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  1. ❈♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s✱ ❈❤❡♥✲❙t❡✐♥ ♠❡t❤♦❞ ❛♥❞ ❡①tr❡♠❡ ✈❛❧✉❡ t❤❡♦r② P❛rt❤❛♥✐❧ ❘♦②✱ ■♥❞✐❛♥ ❙t❛t✐st✐❝❛❧ ■♥st✐t✉t❡ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❆♥✐s❤ ●❤♦s❤ ❛♥❞ ▼❛①✐♠ ❑✐rs❡❜♦♠ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶ ✴ ✸✵

  2. ✶ ✶ ✶ ✶ ✸ ✷ ✸ ✸ ✶ ✶ ✷✹ ✼ ✸ ✸ ✸ ✶ ✼ ✼ ✸ ✷ ✸ ◆♦t❡ t❤❛t ✼ ✸ ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ✳ ❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s ✼ ✷✹ P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

  3. ✶ ✶ ✶ ✸ ✷ ✸ ✸ ✶ ✶ ✸ ✸ ✸ ✶ ✼ ✼ ✸ ✷ ✸ ◆♦t❡ t❤❛t ✼ ✸ ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ✳ ❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s ✼ ✶ ✷✹ = ✷✹ / ✼ P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

  4. ✶ ✶ ✸ ✷ ✸ ✶ ✶ ✸ ✸ ✶ ✼ ✸ ✷ ✸ ◆♦t❡ t❤❛t ✼ ✸ ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ✳ ❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s ✼ ✶ ✶ ✷✹ = ✷✹ / ✼ = ✸ + ✸ ✼ P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

  5. ✶ ✸ ✷ ✸ ✶ ✸ ✶ ✷ ✸ ◆♦t❡ t❤❛t ✼ ✸ ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ✳ ❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s ✼ ✶ ✶ ✶ ✷✹ = ✷✹ / ✼ = = ✸ + ✸ ✶ ✸ + ✼ ✼ / ✸ P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

  6. ✸ ✷ ✸ ◆♦t❡ t❤❛t ✼ ✸ ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ✳ ❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s ✼ ✶ ✶ ✶ ✶ ✷✹ = ✷✹ / ✼ = = = ✸ + ✸ ✶ ✶ ✸ + ✸ + ✷ + ✶ ✼ ✼ / ✸ ✸ P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

  7. ◆♦t❡ t❤❛t ✼ ✸ ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ✳ ❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s ✼ ✶ ✶ ✶ ✶ ✷✹ = ✷✹ / ✼ = = = := [ ✸ , ✷ , ✸ ] ✸ + ✸ ✶ ✶ ✸ + ✸ + ✷ + ✶ ✼ ✼ / ✸ ✸ P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

  8. ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ✳ ❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s ✼ ✶ ✶ ✶ ✶ ✷✹ = ✷✹ / ✼ = = = := [ ✸ , ✷ , ✸ ] ✸ + ✸ ✶ ✶ ✸ + ✸ + ✷ + ✶ ✼ ✼ / ✸ ✸ ◆♦t❡ t❤❛t ✼ > ✸ > ✶✳ P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

  9. ❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s ✼ ✶ ✶ ✶ ✶ ✷✹ = ✷✹ / ✼ = = = := [ ✸ , ✷ , ✸ ] ✸ + ✸ ✶ ✶ ✸ + ✸ + ✷ + ✶ ✼ ✼ / ✸ ✸ ◆♦t❡ t❤❛t ✼ > ✸ > ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ω = p / q ∈ ( ✵ , ✶ ) ✭✇✐t❤ gcd ( p , q ) = ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ✳ P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

  10. ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ ✱ ✶ ✷ ✭✇✐t❤ ✮ ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r ✐♥ ✵ ✶ ✳ ✶ ✷ ❈♦♥✈❡rs❡❧② ✳ ✳ ✳ ❲❤❡♥❡✈❡r A ✶ , A ✷ , A ✸ , A ✹ ∈ N ✱ ✶ [ A ✶ , A ✷ , A ✸ , A ✹ ] := ∈ ( ✵ , ✶ ) ✶ A ✶ + ✶ A ✷ + A ✸ + ✶ A ✹ ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r✳ P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✸ ✴ ✸✵

  11. ❈♦♥✈❡rs❡❧② ✳ ✳ ✳ ❲❤❡♥❡✈❡r A ✶ , A ✷ , A ✸ , A ✹ ∈ N ✱ ✶ [ A ✶ , A ✷ , A ✸ , A ✹ ] := ∈ ( ✵ , ✶ ) ✶ A ✶ + ✶ A ✷ + A ✸ + ✶ A ✹ ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ n ✱ ω = [ A ✶ , A ✷ , . . . A n ] ✭✇✐t❤ A ✶ , A ✷ , . . . A n ∈ N ✮ ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r ✐♥ ( ✵ , ✶ ) ✳ P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✸ ✴ ✸✵

  12. ❋✉rt❤❡r♠♦r❡ ✐♥ t❤✐s ❝❛s❡✱ t❤❡ tr✉♥❝❛t❡ ❛s ✳ ✶ ✷ ❈❛♥♦♥✐❝❛❧ r❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥✿ ✳ ✶ ✷ ✷✷ ✸✺✺ ❊①❛♠♣❧❡s✿ ✼ ❛♥❞ ✶✶✸ ✳ ◆♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ❚❤❡♦r❡♠ ❆ ♥✉♠❜❡r ω ∈ ( ✵ , ✶ ) ❤❛s ❛ ✉♥✐q✉❡ ♥♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ✶ ω = =: [ A ✶ , A ✷ , A ✸ , . . . ] ✶ A ✶ + ✶ A ✷ + A ✸ + ··· ✭✇✐t❤ ❡❛❝❤ A i ∈ N ✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ω / ∈ Q ✳ P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✹ ✴ ✸✵

  13. ❈❛♥♦♥✐❝❛❧ r❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥✿ ✳ ✶ ✷ ✷✷ ✸✺✺ ❊①❛♠♣❧❡s✿ ✼ ❛♥❞ ✶✶✸ ✳ ◆♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ❚❤❡♦r❡♠ ❆ ♥✉♠❜❡r ω ∈ ( ✵ , ✶ ) ❤❛s ❛ ✉♥✐q✉❡ ♥♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ✶ ω = =: [ A ✶ , A ✷ , A ✸ , . . . ] ✶ A ✶ + ✶ A ✷ + A ✸ + ··· ✭✇✐t❤ ❡❛❝❤ A i ∈ N ✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ω / ∈ Q ✳ ❋✉rt❤❡r♠♦r❡ ✐♥ t❤✐s ❝❛s❡✱ t❤❡ n th tr✉♥❝❛t❡ [ A ✶ , A ✷ , . . . A n ] → ω ❛s n → ∞ ✳ P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✹ ✴ ✸✵

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