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P t Prr qs r Pr r rr t tr r Pr

  1. ❈▼P❚✸✷✺✿ ❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣ ❚❡❝❤♥✐q✉❡s ❉r✳ ❇✳ Pr✐❝❡ ✫ ❉r✳ ❘✳ ●r❡✐♥❡r ✶✻t❤ ❙❡♣t❡♠❜❡r ✷✵✵✹ ❉r✳ ❇✳ Pr✐❝❡ ✫ ❉r✳ ❘✳ ●r❡✐♥❡r ❈▼P❚✸✷✺✿ ❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣ ❚❡❝❤♥✐q✉❡s ✶ ❘♦❛❞ ▼❛♣ ❘❡✈✐s✐t❡❞ ◮ ❋✉♥❝t✐♦♥s✿ ❉♦♥❡✦ ◮ ▲✐s♣ ✬s ❋♦✉♥❞❛t✐♦♥s✿ ❉♦♥❡✦ ◮ ❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣ ◮ ❘❡❝✉rs✐♦♥✱ ❱❛r✐❛❜❧❡s✱ ❊✣❝✐❡♥❝②✱ ◮ ❋✉♥❛r❣ Pr♦❜❧❡♠ ✭❙❝♦♣✐♥❣✮ ◮ Pr♦❣r❛♠❂❉❛t❛ ✭❡✈❛❧✱ ♥❧❛♠❜❞❛✱ ♦♦♣✮ ◮ ▲❛♠❜❞❛ ❈❛❧❝✉❧✉s ◮ ❙❊❈❉ ♠❛❝❤✐♥❡ ◮ ✏❊①t❡♥s✐♦♥s✑ t♦ P✉r❡ ▲✐s♣ ◮ ❊①❛♠♣❧❡ ✭♣♦❧②♥♦♠✐❛❧s✮ ❉r✳ ❇✳ Pr✐❝❡ ✫ ❉r✳ ❘✳ ●r❡✐♥❡r ❈▼P❚✸✷✺✿ ❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣ ❚❡❝❤♥✐q✉❡s ✷

  2. ❘❡❝✉rs✐♦♥ ◮ ❘❡❝✉rs✐♦♥ ✐s ❛ ♣r♦❜❧❡♠✲s♦❧✈✐♥❣ t❡❝❤♥✐q✉❡ ✭❛✳❦✳❛✳ ❞✐✈✐❞❡✲❛♥❞✲❝♦♥q✉❡r✮ ◮ ❙t❡♣s ✐♥ ♠❛❣✐❝ ❢♦r♠✉❧❛✿ ◮ ❘❡❞✉❝❡ ♣r♦❜❧❡♠ t♦ s✐♠♣❧❡r✱ s❡❧❢✲s✐♠✐❧❛r ♣r♦❜❧❡♠s ◮ ❙♦❧✈❡ t❤❡ s✐♠♣❧❡r ♣r♦❜❧❡♠s ◮ ❈♦♠♣♦s❡ r❡s✉❧ts t♦ s♦❧✈❡ t❤❡ ♠❛✐♥ ♣r♦❜❧❡♠ ◮ ❉❡❝♦♠♣♦s✐t✐♦♥ ✐s ❛❧s♦ ✉s❡❞ ✐♥ ♣r♦❝❡❞✉r❛❧ ♣r♦❣r❛♠♠✐♥❣ ◮ ■♥ r❡❝✉rs✐♦♥✱ s✉❜♣r♦❜❧❡♠s ❛r❡ s✐♠✐❧❛r t♦ ♦r✐❣✐♥❛❧ ◮ ❘❡❝✉rs✐♦♥ ✐s t❤❡ ❝❡♥tr❛❧ ♠♦❞❡❧ ♦❢ ❝♦♠♣✉t❛t✐♦♥ ✐♥ ♣✉r❡ ❢✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠♠✐♥❣ ❉r✳ ❇✳ Pr✐❝❡ ✫ ❉r✳ ❘✳ ●r❡✐♥❡r ❈▼P❚✸✷✺✿ ❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣ ❚❡❝❤♥✐q✉❡s ✸ ❋❛❝t♦r✐❛❧ ❊①❛♠♣❧❡ ◮ ❈♦✉♥ts ♦r❞❡r❡❞ ♥ ✲t✉♣❧❡s ❞r❛✇❛❜❧❡ ❢r♦♠ ♥ ✐t❡♠s ✇✐t❤♦✉t r❡♣❧❛❝❡♠❡♥t ◮ ❚❤❡ ❢❛❝t♦r✐❛❧ ♦❢ ♥ ✱ ❢❝t ( ♥ ) ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ✜rst ♥ ✐♥t❡❣❡rs✿ � ✐ = ✶ × ✷ × · · · × ( ♥ − ✶ ) × ♥ ✐ = ✶ , ♥ ◮ Pr♦❝❡❞✉r❛❧❧② ✇❡ ❝♦✉❧❞ ✇r✐t❡ t❤✐s ❛s ❛ ❧♦♦♣✿ ✐♥t ❢❝t✭✐♥t ♥✮ ❢❝t ✿❂ ✶ ❋❖❘ ✐ ✿❂ ✶ ❚❖ ♥ ❉❖ ❢❝t ✿❂ ❢❝t ✯ ✐ r❡t✉r♥ ❢❝t ❉r✳ ❇✳ Pr✐❝❡ ✫ ❉r✳ ❘✳ ●r❡✐♥❡r ❈▼P❚✸✷✺✿ ❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣ ❚❡❝❤♥✐q✉❡s ✹

  3. ❋❛❝t♦r✐❛❧✬s ❙❡❧❢✲❙✐♠✐❧❛r ❙✉❜str✉❝t✉r❡ ◮ ■♥ ❣❡♥❡r❛❧ ❝♦♠♣✉t✐♥❣ ❢❝t ( ♥ ) ❢♦r ❞✐✛❡r❡♥t ♥ ✬s r❡♣❡❛ts ❛ ❧♦t ♦❢ ✇♦r❦ ◮ ❢❝t ( ✻ ) = ✶ × ✷ × ✸ × ✹ × ✺ × ✻✱ ❜✉t ❢❝t ( ✺ ) = ✶ × ✷ × ✸ × ✹ × ✺ � �� � = ❢❝t ( ✺ ) ◮ ■❢ ✇❡ ❤❛✈❡ ❝♦♠♣✉t❡❞ ❢❝t ( ✺ ) ✇❡ ❝♦✉❧❞ ❣❡t ❢❝t ( ✻ ) = ✻ × ❢❝t ( ✺ ) ◮ ■♥ ❣❡♥❡r❛❧ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ ❢❝t ( ♥ ) ❛s ♥ × ❢❝t ( ♥ − ✶ ) ❢❝t ( ✺ ) = ✶ × ✷ × ✸ × ✹ × ✺ ❢❝t ( ✺ ) = ✺ × ❢❝t ( ✹ ) ❢❝t ( ✹ ) = ✹ × ❢❝t ( ✸ ) ❢❝t ( ✸ ) = ✸ × ❢❝t ( ✷ ) ❢❝t ( ✷ ) = ✷ × ❢❝t ( ✶ ) ◮ ❢❝t ( ✶ ) ✐s ✉♥❞❡❝♦♠♣♦s❛❜❧❡✳ ❲❡ s♣❡❝✐❢② ❛♥ ❛♥s✇❡r✿ ❢❝t ( ✶ ) = ✶ ❉r✳ ❇✳ Pr✐❝❡ ✫ ❉r✳ ❘✳ ●r❡✐♥❡r ❈▼P❚✸✷✺✿ ❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣ ❚❡❝❤♥✐q✉❡s ✺ ❘❡❝✉rs✐✈❡ ❋❛❝t♦r✐❛❧ ◮ ❙❡❧❢✲s✐♠✐❧❛r s✉❜str✉❝t✉r❡ ✐s ❝❛♣t✉r❡❞ ✇✐t❤ ❛ ❝♦♥❞✐t✐♦♥❛❧ ❢✉♥❝t✐♦♥✿ � ✶ ✐❢ ♥ = ✵ ❢❝t ( ♥ ) = ♥ × ❢❝t ( ♥ − ✶ ) ♦t❤❡r✇✐s❡ ◮ ♥ = ✵ ✐s t❤❡ ✧❜❛s❡ ❝❛s❡✧ ❛♥❞ ♥ > ✵ ✐s t❤❡ ✧r❡❝✉rs✐✈❡ ❝❛s❡✧ ◮ ◆♦t✐❝❡✿ ◮ ❢❝t ( ♥ − ✶ ) ✐s ❛ s✐♠♣❧❡r ♣r♦❜❧❡♠ t❤❛♥ ❢ ( ♥ ) ◮ ♥ − ✶ ✐s ❛ r❡❞✉❝t✐♦♥ ♦♣❡r❛t♦r ✭r❡❞✉❝❡s ♣r♦❜❧❡♠ t♦ ❛ s✐♠♣❧❡r ♦♥❡✮ ◮ ❘❡❞✉❝t✐♦♥ ♦♣❡r❛t♦r ♣r♦❣r❡ss❡s t♦ ❜❛s❡ ❝❛s❡ s♦ r❡❝✉rs✐♦♥ t❡r♠✐♥❛t❡s ◮ ❈♦♠♣♦s✐t✐♦♥ ♦♣❡r❛t♦r × ✐♥ ♥ × ❢❝t ( ♥ − ✶ ) ❝r❡❛t❡s s♦❧✉t✐♦♥ t♦ ♦r✐❣✐♥❛❧ ♣r♦❜❧❡♠ ❢r♦♠ s✉❜♣r♦❜❧❡♠s ❉r✳ ❇✳ Pr✐❝❡ ✫ ❉r✳ ❘✳ ●r❡✐♥❡r ❈▼P❚✸✷✺✿ ❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣ ❚❡❝❤♥✐q✉❡s ✻

  4. ❘❡❝✉rs✐✈❡ ❋❛❝t♦r✐❛❧ ✐♥ P✉r❡ ▲✐s♣ ✭▲❆❇❊▲❙ ✭✭ ❢❛❝t ✭♥✮ ✭■❋ ✭❂ ♥ ✵✮ ✶ ✭✯ ♥ ✭❢❛❝t ✭✲ ♥ ✶✮✮✮✮ ✮✮ ✭▲■❙❚ ✭❢❛❝t ✶✮ ✭❢❛❝t ✹✮ ✭❢❛❝t ✸✸✮ ✮ ✮ → ✭✶ ✷✹ ✽✻✽✸✸✶✼✻✶✽✽✶✶✽✽✻✹✾✺✺✶✽✶✾✹✹✵✶✷✽✵✵✵✵✵✵✵ ✮ ❉r✳ ❇✳ Pr✐❝❡ ✫ ❉r✳ ❘✳ ●r❡✐♥❡r ❈▼P❚✸✷✺✿ ❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣ ❚❡❝❤♥✐q✉❡s ✼ ❘❡❝✉rs✐✈❡ ❋❛❝t♦r✐❛❧ ✐♥ ❙❡♠✐✲♣✉r❡ ▲✐s♣ ◮ ❚❤❡ ❉❊❋❯◆ ❢♦r♠ ❛ss✐❣♥s t❤❡ ❣❧♦❜❛❧ ❢✉♥❝t✐♦♥ s②♠❜♦❧ ❢❛❝t t♦ ❛ ❝❧♦s✉r❡ ✇✐t❤ t❤❡ ❛r❣✉♠❡♥ts ❛♥❞ ❜♦❞② ❣✐✈❡♥ ✭❉❊❋❯◆ ❢❛❝t ✭♥✮ ✧r❡t✉r♥s ❢❛❝t♦r✐❛❧ ♦❢ t❤❡ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡r ♥✧ ✭■❋ ✭❂ ♥ ✵✮ ✶ ✭✯ ♥ ✭❢❛❝t ✭✲ ♥ ✶✮✮✮✮✮ ✭❢❛❝t ✶✮ → ✶ ✭❢❛❝t ✹✮ → ✷✹ ◮ ❇❡ ❝❛r❡❢✉❧ ♥♦t t♦ ❝❧♦❜❜❡r ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ t❤❡ s❛♠❡ ♥❛♠❡ ♦r ✉♥✐♥t❡♥t✐♦♥❛❧❧② ✉s❡ ❛ ♣r❡✈✐♦✉s❧② ❞❡✜♥❡❞ ♣r❡❞✐❝❛t❡✦ ❉r✳ ❇✳ Pr✐❝❡ ✫ ❉r✳ ❘✳ ●r❡✐♥❡r ❈▼P❚✸✷✺✿ ❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣ ❚❡❝❤♥✐q✉❡s ✽

