▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ▼❛st❡r ♦❢ ❙❝✐❡♥❝❡ ✐♥ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❊r✐✈❡❧t♦♥ ●❡r❛❧❞♦ ◆❡♣♦♠✉❝❡♥♦ ❉❡♣❛rt♠❡♥t ♦❢ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❋❡❞❡r❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❙ã♦ ❏♦ã♦ ❞❡❧✲❘❡✐ ❆✉❣✉st ✷✵✶✺ Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✶ ✴ ✽✾
❚❡❛❝❤✐♥❣ P❧❛♥ ❈♦♥t❡♥t ✶ ❚❤❡ ❘❡❛❧ ❛♥❞ ❈♦♠♣❧❡① ◆✉♠❜❡rs ❙②st❡♠s ✷ ❇❛s✐❝ ❚♦♣♦❧♦❣② ✸ ◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ✹ ❈♦♥t✐♥✉✐t② ✺ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✻ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ♦❢ ❋✉♥❝t✐♦♥s ✼ ■❊❊❊ ❙t❛♥❞❛r❞ ❢♦r ❋❧♦❛t✐♥❣✲P♦✐♥t ❆r✐t❤♠❡t✐❝ ✽ ■♥t❡r✈❛❧ ❆♥❛❧②s✐s Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✷ ✴ ✽✾
❘❡❢❡r❡♥❝❡s ❘✉❞✐♥✱ ❲✳ ✭✶✾✼✻✮✱ Pr✐♥❝✐♣❧❡s ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❛❧②s✐s ✱ ▼❝●r❛✇✲❍✐❧❧ ◆❡✇ ❨♦r❦✳ ▲✐♠❛✱ ❊✳ ▲✳ ✭✷✵✶✹✮✳ ❆♥á❧✐s❡ ❘❡❛❧ ✲ ❱♦❧✉♠❡ ✶ ✲ ❋✉♥çõ❡s ❞❡ ❯♠❛ ❱❛r✐á✈❡❧ ✳ ✶✷ ❡❞✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ■▼P❆✳ ❖✈❡rt♦♥✱ ▼✳ ▲✳ ✭✷✵✵✶✮✱ ◆✉♠❡r✐❝❛❧ ❈♦♠♣✉t✐♥❣ ✇✐t❤ ■❊❊❊ ✢♦❛t✐♥❣ ♣♦✐♥t ❛r✐t❤♠❡t✐❝ ✱ ❙■❆▼✳ ■♥st✐t✉t❡ ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝ ❊♥❣✐♥❡❡r✐♥❣ ✭✷✵✵✽✮✱ ✼✺✹✲✷✵✵✽ ✕ ■❊❊❊ st❛♥❞❛r❞ ❢♦r ✢♦❛t✐♥❣✲♣♦✐♥t ❛r✐t❤♠❡t✐❝ ✳ ●♦❧❞❜❡r❣✱ ❉✳ ✭✶✾✾✶✮✱ ❲❤❛t ❊✈❡r② ❈♦♠♣✉t❡r ❙❝✐❡♥t✐st ❙❤♦✉❧❞ ❑♥♦✇ ❆❜♦✉t ❋❧♦❛t✐♥❣✲♣♦✐♥t ❆r✐t❤♠❡t✐❝✱ ❈♦♠♣✉t✐♥❣ ❙✉r✈❡②s ✷✸✭✶✮✱ ✺✕✹✽✳ ▼♦♦r❡✱ ❘✳ ❊✳ ✭✶✾✼✾✮✱ ▼❡t❤♦❞s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ■♥t❡r✈❛❧ ❆♥❛❧②s✐s ✱ P❤✐❧❛❞❡❧♣❤✐❛✿ ❙■❆▼✳ ◆❡♣♦♠✉❝❡♥♦✱ ❊✳ ● ✭✷✵✶✹✮✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥s ♦♥ ❝♦♠♣✉t❡rs✳ ❚❤❡ ❏♦✉r♥❛❧ ♦❢ ❊♥❣✐♥❡❡r✐♥❣ q ✱ ■♥st✐t✉t✐♦♥ ♦❢ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✱ ✶✲✸✳ Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✸ ✴ ✽✾
