . . . é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✐♥t❡r✈❛❧♦s ❢❡❝❤❛❞♦s ❡ ❧✐♠✐t❛❞♦s✱

❡ s❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ In t❡♥❞❡ ❛ ③❡r♦ q✉❛♥❞♦ n t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦✱ ❡♥tã♦ ❡①✐st❡ ✉♠✱ ❡

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

1,0 1,2 1,4 1,6 1,8 2,0

1,6

1+ 5 2

❋✐❣✉r❛ ✶✳✹✿ ■♥t❡r✈❛❧♦s ❡♥❝❛✐①❛♥t❡s

P❛r❛ ✈❡r q✉❡In é ❡♥❝❛✐①❛♥t❡✱ ♦❜s❡r✈❡ q✉❡ ❛ s❡q✉ê♥❝✐❛rn ♣♦ss✉✐ ❞✉❛s s✉❜s❡q✉ê♥❝✐❛s

♠♦♥ót♦♥❛s✱ ❛ s❛❜❡r✿

r2 > r4 >· · ·> r2n> r2n+2 >· · · ❡ r1 < r3 <· · ·< r2n−1 < r2n+1 <· · ·

▼♦str❛r ♦ ❢❛t♦ r2n > r2n+2 é ❡q✉✐✈❛❧❡♥t❡ ❛ ❞❡♠♦♥str❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡

F2n+1/F2n > F2n+3/F2n+2✱ ♦✉ s❡❥❛✱ ✈❛♠♦s ♠♦str❛r q✉❡ F2n+2F2n+1 −F2nF2n+3 > 0✳

P❛r❛ ✐ss♦✱ ❜❛st❛ ❞✐✈✐❞✐r ❛ Pr♦♣♦s✐çã♦ ✶✳✶✹ ♣♦rF2n+2F2n✳ ❖❜s❡r✈❡✳

F2n+2F2n+1

F2n+2F2n −

F2nF2n+3

F2n+2F2n

= 1

F2n+2F2n ⇔

F2n+1

F2n −

F2n+3

F2n+2

= 1

F2n+2F2n

>0_⇒r2n> r2n+2✳

❆♥❛❧♦❣❛♠❡♥t❡✱ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ r2n−1 < r2n+1 ♣❛r❛ t♦❞♦n ∈N✳

❆❧é♠ ❞✐ss♦✱ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ r2n−1 < r2n ♣❛r❛ t♦❞♦ n ♥❛t✉r❛❧✳ ❉❡ ❢❛t♦✱ ❝♦♠

❛✉①í❧✐♦ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶✺✱ t❡♠♦sF2

2n=F2n−1F2n+1−(−1)2n✳

❉✐✈✐❞✐♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ ❛♥t❡r✐♦r ♣♦rF2nF2n−1✱ t❡♠♦s✿

F2n+1

F2n −

F2n

F2n−1

= 1

F2n−1F2n+1✳

P♦rt❛♥t♦✱ ❛ ♣❛rt✐r ❞❛ ✐❣✉❛❧❞❛❞❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❛❝✐♠❛✱ ♦❜t❡♠♦s q✉❡✿

r2n−r2n−1 =

1

F2n−1F2n+1

✶✳✹ ◆ú♠❡r♦s ❞❡ ▲✉❝❛s ✶✽

r2n−r2n−1 =

1

F2n−1F2n+1 →

0 q✉❛♥❞♦n _{→ ∞}✳

❯♠❛ ✈❡③ q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞♦s ✐♥t❡r✈❛❧♦s ❢❡❝❤❛❞♦s [r1, r2],[r3, r4],[r5, r6], . . . é ❡♥❝❛✐✲

①❛♥t❡ ❡ ♦ t❛♠❛♥❤♦ ❞❡ In = [r2n−1, r2n] t❡♥❞❡ ❛ ③❡r♦ q✉❛♥❞♦ n t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦✱ ❡♥tã♦

❡①✐st❡ L_∈R t❛❧ q✉❡

L= lim

n→∞ Fn+1

Fn

= lim

n→∞

Fn+Fn−1

Fn

= lim

n→∞

1 + Fn−1

Fn

= 1 + 1

L.

