MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS Mai Thu Hien REPRESENTATION OF REAL NUMBERS BY CONTINUED FRACTION GRADUATION THESIS HaNoi, 2019 ▼■◆■❙❚❘❨ ❖❋ ❊❉❯❈❆❚■❖◆ ❆◆❉ ❚❘❆■◆■◆● ❍❆◆❖■ P❊❉❆●❖●■❈❆▲ ❯◆■❱❊❘❙■❚❨ ✷ ❉❊P❆❘❚▼❊◆❚ ❖❋ ▼❆❚❍❊▼❆❚■❈❙ ▼❛✐ ❚❤✉ ❍✐❡♥ ❘❊P❘❊❙❊◆❚❆❚■❖◆ ❖❋ ❘❊❆▲ ◆❯▼❇❊❘❙ ❇❨ ❈❖◆❚■◆❯❊❉ ❋❘❆❈❚■❖◆ ▼❛❥♦r✿ ❆❧❣❡❜r❛ ❈♦❞❡ ♥✉♠❜❡r✿ ●❘❆❉❯❆❚■❖◆ ❚❍❊❙■❙ ❙✉♣❡r✈✐s♦r✿ ❉r✳ ❚r❛♥ ◆❛♠ ❚r✉♥❣ ❍❛◆♦✐✱ ✷✵✶✾ ❚❤❡s✐s ❆❝❦♥♦✇❧❡❞❣❡♠❡♥t ❋✐rst❧②✱ ■ ✇✐s❤ t♦ ❡①♣r❡ss ♠② s✐♥❝❡r❡ t❤❛♥❦ t♦ t❤❡ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡✲ ♠❛t✐❝s✱ ❍❛ ◆♦✐ P❡❞❛❣♦❣✐❝❛❧ ❯♥✐✈❡rs✐t② ✷ ❢♦r ♣r♦✈✐❞✐♥❣ ♠❡ ✇✐t❤ ❛❧❧ ♥❡❝❡ss❛r② ❢❛❝✐❧✐t✐❡s ❢♦r t❤❡ r❡s❡❛r❝❤✳ ■ ❛♠ ❛❧s♦ ❣r❛t❡❢✉❧ t♦ ❧❡❝t✉r❡rs ✐♥ ❚❤❡ ❆♥❣❡❜r❛ ●r♦✉♣ ❛♥❞ t❤❡ ❧❡❝t✉r❡rs ✇❤♦ ❤❛✈❡ t❛✉❣❤t ♠❡ ❢♦r ✹ ②❡❛rs✳ ■ ❛♠ ❡①tr❡♠❧② t❤❛♥❦❢✉❧ ❛♥❞ ✐♥❞❡❜t❡❞ t♦ t❤❡♠ ❢♦r s❤❛r✐♥❣ ❡①♣❡rt✐s❡ ❛♥❞ ❣✐✈✐♥❣ ♠❡ ❣r❡❛t ❛♥❞ ✈❛❧✉❛❜❧❡ ❣✉✐❞❛♥❝❡ ❛♥❞ ❡♥❝♦✉r❛❣❡♠❡♥t✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ ❡①♣r❡ss ♠② ❞❡❡♣ ❣r❛t✐t✉❞❡ t♦ ♠② ❙✉♣❡r✈✐s♦r ❉r✳ ❚r❛♥ ◆❛♠ ❚r✉♥❣ ❢♦r ❤✐s ❝♦♥t✐♥✉♦✉s s✉♣♣♦rt t♦ ♠② ❣r❛❞✉❛t✐♦♥ st✉❞② ❛♥❞ r❡❧❛t❡❞ r❡s❡❛r❝❤✱ ❢♦r ❤✐s ♣❛t✐❡♥❝❡✱ ♠♦t✐✈❛t✐♦♥✱ ❛♥❞ ✐♠♠❡♥s❡ ❦♥♦✇❧❡❞❣❡✳ ❍✐s ❣✉✐❞❛♥❝❡ s✐❣♥✐✜❝❛♥t❧② ❤❡❧♣❡❞ ♠❡ ❞✉r✐♥❣ t❤❡ r❡s❡❛r❝❤ ❛♥❞ t❤❡ ✇r✐t✐♥❣ ♦❢ t❤✐s t❤❡s✐s✳ ❉✉❡ t♦ ❧✐♠✐t❡❞ t✐♠❡✱ ❝❛♣❛❝✐t② ❛♥❞ ❝♦♥❞✐t✐♦♥s✱ t❤❡ t❤❡s✐s ❝❛♥ ♥♦t ❛✈♦✐❞ ❡rr♦rs✳ ❆❢t❡r t❤❛t✱ ■ ❤♦♣❡ t♦ ❣❡t ❝♦♥str✉❝t✐✈❡ ❝♦♠♠❡♥ts ❢r♦♠ t❡❛❝❤❡rs ❛♥❞ ❢r✐❡♥❞s✳ ❍❛◆♦✐✱ ❆✉t❤♦r ▼❛✐ ❚❤✉ ❍✐❡♥ ✐ ❚❤❡s✐s ❆ss✉r❛♥❝❡ ■ ❣✉❛r❛♥t❡❡ t❤❛t t❤❡ ❞❛t❛ ❛♥❞ r❡s✉❧ts ♦❢ t❤✐s t❤❡s✐s ❛r❡ ❝♦rr❡❝t ❛♥❞ ♥♦t ✐❞❡♥t✐❝❛❧ t♦ ♦t❤❡r t♦♣✐❝s✳ ■ ❛❧s♦ ♠❛❦❡ s✉r❡ t❤❛t ❛❧❧ t❤❡ ❤❡❧♣ ❢♦r t❤✐s t❤❡s✐s ❤❛s ❜❡❡♥ ❛❝❦♥♦✇❧❡❞❣❡❞ ❛♥❞ t❤❡ r❡s✉❧ts ♣r❡s❡♥t❡❞ ✐♥ t❤❡ t❤❡s✐s ❤❛✈❡ ❤❛s ❜❡❡♥ ❝❧❡❛r❧② ❞❡✜♥❡❞✳ ❍❛◆♦✐✱ ❆✉t❤♦r ▼❛✐ ❚❤✉ ❍✐❡♥ ✐✐ ❈♦♥t❡♥ts ❚❤❡s✐s ❆❝❦♥♦✇❧❡❞❣❡♠❡♥t ✐ ❚❤❡s✐s ❆ss✉r❛♥❝❡ ✐✐ Pr❡❢❛❝❡ ✶ ✶ ❈♦♥t✐♥✉❡❞ ❋r❛❝t✐♦♥s ✷ ✶✳✶ ✶✳✷ ❋✐♥✐t❡ ❈♦♥t✐♥✉❡❞ ❋r❛❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✶ ❋✐♥✐t❡ ❈♦♥t✐♥✉❡❞ ❋r❛❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✷ ❯♥✐q✉❡♥❡ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ■♥✜♥✐t❡ s✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✶ ■♥✜♥✐t❡ s✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✷ ■rr❛t✐♦♥❛❧ ♥✉♠❜❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷✳✸ ❆♣♣r♦①✐♠❛t✐♦♥s t♦ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷ ❘❙❆ ✶✼ ✷✳✶ ▲✐♥❡❛r ❝♦♥❣r✉❡♥❝❡ ❡q✉❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷✳✶ ❘❙❆ ❙②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷✳✷ ❆tt❛❝❦ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ❈♦♥❝❧✉s✐♦♥ ✷✹ ✐✐✐ Pr❡❢❛❝❡ ❚❤❡r❡ ❛r❡ ♠❛♥② ✇❛②s t♦ ❡①♣r❡ss ❛ ♥✉♠❜❡r s✉❝❤ ❛s ✉s✐♥❣ ❞✐✛❡r❡♥t ♥✉♠✲ ❜❡r ❜❛s❡ s②st❡♠s✱ ❢r❛❝t✐♦♥✱ ❞❡❝✐♠❛❧ ♥✉♠❜❡rs✱ ❧♦❣❛r✐t❤♠✱ ♣♦✇❡r ♦r ✇♦r❞s✳ ❊❛❝❤ ✇❛② ✐s ❝♦♥✈❡♥✐❡♥t ❢♦r ✈❛r✐❛♥t ❝❛s❡s ❛♥❞ ❛✐♠s✳ ■♥ ❝♦♠♣✉t✐♥❣✱ ✇❡ ♦❢t❡♥ ✉s❡ ❛♣♣r♦①✐♠❛t❡ ✈❛❧✉❡s t♦ r❡♣r❡s❡♥t ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡rs✳ ■t ✐s ❛❧s♦ ❡❛s② t♦ r❡♣r❡s❡♥t ❛♥ ✐♥t❡❣❡r ♦r ❛ r❛t✐♦♥❛❧ ✐♥ ❢r❛❝t✐♦♥ ❢♦r♠s✳ ❚❤❡ ✐ss✉❡ ♣♦s❡❞ ✐s ❤♦✇ ❛r❜✐tr❛r② r❡❛❧ ♥✉♠❜❡r ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛♣♣r♦①✐♠❛t✐♥❣ ❢r❛❝t✐♦♥s ❝❧❡❛r❧②❄ ❈♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s ❛r❡ ✐♠♣♦rt❛♥t ✐♥ ♠❛♥② ❜r❛♥❝❤❡s ♦❢ ♠❛t❤❡♠❛t✐❝s✳ ❚❤❡② ❛r✐s❡ ♥❛t✉r❛❧❧② ✐♥ ❧♦♥❣ ❞✐✈✐s✐♦♥ ❛♥❞ ✐♥ t❤❡ t❤❡♦r② ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ r❡❛❧ ♥✉♠❜❡rs ❜② r❛t✐♦♥❛❧s✳ ❚❤❡s❡ ♦❜❥❡❝ts t❤❛t ❛r❡ r❡❧❛t❡❞ t♦ ♥✉♠❜❡r t❤❡♦r② ❤❡❧♣ ✉s ✜♥❞ ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r r❡❛❧ ❧✐❢❡ ❝♦♥st❛♥ts✳ ■t ✐s t❤❡ r❡❛s♦♥ ✇❤② ■ ❝❤♦♦s❡ t❤❡ t❤❡♠❡✳ ❚❤❡ ♠❛✐♥ ❛✐♠ ♦❢ ❣r❛❞✉❛✲ t✐♦♥ ♠✐♥✐✲ t❤❡s✐s ✐s t♦ st✉❞② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜② ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s✳ ❚❤❡r❡ ❛r❡ t✇♦ ❝❤❛♣t❡rs ❛s ❢♦❧❧♦✇s✳ ❈❤❛♣t❡r ✶ ✧❈♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s✧ ✇❡ ❞✐s❝✉ss t❤❡ ❣❡♥❡r❛❧ ❢♦r♠ ♦❢ ❛ ❝♦♥✲ t✐♥✉❡❞✳ ❈❤❛♣t❡r ✷ ✧❘❙❆✧ ✇❡ r❡♣r❡s❡♥t t❤❡ ❘❙❆ s②st❡♠ ❛♥❞ ❞✐s❝✉ss ❤♦✇ t♦ ❛tt❛❝❦ t❤✐s s②st❡♠ ❜② ✉s✐♥❣ t❤❡ t❤❡♦r② ♦❢ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s✳ ✶ ❈❤❛♣t❡r ✶ ❈♦♥t✐♥✉❡❞ ❋r❛❝t✐♦♥s ■♥ t❤✐s ❝❤❛♣t❡r ✇❡ r❡♣r❡s❡♥t t❤❡ t❤❡♦r② ♦❢ ✜♥✐t❡ ❛♥❞ ✐♥✜♥✐t❡ s✐♠♣❧❡ ❝♦♥✲ t✐♥✉❡❞ ❢r❛❝t✐♦♥s✱ ❛♥❞ r❡♣r❡s❡♥t r❡❛❧ ♥✉♠❜❡rs ❜② ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s✳ ❚❤❡ r❡❢❡r❡♥❝❡ ✐s t❤❡ ❜♦♦❦ ❬✷❪✳ ✶✳✶ ❋✐♥✐t❡ ❈♦♥t✐♥✉❡❞ ❋r❛❝t✐♦♥s ✶✳✶✳✶ ❋✐♥✐t❡ ❈♦♥t✐♥✉❡❞ ❋r❛❝t✐♦♥s ●✐✈❡♥ ❛♥② r❛t✐♦♥❛❧ ❢r❛❝t✐♦♥ u0 /u1 ✱ ✐♥ t❤❡ ❧♦✇❡st t❡r♠✱ s♦ t❤❛t (u0 , u1 ) = ❛♥❞ u1 > 0✱ ✇❡ ❛♣♣❧② t❤❡ ❊✉❝❧✐❞❡❛♥ ❛❧❣♦r✐t❤♠ t♦ ✜♥❞ t❤❡ ❣r❡❛t❡st ❝♦♠♠♦♥ ❞✐✈✐s♦r ♦❢ u0 ❛♥❞ u1 ❛s ❢♦❧❧♦✇s✿ u0 = u1 a0 + u2 u1 = u2 a1 + u3 u2 = u3 a2 + u4 uj−1 = uj aj−1 + uj+1 uj = uj+1 aj ✷ ■❢ ✇❡ ♣✉t ξi = ui /ui+1 ❢♦r i = 0, , j ✱ t❤❡♥ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥s ❜❡❝♦♠❡ ξ i = + ξi+1 , ≤ i ≤ j − 1; ξj = aj ✭✶✳✶✮ ❇② ✉s✐♥❣ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r