❆ss❡ss♠❡♥t ■t❡♠ ❱❛❧✉❡ ❉❛t❡ ❖❜s❡r✈❛t✐♦♥ N 1 ✲ ❊①❛♠ ✶ ✶✵✵ ✵✷✴✵✾✴✷✵✶✺ ❈❤❛♣t❡rs ✶ ❛♥❞ ✷ N 2 ✲ ❊①❛♠ ✷ ✶✵✵ ✶✹✴✶✵✴✷✵✶✺ ❈❤❛♣t❡rs ✸ ❛♥❞ ✹✳ N 3 ✲ ❊①❛♠ ✸ ✶✵✵ ✵✹✴✶✶✴✷✵✶✺ ❈❤❛♣t❡rs ✺ ❛♥❞ ✻✳ N 4 ✲ ❊①❛♠ ✹ ✶✵✵ ✵✷✴✶✷✴✷✵✶✺ ❈❤❛♣t❡rs ✼ ❛♥❞ ✽✳ N s ✲ ❙❡♠✐♥❛r ✶✵✵ ✵✾✴✶✷✴✷✵✶✺ P❛♣❡r ✰ Pr❡s❡♥t❛t✐♦♥✳ N e ✲ ❊s♣❡❝✐❛❧ ✶✵✵ ✶✻✴✶✷✴✷✵✶✺ ❊s♣❡❝✐❛❧ ❊①❛♠ ❚❛❜❧❡ ✶✿ ❆ss❡s♠❡♥t ❙❝❤❡❞✉❧❡ ❙❝♦r❡✿ S = 2( N 1 + N 2 + N 3 + N 4 + N s ) 500 ❲✐t❤ N e t❤❡ ✜♥❛❧ s❝♦r❡ ✐s✿ S f = S + N e ✱ ♦t❤❡r✇✐s❡ S f = S. 2 ■❢ S f ≥ 6 . 0 t❤❡♥ ❙✉❝❝❡❡❞ ✳ ■❢ S f < 6 . 0 t❤❡♥ ❋❛✐❧❡❞ ✳ Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✹ ✴ ✽✾
❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ■♥tr♦❞✉❝t✐♦♥ ✶✳ ❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ✶✳✶ ■♥tr♦❞✉❝t✐♦♥ ❆ ❞✐s❝✉ss✐♦♥ ♦❢ t❤❡ ♠❛✐♥ ❝♦♥❝❡♣ts ♦❢ ❛♥❛❧②s✐s ✭s✉❝❤ ❛s ❝♦♥✈❡r❣❡♥❝❡✱ ❝♦♥t✐♥✉✐t②✱ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❛♥❞ ✐♥t❡❣r❛t✐♦♥✮ ♠✉st ❜❡ ❜❛s❡❞ ♦♥ ❛♥ ❛❝❝✉r❛t❡❧② ❞❡✜♥❡❞ ♥✉♠❜❡r ❝♦♥❝❡♣t✳ ◆✉♠❜❡r✿ ❆♥ ❛r✐t❤♠❡t✐❝❛❧ ✈❛❧✉❡ ❡①♣r❡ss❡❞ ❜② ❛ ✇♦r❞✱ s②♠❜♦❧✱♦r ✜❣✉r❡✱ r❡♣r❡s❡♥t✐♥❣ ❛ ♣❛rt✐❝✉❧❛r q✉❛♥t✐t② ❛♥❞ ✉s❡❞ ✐♥ ❝♦✉♥t✐♥❣ ❛♥❞ ♠❛❦✐♥❣ ❝❛❧❝✉❧❛t✐♦♥s✳ ✭❖①❢♦r❞ ❉✐❝t✐♦♥❛r②✮✳ ▲❡t ✉s s❡❡ ✐❢ ✇❡ r❡❛❧❧② ❦♥♦✇ ✇❤❛t ❛ ♥✉♠❜❡r ✐s✳ ❚❤✐♥❦ ❛❜♦✉t t❤✐s q✉❡st✐♦♥✿ ✶ Is 0 . 999 . . . = 1? ✭✶✮ ✶ ❘✐❝❤♠❛♥✱ ❋✳ ✭✶✾✾✾✮ ■s ✵✳✾✾✾ ✳✳✳ ❂ ✶❄ ▼❛t❤❡♠❛t✐❝s ▼❛❣❛③✐♥❡ ✳ ✼✷✭✺✮✱ ✸✽✻✕✹✵✵✳ Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✺ ✴ ✽✾
❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ s❡t N ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs ✐s ❞❡✜♥❡❞ ❜② t❤❡ P❡❛♥♦ ❆①✐♦♠s✿ ✶ ❚❤❡r❡ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ s : N → N . ❚❤❡ ✐♠❛❣❡ s ( n ) ♦❢ ❡❛❝❤ ♥❛t✉r❛❧ ♥✉♠❜❡r n ∈ N ✐s ❝❛❧❧❡❞ s✉❝❝❡ss♦r ♦❢ n. ✷ ❚❤❡r❡ ✐s ❛♥ ✉♥✐q✉❡ ♥❛t✉r❛❧ ♥✉♠❜❡r 1 ∈ N s✉❝❤ t❤❛t 1 � = s ( n ) ❢♦r ❛❧❧ n ∈ N . ✸ ■❢ ❛ s✉❜s❡t X ⊂ N ✐s s✉❝❤ t❤❛t 1 ∈ X ❛♥❞ s ( X ) ⊂ X ✭t❤❛t ✐s✱ n ∈ X ⇒ s ( n ) ∈ X ) t❤❡♥ X = N . ❚❤❡ s❡t Z = { . . . , − 2 , − 1 , 0 , 1 , 2 . . . } ♦❢ ✐♥t❡❣❡rs ✐s ❛ ❜✐❥❡❝t✐♦♥ f : N → Z s✉❝❤ t❤❛t f ( n ) = ( n − 1) / 2 ✇❤❡♥ n ✐s ♦❞❞ ❛♥❞ f ( n ) − n/ 2 ✇❤❡♥ n ✐s ❡✈❡♥✳ ❚❤❡ s❡t Q = { m/n ; m, n ∈ Z , n � = 0 } ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ♠❛② ❜❡ ✇r✐tt❡♥ ❛s f : Z × Z ∗ → Q s✉❝❤ t❤❛t Z ∗ = Z − { 0 } ❛♥❞ f ( m, n ) = m/n ✳ Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✻ ✴ ✽✾
❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ❛r❡ ✐♥❛❞❡q✉❛t❡ ❢♦r ♠❛♥② ♣✉r♣♦s❡s✱ ❜♦t❤ ❛s ❛ ✜❡❧❞ ❛♥❞ ❛s ❛♥ ♦r❞❡r❡❞ s❡t✳ ❋♦r ✐♥st❛♥❝❡✱ t❤❡r❡ ✐s ♥♦ r❛t✐♦♥❛❧ p s✉❝❤ t❤❛t p 2 = 2 ✳ ❆♥ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r ✐s ✇r✐tt❡♥ ❛s ✐♥✜♥✐t❡ ❞❡❝✐♠❛❧ ❡①♣❛♥s✐♦♥✳ √ ❚❤❡ s❡q✉❡♥❝❡ ✶✱ ✶✳✹✱ ✶✳✹✶✱ ✶✳✹✶✹✱ ✶✳✹✶✹✷ . . . t❡♥❞s t♦ 2 ✳ ❲❤❛t ✐s ✐t t❤❛t t❤✐s s❡q✉❡♥❝❡ t❡♥❞s t♦ ❄ ❲❤❛t ✐s ❛♥ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r❄ ❚❤✐s s♦rt ♦❢ q✉❡st✐♦♥ ❝❛♥ ❜❡ ❛♥s✇❡r❡❞ ❛s s♦♦♥ ❛s t❤❡ s♦✲❝❛❧❧❡❞ ✏r❡❛❧ ♥✉♠❜❡r s②st❡♠✑ ✐s ❝♦♥str✉❝t❡❞✳ Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✼ ✴ ✽✾
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