❘❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ L = 1 + 1/L ❡ ♦❜s❡r✈❛♥❞♦✱ ♣❡❧♦ ❝♦♥t❡①t♦ ❞♦ ♣r♦❜❧❡♠❛✱ ❛ r❛✐③ ♣♦s✐t✐✈❛ L = (1 + √5)/2✱ t❡♠♦s q✉❡ ❛ r❛③ã♦ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐

❝♦♥s❡❝✉t✐✈♦s t❡♥❞❡ ❛ (1 +√5)/2✱ ✐st♦ é✱ lim

n→∞ Fn+1

Fn

= 1 +

√

5 2 .

❊①❡♠♣❧♦ ✶✳✶✻ ✭❋✐❜♦♥❛❝❝✐ ❡ tr✐♣❧❛s ♣✐t❛❣ór✐❝❛s✮✳ ◗✉❛tr♦ ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❝♦♥✲ s❡❝✉t✐✈♦s Fk, Fk+1, Fk+2 ❡ Fk+3 ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❛ ✉♠❛ tr✐♣❧❛ ♣✐t❛❣ór✐❝❛ ♣r✐♠✐t✐✈❛ s❡

Fk+1 ❡ Fk+2 tê♠ ♣❛r✐❞❛❞❡s ❞✐st✐♥t❛s✱ ❡ r❡❧❛❝✐♦♥❛❞♦s ❛ ✉♠❛ tr✐♣❧❛ ♣✐t❛❣ór✐❝❛ s❡ Fk+1 ❡

Fk+2 tê♠ ♣❛r✐❞❛❞❡s ✐❣✉❛✐s✱ ✐st♦ é✱ s❡ Fk+1 ≡Fk+2 (mod 2)✳

❙♦❧✉çã♦✳ ❙❛❜❡♠♦s q✉❡ ❛s tr✐♣❧❛s ♣✐t❛❣ór✐❝❛s ♣r✐♠✐t✐✈❛s ✭✈❡r ❬✶✽❪✮ sã♦ ❞❛ ❢♦r♠❛ a=m2_+n2_{, b= 2mn❡ c=m}2_−n2_{✱ ❝♦♠ (m, n) = 1 ❡m+n í♠♣❛r✳}

❖ ❢❛t♦ ❞❡ m ❡ n t❡r❡♠ ♣❛r✐❞❛❞❡s ❞✐st✐♥t❛s é ♣❛r❛ ❣❛r❛♥t✐r q✉❡ ❛ tr✐♣❧❛ ♣✐t❛❣ór✐❝❛ s❡❥❛ ♣r✐♠✐t✐✈❛✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ (m, n) = 1 t❡♠♦s q✉❡ (m2_{, m}2 _+n2_{) = 1 ❡ ♣♦rt❛♥t♦}

(a, c) = (m2_+n2_{, m}2_−n2_{) = (2m}2_{, m}2_+n2_{) = (2, m}2_+n2_{)✱ s❡rá ✐❣✉❛❧ ❛ ✶ s❡✱ ❡ s♦♠❡♥t❡}

s❡✱m2_+n2 _{é í♠♣❛r✱ ♦✉ s❡❥❛✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ m+n é í♠♣❛r✳}

❋❛③❡♥❞♦ Fk+1 = n ❡ Fk+2 = m✱ t❡♠♦s a = (Fk+2)2 + (Fk+1)2, b = 2Fk+2Fk+1 ❡

c= (Fk+2)2−(Fk+1)2 =FkFk+3✱ ❡①❡♠♣❧✐✜❝❛♥❞♦ ❛ r❡❧❛çã♦ ❝✐t❛❞❛ ❛❝✐♠❛✳

❖❜s❡r✈❡ q✉❡ ♦s q✉❛tr♦ ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ 1,1,2 ❡ 3 ❢♦r♠❛♠ ❛ tr✐♣❧❛ ♣✐t❛❣ór✐❝❛

♣r✐♠✐t✐✈❛ (3,4,5) ❡ ♦s q✉❛tr♦ ♥ú♠❡r♦s s❡❣✉✐♥t❡s 1,2,3 ❡ 5 ❢♦r♠❛♠ ❛ tr✐♣❧❛ ♣r✐♠✐t✐✈❛ (5,12,13)✳ ❏á ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐2,3,5 ❡8✱ ❝♦♠5_≡3✭♠♦❞ 2✮✱ ❢♦r♠❛♠ ❛ tr✐♣❧❛

♣✐t❛❣ór✐❝❛(16,30,34)✳

✶✳✹ ◆ú♠❡r♦s ❞❡ ▲✉❝❛s