  5. ❘❡❝✉rs✐♦♥s ✇✐t❤ ▲✐sts✿ ❝♦♥t❛✐♥s ◮ ❉❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ ✧❝♦♥t❛✐♥s✭s✱❛✮✧ ✇❤✐❝❤ r❡t✉r♥s tr✉❡ ⇔ t❤❡ ❛t♦♠ ❛ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❧✐st s ✳ ◮ ❈❛♥ ✇❡ s❡❡ s❤❛r❡❞ s✉❜♣r♦❜❧❡♠s ❤❡r❡❄ ✭❝♦♥t❛✐♥s ✬✭✮ ✸✮ → ◆■▲ ✭❝♦♥t❛✐♥s ✬✭✸✮ ✸✮ → ❚ ✭❝♦♥t❛✐♥s ✬✭✷ ✸✮ ✸✮ ✭❝♦♥t❛✐♥s ✬✭✶ ✷ ✸✮ ✸✮ ❀✭❖❘ ✭❊◗ ✶ ✸✮ ✭❝♦♥t❛✐♥s ✬✭✶ ✷✮ ✸✮ ◮ ❆s ▲✐s♣ ❝♦❞❡ ✭❉❊❋❯◆ ❝♦♥t❛✐♥s ✭s ❛✮ ✭❈❖◆❉ ✭✭◆❯▲▲ s✮ ♥✐❧✮ ✭✭❊◗❯❆▲ ✭❈❆❘ s✮ ❛✮ t✮ ✭ t ✭❝♦♥t❛✐♥s ✭❈❉❘ s✮ ❛✮✮✮✮ ❉r✳ ❇✳ Pr✐❝❡ ✫ ❉r✳ ❘✳ ●r❡✐♥❡r ❈▼P❚✸✷✺✿ ❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣ ❚❡❝❤♥✐q✉❡s ✾ ❆❧t❡r♥❛t✐✈❡ ❱❡rs✐♦♥ ♦❢ ❝♦♥t❛✐♥s ◮ ❖r✐❣✐♥❛❧ ❱❡rs✐♦♥ ✭❉❊❋❯◆ ❝♦♥t❛✐♥s ✭s ❛✮ ✭❈❖◆❉ ✭✭◆❯▲▲ s✮ ♥✐❧✮ ✭✭❊◗❯❆▲ ✭❈❆❘ s✮ ❛✮ t✮ ✭ t ✭❝♦♥t❛✐♥s ✭❈❉❘ s✮ ❛✮✮✮✮ ◮ ❆❧t❡r♥❛t✐✈❡ ❱❡rs✐♦♥ ❡♠♣❤❛s✐③✐♥❣ ❢✉♥❝t✐♦♥❛❧ ♣❡rs♣❡❝t✐✈❡ ✭❉❊❋❯◆ ❝♦♥t❛✐♥s ✭s ❛✮ ✭❆◆❉ ✭◆❖❚ ✭◆❯▲▲ s✮✮ ✭❖❘ ✭❊◗❯❆▲ ✭❈❆❘ s✮ ❛✮ ✭❝♦♥t❛✐♥s ✭❈❉❘ s✮ ❛✮✮ ✮✮ ◮ ❊✛❡❝t✐✈❡❧②✱ ✇❡ ❛r❡ ✉s✐♥❣ ♦r t♦ ❝♦♠♣♦s❡ ✈❛❧✉❡ ♦❢ s✉❜♣r♦❜❧❡♠s ◮ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ ❝♦♠♣❛❝t ✐♥t✉✐t✐✈❡ ❢♦r♠ ❉r✳ ❇✳ Pr✐❝❡ ✫ ❉r✳ ❘✳ ●r❡✐♥❡r ❈▼P❚✸✷✺✿ ❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣ ❚❡❝❤♥✐q✉❡s ✶✵

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