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tt t t ts t rs t ssr t r t rr

  1. ❙t❛t❡ ❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ▼✉❧t✐♣❧❡s ♦❢ t❤❡ ❚❤✉❡✲▼♦rs❡ ❙❡t ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❏♦✐♥t ✇♦r❦ ✇✐t❤ ➱♠✐❧✐❡ ❈❤❛r❧✐❡r ❛♥❞ ❈é❧✐❛ ❈✐st❡r♥✐♥♦ ❙❡♣t❡♠❜❡r ✷ ♥❞ ✱ ✷✵✶✾ ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✴✸✵

  2. ■t ✐s ❡q✉✐✈❛❧❡♥t t♦ ✇♦r❦ ✇✐t❤ ✵ r❡♣ ✳ ❚❤❡♦r❡♠ ▲❡t ✷ ❛♥❞ ✳ ■❢ ✐s ✲r❡❝♦❣♥✐③❛❜❧❡✱ s♦ ✐s ✳ ❲❤❛t ❞♦ ✇❡ ✇❛♥t t♦ ❞♦ ❄ ❉❡✜♥✐t✐♦♥ ▲❡t b ∈ N ≥ ✷ ✳ ❆ s✉❜s❡t X ♦❢ N ✐s b ✲r❡❝♦❣♥✐③❛❜❧❡ ✐❢ r❡♣ b ( X ) ✐s r❡❣✉❧❛r✳ ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✴✸✵

  3. ❚❤❡♦r❡♠ ▲❡t ✷ ❛♥❞ ✳ ■❢ ✐s ✲r❡❝♦❣♥✐③❛❜❧❡✱ s♦ ✐s ✳ ❲❤❛t ❞♦ ✇❡ ✇❛♥t t♦ ❞♦ ❄ ❉❡✜♥✐t✐♦♥ ▲❡t b ∈ N ≥ ✷ ✳ ❆ s✉❜s❡t X ♦❢ N ✐s b ✲r❡❝♦❣♥✐③❛❜❧❡ ✐❢ r❡♣ b ( X ) ✐s r❡❣✉❧❛r✳ ■t ✐s ❡q✉✐✈❛❧❡♥t t♦ ✇♦r❦ ✇✐t❤ ✵ ∗ r❡♣ b ( X ) ✳ ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✴✸✵

  4. ❲❤❛t ❞♦ ✇❡ ✇❛♥t t♦ ❞♦ ❄ ❉❡✜♥✐t✐♦♥ ▲❡t b ∈ N ≥ ✷ ✳ ❆ s✉❜s❡t X ♦❢ N ✐s b ✲r❡❝♦❣♥✐③❛❜❧❡ ✐❢ r❡♣ b ( X ) ✐s r❡❣✉❧❛r✳ ■t ✐s ❡q✉✐✈❛❧❡♥t t♦ ✇♦r❦ ✇✐t❤ ✵ ∗ r❡♣ b ( X ) ✳ ❚❤❡♦r❡♠ ▲❡t b ∈ N ≥ ✷ ❛♥❞ m ∈ N ✳ ■❢ X ⊆ N ✐s b ✲r❡❝♦❣♥✐③❛❜❧❡✱ s♦ ✐s mX ✳ ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✴✸✵

  5. ❚❤❡♦r❡♠ ❬❆❧❡①❡❡✈✱ ✷✵✵✹❪ ❚❤❡ st❛t❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ❧❛♥❣✉❛❣❡ ✵ ∗ r❡♣ b ( m N ) ✐s � N − ✶ � b n m � min ❣❝❞ ( m , b N ) + ❣❝❞ ( b n , m ) N ≥ ✵ n = ✵ ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✸✴✸✵

  6. ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣ ✷ ✷ ✶ ❚❤✉❡✲▼♦rs❡ ✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

  7. ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣ ✷ ✷ ✶ ❚❤✉❡✲▼♦rs❡ ✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

  8. ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣ ✷ ✷ ✶ ❚❤✉❡✲▼♦rs❡ ✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

  9. ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣ ✷ ✷ ✶ ❚❤✉❡✲▼♦rs❡ ✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

  10. ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣ ✷ ✷ ✶ ❚❤✉❡✲▼♦rs❡ ✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

  11. ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣ ✷ ✷ ✶ ❚❤✉❡✲▼♦rs❡ ✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

  12. ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣ ✷ ✷ ✶ ❚❤✉❡✲▼♦rs❡ ✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

  13. ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣ ✷ ✷ ✶ ❚❤✉❡✲▼♦rs❡ ✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

  14. ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣ ✷ ✷ ✶ ❚❤✉❡✲▼♦rs❡ ✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

  15. ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣ ✷ ✷ ✶ ❚❤✉❡✲▼♦rs❡ ✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

  16. ❚❤✉❡✲▼♦rs❡ ✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t T = { n ∈ N : | r❡♣ ✷ ( n ) | ✶ ∈ ✷ N } . ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

  17. ❚❤✉❡✲▼♦rs❡ ✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t T = { n ∈ N : | r❡♣ ✷ ( n ) | ✶ ∈ ✷ N } . ✵ T ✶ ✶ B ✵ ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

  18. ❚❤✉❡✲▼♦rs❡ ✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t T = { n ∈ N : | r❡♣ ✷ ( n ) | ✶ ∈ ✷ N } . ✵ , ✸ T ✶ , ✷ ✶ , ✷ B ✵ , ✸ ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

  19. ❚❤❡♦r❡♠ ❬❈♦❜❤❛♠✱ ✶✾✻✾❪ ▲❡t t✇♦ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ✐♥❞❡♣❡♥❞❡♥t ❜❛s❡s✳ ❆ s✉❜s❡t ♦❢ ✐s ❜♦t❤ ✲r❡❝♦❣♥✐③❛❜❧❡ ❛♥❞ ✲r❡❝♦❣♥✐③❛❜❧❡ ✐✛ ✐t ✐s ❛ ✜♥✐t❡ ✉♥✐♦♥ ♦❢ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s✳ ▲❡t t✇♦ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ❞❡♣❡♥❞❡♥t ❜❛s❡s✳ ❆ s✉❜s❡t ♦❢ ✐s ✲r❡❝♦❣♥✐③❛❜❧❡ ✐✛ ✐t ✐s ✲r❡❝♦❣♥✐③❛❜❧❡✳ ❉❡✜♥✐t✐♦♥ ▲❡t p , q ∈ N ≥ ✷ ✳ ❲❡ s❛② t❤❛t p ❛♥❞ q ❛r❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ✐♥❞❡♣❡♥❞❡♥t ✐❢ p a = q b ⇒ a = b = ✵ . ❚❤❡② ❛r❡ s❛✐❞ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ❞❡♣❡♥❞❡♥t ♦t❤❡r✇✐s❡✳ ❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✺✴✸✵

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