ξi ✐♥ ❊q✉❛t✐♦♥ ✭✶✳✶✮ ✇❡ ♦❜t❛✐♥ ξ0 = a0 + a1 + a2 + ✭✶✳✷✮ ✳✳✳ + ❚❤✐s ✐s ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ ξ0 ✱ ♦r ♦❢ aj u0 ✳ u1 ❉❡✜♥✐t✐♦♥ ✶✳✶✳✶ ✭❋✐♥✐t❡ ❈♦♥t✐♥✉❡❞ ❋r❛❝t✐♦♥✮✳ ❆ ✜♥✐t❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ✐s ❛♥ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❢♦r♠ a0 + a1 + a2 + ✳✳✳ + aj ✇❤❡r❡ ❡❛❝❤ aj ✐s ❛ r❡❛❧ ♥✉♠❜❡r ❛♥❞ aj > ❢♦r j ≥ ❲❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥ [a0 , a1 , a2 , , aj ] = ξ0 ✳ ❉❡✜♥✐t✐♦♥ ✶✳✶✳✷ ✭❙✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥✮✳ ❆ ✜♥✐t❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ξ0 = [a0 , a1 , , aj ] ✐s s✐♠♣❧❡ ✐❢ ❛❧❧ ❛r❡ ✐♥t❡❣❡rs✳ ❊①❛♠♣❧❡ ✶✳✶✳✶✳ ❊①♣❛♥❞ 225 ❛s ❛ s✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥✳ 157 ✸ ❙♦❧✉t✐♦♥✳ ❲❡ ❤❛✈❡ 225 = 157 · + 68 157 = 68 · + 21 68 = 21 · + 21 = · + = · ❚❤✉s✱ 225 = [1, 2, 3, 4, 5]✳ 157 ✶✳✶✳✷ ❯♥✐q✉❡♥❡ss 225 ✐♥t♦ ❛ s✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ 157 225 ❢r❛❝t✐♦♥ = [1, 2, 3, 4, 5]✳ ❲❡ ❛❧s♦ s❡❡ t❤❛t t❤✐s ❢r❛❝t✐♦♥ ❝❛♥ ❜❡ ❛❧s♦ 157 ❡①♣r❡ss❡❞ ❛s [1, 2, 3, 4, 4, 1]✳ ■♥ ❣❡♥❡r❛❧✱ ✇❡ ♥♦t❡ t❤❛t t❤❡ s✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ ■♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ ✇❡ ❡①♣❛♥❞ t❤❡ ❢r❛❝t✐♦♥ ❢r❛❝t✐♦♥ ξ0 = [a0 , a1 , , aj ] ✇✐t❤ aj > ❤❛s ❛♥ ❛❧t❡r♥❛t✐✈❡ ❢♦r♠ ξ0 = [a0 , a1 , , aj ] = [a0 , a1 , , aj − 1, 1] ✭✶✳✸✮ ■♥ ❢❛❝t✱ t❤❡s❡ ❛r❡ ♦♥❧② t✇♦ s✉❝❤ ✇❛②s✱ ❚❤❡♦r❡♠ ✶✳✶✳ ■❢ [a0, a1, , aj ] = [b0, b1, , bn] ✇❤❡r❡ t❤❡s❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s ❛r❡ s✐♠♣❧❡✱ ❛♥❞ ✐❢ aj > ❛♥❞ bn > 1✱ t❤❡♥ j = n ❛♥❞ = bi ❢♦r i = 0, 1, , j ✳ Pr♦♦❢✳ ▲❡t yi = [bi, bi+1, , bn] ❢♦r i = 0, , n✳ ❚❤❡♥✱ yi = bi + [bi+1 , bi+2 , bn ] = bi + yi+1 ✭✶✳✹✮ ❚❤✉s ✇❡ ❤❛✈❡ yi > bi ❛♥❞ yi > ❢♦r i = 1, , n − 1✱ ❛♥❞ yn = bn > 1✳ ✹ ❈♦♥s❡q✉❡♥t❧②✱ bi = [yi ] ❢♦r ❛❧❧ i = 0, , n✳ ❙✐♥❝❡ ξ0 = [a0 , a1 , , aj ] = [b0 , b1 , , bn ]✱ ✇❡ ❤❛✈❡ y0 = ξ0 ✱ ✇❤❡r❡ ✇❡ ❛r❡ ✉s✐♥❣ t❤❡ ♥♦t❛t✐♦♥ ✐♥ ❊q✉❛✲ t✐♦♥ ✭✶✳✶✮✳ ◆♦✇ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ξi ❛s ui /ui+1 ✐♠♣❧✐❡s t❤❛t ξi+1 > ❢♦r ❛❧❧ i = 0, , j − 1✳ ❋r♦♠ ❊q✉❛t✐♦♥ ✭✶✳✶✮ ✇❡ ❤❛✈❡ = [ξi ] ❢♦r i = 0, , j ✳ ❚❤✉s✱ b0 = [y0 ] = [ξ0 ] = a0 ✳ ❚♦❣❡t❤❡r ✇✐t❤ ❊q✉❛t✐♦♥s ✭✶✳✶✮ ❛♥❞ ✭✶✳✹✮ ✇❡ ❣❡t 1 = ξ0 − a0 = y0 − b0 = , a1 = [ξ0 ] = [y0 ] = b1 ξ1 y1 ❲❡ ♥♦✇ ❝❧❛✐♠ t❤❛t = bi ❛♥❞ ξi = yi ❢♦r ❛❧❧ i min{j, n}✳ ❲❡ ✇✐❧❧ ♣r♦✈❡ ❜② ✐♥❞✉❝t✐♦♥✳ ■❢ i = 0✱ ✇❡ ❤❛✈❡ ♣r♦✈❡❞ ❛❜♦✈❡✳ ❆ss✉♠❡ t❤❛t = bi ❛♥❞ i < min{j, n}✳ ❚❤❡♥✱ ❢r♦♠ ❊q✉❛t✐♦♥s ✭✶✳✶✮ ❛♥❞ ✭✶✳✹✮ ✇❡ ❤❛✈❡ ξi+1 = ξi − = yi − = yi+1 , ξi+1 = yi+1 , ai+1 = [ξi+1 ] = [yi+1 ] = bi+1 , ❛♥❞ t❤❡ ❝❧❛✐♠ ❢♦❧❧♦✇s✳ ■t r❡♠❛✐♥s t♦ ♣r♦✈❡ j = n✳ ❆ss✉♠❡ ♦♥ t❤❡ ❝♦♥tr❛r② t❤❛t j < n✳ ❇② t❤❡ ❝❧❛✐♠ ❛❜♦✈❡ ξj = yj ❛♥❞ aj = bj ✳ ❇✉t aj = ξj ❜② ❊q✉❛t✐♦♥ ✭✶✳✶✮ ❛♥❞ yj > bj ❜② ❊q✉❛t✐♦♥ ✭✶✳✹✮✱ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❚❤✐s s❤♦✇s t❤❛t j = n✱ ❛♥❞ t❤❡ t❤❡♦r❡♠ ✐s ♣r♦✈❡❞✳ ❚❤❡♦r❡♠ ✶✳✷✳ ❆♥② ✜♥✐t❡ s✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ r❡♣r❡s❡♥ts ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r✳ ❈♦♥✈❡rs❡❧②✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ❛ ✜♥✐t❡ s✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s✱ ❛♥❞ ✐♥ ❡①❛❝t❧② t✇♦ ✇❛②s✳ Pr♦♦❢✳ ❚❤❡ ✜rst ❛ss❡rt✐♦♥ ❝❛♥ ❜❡ ♣r♦✈❡❞ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ t❡r♠s ♦❢ t❤❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥✱ ❜② ✉s✐♥❣ t❤❡ ❢♦r♠✉❧❛ [a0 , a1 , , aj ] = a0 + ✺ [a1 , a2 , , aj ] ❲❡ ♥♦✇ ♣r♦✈❡ t❤❛t ξ = [a0 , a1 , a2 , ]✳ ❚❤❡♦r❡♠ ✶✳✽✳ ❆♥② ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r ξ ✐s ✉♥✐q✉❡❧② ❡①♣r❡ss✐❜❧❡✱ ❜② t❤❡ ♣r♦✲ ❝❡❞✉r❡ t❤❛t ❣❛✈❡ ❡q✉❛t✐♦♥s ✭✶✳✻✮✱ ❛s ❛♥ ✐♥✜♥✐t❡ s✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ [a0 , a1 , a2 , ]✳ ▼♦r❡♦✈❡r✱ ❧❡t ξi ❞❡✜♥❡❞ ❜② ✭✶✳✻✮✱ ✇❡ ❤❛✈❡ [a0 , a1 , a2 , ] = [a0 , a1 , , an−1 , ξn ] ❛♥❞ ξn = [an , an+1 , ]✳ Pr♦♦❢✳ ❇② ❚❤❡♦r❡♠ ✶✳✸ ✇❡ ❤❛✈❡ ξ = [a0 , a1 , , an−1 , ξn ] = ξn hn−1 + hn−2 ξn kn−1 + kn−2 ✭✶✳✼✮ ✇❤❡r❡ hi ❛♥❞ ki ❞❡✜♥❡❞ ❛s ✐♥ ✭✶✳✺✮✳ ❇② ❚❤❡♦r❡♠ ✶✳✹ ✇❡ ❣❡t hn−1 ξn hn−1 + hn−2 hn−1 = − kn−1 ξn kn−1 + kn−2 kn−1 −(hn−1 kn−2 − hn−2 kn−1 ) (−1)n−1 = = kn−1 (ξn kn−1 + kn−2 ) kn−1 (ξn kn−1 + kn−2 ) ξ − rn−1 = ξ − ❚❤✐s ❢r❛❝t✐♦♥ t❡♥❞s t♦ ③❡r♦ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t② ❛♥❞ t❤❡♥✱ ❜② ❉❡✜♥✐t✐♦♥ ✶✳✷✳✷ ✇❡ ❤❛✈❡ ξ = lim rn = lim [a0 , a1 , , an ] = [a0 , a1 , a2 , ] n→∞ n→∞ ◆♦t❡ t❤❛t ξn = [an , an+1 , ] ✐s ♦❜✈✐♦✉s ✐❢ ✇❡ ❛♣♣❧② t♦ ξn t❤❡ ♣r♦❝❡ss ❛s ✐♥ ❡q✉❛t✐♦♥s ✭✶✳✻✮✳ ❚❤❡ ♣r♦♦❢ ✐s ❝♦♠♣❧❡t❡✳ √ ❊①❛♠♣❧❡ ✶✳✷✳✶✳ ❊①♣❛♥❞ ❛s ❛♥ ✐♥✜♥✐t❡ s✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥✳ ❙♦❧✉t✐♦♥✳ ❲❡ s❡❡ t❤❛t ξ0 = √ √ 5, a0 = [ξ0 ] = [ 5] = 2, √ √ ξ1 = 1/(ξ0 − a0 ) = 1/( − 2) = + 2, a1 = [ξ1 ] = 4, ✶✷ √ ξ2 = 1/(ξ1 − a1 ) = 1/( − 2) = ξ1 , a2 = [ξ2 ] = a1 = ❚❤✉s ξi = ξ1 ❛♥❞ = a1 ❢♦r ❛❧❧ i ≥ 1✱ ❛♥❞ t❤✉s √ = [2, 4, 4, 4, ]✳ ❊①❛♠♣❧❡ ✶✳✷✳✷✳ ❊①♣❛♥❞ e ❛s ✐♥✜♥✐t❡ s✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥✳ ❚❤✐s ✐s ❊✉❧❡r✬s ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ✭s❡❡ ❬✶❪✮✿ e = [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ] ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ♣❛rt 2, 1, 2✱ ❛♥❞ r❡♣❡❛t❡❞ ♣❛rts 1, 1, 2n✱ ✇❤❡r❡ n 2✳ ✶✳✷✳✸ ❆♣♣r♦①✐♠❛t✐♦♥s t♦ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡rs ■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t t❤❡ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ✐s ❞❡♥s❡ ✐♥ R✱ ✐t ♠❡❛♥s t❤❛t ❛♥② r❡❛❧ ♥✉♠❜❡r ✐s t❤❡ ❧✐♠✐t ♦❢ s♦♠❡ r❛t✐♦♥❛❧ s❡q✉❡♥❝❡✳ ■♥ t❤✐s s✉❜s❡❝✲ t✐♦♥ ✇❡ s❤♦✇ t❤❛t t❤❡ ❝♦♥✈❡r❣❡♥ts hn /kn ❢♦r♠ ❛ s❡q✉❡♥❝❡ ✧❜❡st✧ r❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s t♦ ❛♥ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r ξ ✳ ❚❤❡♦r❡♠ ✶✳✾✳ ❋♦r ❛♥② n ≥ ✇❡ ❤❛✈❡ ξ− hn < kn kn kn+1 Pr♦♦❢✳ ◆♦t❡ t❤❛t ξ = [a0, a1, , an, ξn+1] ❜② ❚❤❡♦r❡♠ ✶✳✽✳ ❚♦❣❡t❤❡r ✇✐t❤ ❚❤❡♦r❡♠ ✶✳✹ ✇❡ ❣❡t ξ− hn ξn+1 hn + hn−1 hn = − kn ξn+1 kn + kn−1 kn −(hn kn−1 − hn−1 kn ) (−1)n = = , kn (ξn+1 kn + kn−1 ) kn (ξn+1 kn + kn−1 ) s♦ ξ− hn = kn kn (ξn+1 kn + kn−1 ) ✶✸ ❙✐♥❝❡ ξn+1 > an+1 ❜② ✭✶✳✻✮✱ ✇❡ ❤❛✈❡ ξ− hn 1 = < = kn kn (ξn+1 kn + kn−1 ) kn (an+1 kn + kn−1 ) kn kn+1 ❚❤❡ ❝♦♥✈❡r❣❡♥t hn /kn ✐s t❤❡ ❜❡st ❛♣♣r♦①✐♠❛t✐♦♥ t♦ ξ ♦❢ ❛❧❧ t❤❡ r❛t✐♦♥❛❧ ❢r❛❝t✐♦♥s ✇✐t❤ ❞❡♥♦♠✐♥❛t♦r kn ♦r ❧❡ss✳ ❚❤❡♦r❡♠ ✶✳✶✵✳ ■❢ a/b ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r ✇✐t❤ ♣♦s✐t✐✈❡ ❞❡♥♦♠✐♥❛t♦r s✉❝❤ t❤❛t |ξ − a/b| < |ξ − hn/kn| ❢♦r s♦♠❡ n ≥ 1✱ t❤❡♥ b > kn✳ ■♥ ❢❛❝t t❤❛t ✐❢ |ξb − a| < |ξkn − hn| ❢♦r s♦♠❡ n ≥ 0✱ t❤❡♥ b ≥ kn+1✳ Pr♦♦❢✳ ❋✐rst ✇❡ s❤♦✇ t❤❛t t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠ ✐♠♣❧✐❡s t❤❡ ✜rst✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ✜rst ♣❛rt ✐s ❢❛❧s❡ s♦ t❤❛t t❤❡r❡ ✐s ❛♥ a/b ✇✐t❤ ξ− a hn < ξ− b kn ❛♥❞ b ≤ kn ❚❤❡ ♣r♦❞✉❝t ♦❢ t❤❡s❡ ✐♥❡q✉❛❧✐t✐❡s ❣✐✈❡ |ξb − a| < |ξkn − hn |✳ ❇✉t t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡s❡ t❤❡♦r❡♠ s❛②s t❤❛t t❤✐s ✐♠♣❧✐❡s b ≥ kn+1 ✱ s♦ ✇❡ ❤❛✈❡ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✱ s✐♥❝❡ kn < kn+1 ❢♦r n ≥ 1✳ ❚♦ ♣r♦✈❡ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠ ✇❡ ♣r♦❝❡❡❞ ❛❣❛✐♥ ❜② ✐♥❞✐r❡❝t ❛r❣✉♠❡♥t✱ ❛ss✉♠✐♥❣ t❤❛t |ξb − a| < |ξkn − hn | ❛♥❞ b < kn+1 ✳ ❈♦♥s✐❞❡r t❤❡ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✐♥ x ❛♥❞ y ✱ xkn + ykn+1 = b, xhn + yhn+1 = a ❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❝♦❡✣❝✐❡♥ts ✐s ±1 ❜② ❚❤❡♦r❡♠ ✶✳✹✱ ❛♥❞ ❝♦♥s❡q✉❡♥t❧② t❤❡s❡ ❡q✉❛t✐♦♥s ❤❛✈❡ ❛♥ ✐♥t❡❣r❛❧ s♦❧✉t✐♦♥ x, y ✳ ▼♦r❡♦✈❡r✱ ♥❡✐t❤❡r x ♥♦r y ✐s ③❡r♦✳ ❋♦r ✐❢ x = t❤❡♥ b = ykn+1 ✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t y = 0✱ ✐♥ ❢❛❝t t❤❛t y > ❛♥❞ b ≥ kn+1 ✱ ✐♥ ❝♦♥tr❛❞✐❝t✐♦♥ t♦ b < kn+1 ✳ ■❢ y = t❤❡♥ a = xhn ✱ ✶✹ b = xkn ✱ ❛♥❞ |ξb − a| = |ξxkn − xhn | = |x||ξkn − hn | ≥ |kn ξ − hn |, s✐♥❝❡ |x| ≥ 1✱ ❛♥❞ ❛❣❛✐♥ ✇❡ ❤❛✈❡ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ◆❡①t ✇❡ ♣r♦✈❡ t❤❛t x ❛♥❞ y ❤❛✈❡ ♦♣♣♦s✐t❡ s✐❣♥s✳ ❋✐rst✱ ✐❢ y < t❤❡♥ xkn = b − ykn+1 ✱ s❤♦✇s t❤❛t x > 0✳ ❙❡❝♦♥❞✱ ✐❢ y > 0✱ t❤❡♥ b < kn+1 ✐♠♣❧✐❡s t❤❛t b < ykn+1 ❛♥❞ s♦ ykn ✐s ♥❡❣❛t✐✈❡✱ ✇❤❡♥❝❡ x < 0✳ ◆♦✇ ✐t ❢♦❧❧♦✇s ❢r♦♠ ❚❤❡♦r❡♠ ✶✳✺ t❤❛t ξkn − hn ❛♥❞ ξkn+1 − hn+1 ❤❛✈❡ ♦♣♣♦s✐t❡ s✐❣♥s✱ ❛♥❞ ❤❡♥❝❡ x(ξkn − hn ) ❛♥❞ y(ξkn+1 − hn+1 ) ❤❛✈❡ t❤❡ s❛♠❡ s✐❣♥✳ ❋r♦♠ t❤❡ ❡q✉❛t✐♦♥s ❞❡✜♥✐♥❣ x ❛♥❞ y ✇❡ ❣❡t ξb − a = x(ξkn − hn ) + y(ξkn+1 − hn+1 ) ❙✐♥❝❡ t❤❡ t✇♦ t❡r♠s ♦♥ t❤❡ r✐❣❤t ❤❛✈❡ t❤❡ s❛♠❡ s✐❣♥✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t |ξb − a| = |x(ξkn − hn ) + y(ξkn+1 − hn+1 )| = |x(ξkn − hn )| + |y(ξkn+1 − hn+1 )| > |x(ξkn − hn )| = |x||ξkn − hn | ≥ |ξkn − hn | ❚❤✐s ✐s ❛ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❛♥❞ s♦ t❤❡ t❤❡♦r❡♠ ✐s ❡st❛❜❧✐s❤❡❞✳ ❚❤❡♦r❡♠ ✶✳✶✶✳ ▲❡t ξ ❜❡ ❛♥② ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r✳ ■❢ t❤❡r❡ ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r a/b ✇✐t❤ b ≥ s✉❝❤ t❤❛t ξ− a < 2, b 2b a ❡q✉❛❧s ♦♥❡ ♦❢ t❤❡ ❝♦♥✈❡r❣❡♥ts ♦❢ t❤❡ s✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①✲ b ♣❛♥s✐♦♥ ♦❢ ξ t❤❡♥ Pr♦♦❢✳ ■t s✉✣❝❡s t♦ ♣r♦✈❡ t❤❡ r❡s✉❧t ✐♥ t❤❡ ❝❛s❡ (a, b) = 1✳ ▲❡t t❤❡ ❝♦♥✈❡r✲ ✶✺ hj ✱ ❛♥❞ s✉♣♣♦s❡ kj a t❤❛t ✐s ♥♦t ❛ ❝♦♥✈❡r❣❡♥t✳ ❚❤❡ ✐♥❡q✉❛❧✐t✐❡s kn ≤ b < kn+1 ❞❡t❡r♠✐♥❡ ❛♥ b ✐♥t❡❣❡r n✳ ❋♦r t❤✐s n✱ t❤❡ ✐♥❡q✉❛❧✐t② |ξb − a| < |ξkn − hn | ✐s ✐♠♣♦ss✐❜❧❡ ❣❡♥ts ♦❢ t❤❡ s✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ ξ ❜❡ ❜❡❝❛✉s❡ ♦❢ ❚❤❡♦r❡♠ ✶✳✶✵✳ ❚❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ |ξkn − hn | ≤ |ξb − a| < ξ− ❯s✐♥❣ t❤❡ ❢❛❝t t❤❛t , 2b hn < kn 2bkn a hn = ❛♥❞ t❤❛t bhn − akn ✐s ❛♥ ✐♥t❡❣❡r✱ ✇❡ ✜♥❞ t❤❛t b kn |bhn − akn | hn a hn a 1 ≤ = − + ξ− + ≤ ξ− < bkn bkn kn b kn b 2bkn 2b ❚❤✐s ✐♠♣❧✐❡s b < qn ✇❤✐❝❤ ✐s ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ✶✻ ❈❤❛♣t❡r ✷ ❘❙❆ ✷✳✶ ▲✐♥❡❛r ❝♦♥❣r✉❡♥❝❡ ❡q✉❛t✐♦♥ ❚❤❡ s✉❜❥❡❝t ♦❢ t❤✐s s❡❝t✐♦♥ ✐s ❤♦✇ t♦ s♦❧✈❡ ❛♥② ❧✐♥❡❛r ❝♦♥❣r✉❡♥❝❡ ax ≡ b (mod m) ✭✷✳✶✮ ✇❤❡r❡ a, b ❛r❡ ❣✐✈❡♥ ✐♥t❡❣❡rs ❛♥❞ m ✐s ❛ ❣✐✈❡♥ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❲❡ ♦♥❧② st❛t❡ ❤❡r❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❛❜♦✈❡✳ ❚❤❡♦r❡♠ ✷✳✶✳ ❚❤❡ ❧✐♥❡❛r ❝♦♥❣r✉❡♥❝❡ ax ≡ b (mod m) ❤❛s ❛ s♦❧✉t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ d | b✱ ✇❤❡r❡ d = (a, m)✳ ■❢ d | b✱ t❤❡♥ ✐t ❤❛s d ✐♥❝♦♥❣r✉❡♥t s♦❧✉t✐♦♥s ♠♦❞✉❧♦ m✳ ❲❤❡♥ d = 1✱ t❤❡ ❡q✉❛t✐♦♥ ❤❛s ♦♥❧② ♦♥ s♦❧✉t✐♦♥✳ ■♥ t❤✐s ❝❛s❡✱ ✇❡ ❝❛♥ ✜♥❞ t❤❡ s♦❧✉t✐♦♥ ❜② ✉s✐♥❣ t❤❡ t❤❡♦r② ♦❢ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s✳ ❊①♣❛♥❞ m/a ✐♥t♦ t❤❡ s✐♠♣❧❡ ✜♥✐t❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥✱ s❛② m = [a0 , a1 , , an ] a ❚✇♦ ❧❛st ❝♦♥✈❡r❣❡♥ts ❛r❡ hn−1 /kn−1 ❛♥❞ hn /kn ✳ ❇② ❚❤❡♦r❡♠ ✶✳✹ ✇❡ ✶✼ ❤❛✈❡ hn−1 kn − hn kn−1 = (−1)n ◆♦t❡ ❛❧s♦ t❤❛t m hn = a kn ❛♥❞ t❤❡② ❛r❡ t✇♦ r❡❞✉❝❡❞ ❢r❛❝t✐♦♥s✱ s♦ t❤❛t m = hn ❛♥❞ a = kn ✳ ❚❤✉s✱ ahn−1 − mkn = (−1)n−1 ✱ ❛♥❞ t❤✉s a((−1)n bhn−1 ) ≡ b (mod m)✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ s♦❧✉t✐♦♥ ♦❢ ax ≡ b (mod m) ✐s x ≡ (−1)n bhn−1 (mod m) ■❢ d > 1✱ t❤❡♥ ❞✐✈✐❞❡s ❜♦t❤ s✐❞❡s ♦❢ t❤❡♠ ❡q✉❛t✐♦♥ ❜② d t♦ ❣❡t t❤❡ ❡q✉❛t✐♦♥ a1 x ≡ b1 (mod m1 ) ✇❤❡r❡ a1 = a/d, b1 = b/d ❛♥❞ m1 = m/d✳ ❚❤✐s ❡q✉❛t✐♦♥ ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥✱ s❛② x ≡ x0 (mod m1 )✳ ❚❤❡♥✱ t❤❡ ❡q✉❛t✐♦♥ ✭✷✳✶✮ ❤❛s d s♦❧✉t✐♦♥s✿ x ≡ x0 , x0 + m1 , , x0 + (d − 1)m1 (mod m) ❊①❛♠♣❧❡ ✷✳✶✳✶✳ ❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❣r✉❡♥❝❡ 5x ≡ (mod 13)✳ ❙♦❧✉t✐♦♥✳ ❲❡ ❤❛✈❡ 13 = [2, 1, 1, 2]✱ ❛♥❞ h0 = 2, h1 = a1 h0 + h−1 = 1.2 + = 3, h2 = a2 h1 + h0 = 1.3 + = ❚❤✉s✱ t❤❡ s♦❧✉t✐♦♥ ✐s x ≡ (−1)3 · · ≡ −40 ≡ −1 ≡ 12 ✶✽ (mod 13) ❊①❛♠♣❧❡ ✷✳✶✳✷✳ ❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❣r✉❡♥❝❡ 186x ≡ 374 (mod 422)✳ ❙♦❧✉t✐♦♥✳ ❙✐♥❝❡ (186, 422) = ❛♥❞ ✐s ❛ ❞✐✈✐s♦r ♦❢ 374✱ t❤❡ ❝♦♥❣r✉❡♥❝❡ ❤❛s s♦❧✉t✐♦♥s✳ ❉✐✈✐❞❡ ❜♦t❤ s✐❞❡s ♦❢ ❡q✉❛t✐♦♥ ❜② 2✱ ✇❡ ❤❛✈❡ 93x ≡ 187 (mod 211)✳ ❲❡ ❤❛✈❡ 211 = [2, 3, 1, 2, 1, 1, 3]✱ ❛♥❞ 93 h0 = h1 = a1 h0 + h−1 = 3.2 + = h2 = a2 h1 + h0 = 1.7 + = h3 = a3 h2 + h1 = 2.9 + = 25 h4 = a4 h3 + h2 = 1.25 + = 34 h5 = a5 h4 + h3 = 1.34 + 25 = 59 ❚❤❡ ❝♦♥❣r✉❡♥❝❡ 93x ≡ 187 (mod 211) ❤❛s s♦❧✉t✐♦♥ x ≡ (−1)6 · 59 · 187 ≡ 11033 ≡ 61 (mod 211) ❚❤✉s✱ t❤❡ ❝♦♥❣r✉❡♥❝❡ 186x ≡ 374 (mod 422) ❤❛s t✇♦ s♦❧✉t✐♦♥s✿ x ≡ 61, 272 (mod 422) ✷✳✷ ❘❙❆ ✷✳✷✳✶ ❘❙❆ ❙②st❡♠s ❚❤❡ ❘❙❆ ✭❘✐✈❡st ✕ ❙❤❛♠✐r ✕ ❆❞❧❡♠❛♥✮ ❡♥❝r②♣t✐♦♥ ❛❧❣♦r✐t❤♠ ✇❛s ✜rst ♣✉❜❧✐s❤❡❞ ✐♥ 1978 ❜② t❤r❡❡ ♠❛t❤❡♠❛t✐❝✐❛♥s ❛♥❞ ❝♦♠♣✉t❡r s❝✐❡♥t✐sts ❘♦♥ ✶✾ ❘✐✈❡st✱ ❆❞✐ ❙❤❛♠✐r ❛♥❞ ▲❡♦♥❛r❞ ❆❞❧❡♠❛♥✳ ■t ✐s ♦♥❡ ♦❢ t❤❡ ✜rst ♣✉❜❧✐❝✲ ❦❡② ❝r②♣t♦s②st❡♠s ❛♥❞ ✐s ✇✐❞❡❧② ✉s❡❞ ❢♦r s❡❝✉r❡ ❞❛t❛ tr❛♥s♠✐ss✐♦♥✳ ■ts s❡❝✉r✐t② ✐s ❜❛s❡❞ ♦♥ t❤❡ ❞✐✣❝✉❧t② ♦❢ ✜♥❞✐♥❣ t❤❡ ♣r✐♠❡ ❢r❛❝t♦rs ♦❢ ❧❛r❣❡ ✐♥t❡❣❡rs✳ ❋✐rst ✇❡ ❣✐✈❡ t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❘❙❆ ❛s ❢♦❧❧♦✇s✳ ❉❡s❝r✐♣t✐♦♥ ♦❢ ❘❙❆ ❙✉♣♣♦s❡ t❤❛t ❇♦❜ ✇❛♥t t♦ s❡♥❞ ✐♥❢♦r♠❛t✐♦♥ t♦ ❆❧✐❝❡✳ ■❢ t❤❡② ❞❡❝✐❞❡ t♦ ✉s❡ ❘❙❆✱ ❇♦❜ ♠✉st ❦♥♦✇ ❆❧✐❝❡✬s ♣✉❜❧✐❝ ❦❡② t♦ ❡♥❝r②♣t t❤❡ ♠❡ss❛❣❡ ❛♥❞ ❆❧✐❝❡ ♠✉st ✉s❡ ❤❡r ♣r✐✈❛t❡ ❦❡② t♦ ❞❡❝r✉♣t t❤❡ ♠❡ss❛❣❡✳ ❚♦ ❡♥❛❜❧❡ ❇♦❜ t♦ s❡♥❞ ❤✐s ❡♥❝r②♣t❡❞ ♠❡ss❛❣❡s✱ ❆❧✐❝❡ tr❛♥s♠✐ts ❤❡r ♣✉❜❧✐❝ ❦❡② (n, e) t♦ ❇♦❜ ✈✐❛ ❛ r❡❧✐❛❜❧❡✱ ❜✉t ♥♦t ♥❡❝❡ss❛r② s❡❝r❡t✱ r♦✉t❡ ❆❧✐❝❡✬s ♣r✐✈❛t❡ ❦❡② (d) ✐s ♥❡✈❡r ❞✐str✐❜✉t❡❞✳ ❊♥❝r②♣t✐♦♥ ❆❢t❡r ❇♦❜ ♦❜t❛✐♥s ❆❧✐❝❡✬s ♣✉❜❧✐❝ ❦❡②✱ ❤❡ ❝❛♥ s❡♥❞ ❛ ♠❡ss❛❣❡ M t♦ ❆❧✐❝❡ ❚♦ ❞♦ ✐t✱ ❤❡ ✜rst t✉r♥s M ✐♥t♦ ❛♥ ✐♥t❡❣❡r m✱ s✉❝❤ t❤❛t ≤ m < n✳ ❍❡ t❤❡♥ ❝♦♠♣✉t❡s t❤❡ ❝✐♣❤❡rt❡①t c✱ ✉s✐♥❣ ❆❧✐❝❡✬s ♣✉❜❧✐❝ ❦❡② e ❜② c ≡ me ( mod n) ❚❤✐s ❝❛♥ ❜❡ ❞♦♥❡ r❡❛s♦♥❛❜❧② q✉✐❝❦❧②✱ ❡✈❡♥ ❢♦r ✺✵✵✲❜✐t ♥✉♠❜❡rs✱ ✉s✐♥❣ ♠♦❞✲ ✉❧❛s ❡①♣♦♥❡♥t✐❛t✐♦♥ ❇♦❜ t❤❡♥ tr❛♥s♠✐ts c t♦ ❆❧✐❝❡✳ ❉❡❝r②t✐♦♥ ❆❧✐❝❡ ❝❛♥ r❡❝♦✈❡r m ❢r♦♠ c ❜② ✉s✐♥❣ ❤❡r ♣r✐✈❛t❡ ❦❡② ❡①♣♦♥❡♥t d ❜② ❝♦♠♣✉t✐♥❣ cd ≡ (me )d ≡ m (mod n) ●✐✈❡♥ m✱ s❤❡ ❝❛♥ r❡❝♦✈❡r t❤❡ ♦r✐❣✐♥❛❧ ♠❡ss❛❣❡ M ❜② r❡✈❡rs✐♥❣ t❤❡ ♣❛❞❞✐♥❣ s❝❤❡♠❡✳ ❘❙❆ ❑❡② ●❡♥❡r❛t✐♦♥ ❆❧❣♦r✐t❤♠✿ ✷✵ ✶✳ ❈❤♦♦s❡ t✇♦ ❞✐st✐♥❝t ♣r✐♠❡ ♥✉♠❜❡rs p ❛♥❞ q ✳ ✷✳ ❈♦♠♣✉t❡ n = pq ✱ φ(n) = (p − 1)(q − 1)✳ ✸✳ ❈❤♦♦s❡ e st < e < φ(n) ❛♥❞ (e, φ(n)) = 1✳ ✹✳ ❉❡t❡r♠✐♥❡ de ≡ (mod φ(n))✳ ✺✳ ❑❡❡♣ p, q, d s❡❝r❡t✳ ✻✳ P✉❜❧✐s❤ n, e✳ ❆♥❞ ♥♦✇ ✇❡ ♣r♦✈❡ t❤❡ r❡❛❧✐t② ♦❢ t❤❡ ❘❙❆ s②st❡♠✿ Pr♦♦❢✳ ❲❡ ❤❛✈❡ cd ≡ (me)d ≡ med (mod n)✳ ❙✐♥❝❡ ed ≡ (mod φ(n))✱ ✇❡ ❤❛✈❡ de = + kφ(n) ❢♦r k ∈ Z✳ ❆ss✉♠❡ (m, n) = 1✳ ❇② ❊✉❧❡r✬s t❤❡♦r❡♠ mde ≡ m1+kφ(n) ≡ (mφ(n) )k m ≡ m (mod n) ❚❤✉s✱ cd ≡ m( mod n)✳ ❊①❛♠♣❧❡ ✷✳✷✳✶✳ ❇♦❜ ❝❤♦♦s❡s p = 17, q = 23✳ ❈❛❧❝✉❧❛t❡ n = 17.23 = 391 ❛♥❞ φ(n) = (17 − 1)(23 − 1) = 352✳ ❈❤♦♦s❡ e = ❛♥❞ d = 141✳ ❙♦ ed = 5.141 = 705 ≡ (mod 352)✳ ❇♦❜ ♥♦✇ ✇♦✉❧❞ ❧✐❦❡ t♦ s❡♥❞ t❤❡ ✐♥❢♦r♠❛t✐♦♥ m = 89 t♦ ❆❧✐❝❡✳ ❇♦❜ ❝❛❧❝✉❧❛t❡s 895 ≡ 378 (mod 391) ❛♥❞ s❡♥❞s t❤✐s ♥✉♠❜❡r t♦ ❆❧✐❝❡✳ ❙❤❡ ❡♥❝♦❞❡s 378141 ≡ 89 (mod 391)✳ ■♥ t❤✐s ❡①❛♠♣❧❡✱ ❢♦r s✐♠♣❧✐❝✐t②✱ ✇❡ ❝❤♦♦s❡ s♠❛❧❧ ♥✉♠❜❡rs p ❛♥❞ q ✱ ✐♥ ❢❛❝t p, q ❛r❡ ✈❡r② ❧❛r❣❡ ♣r✐♠❡s✱ s♦ ✜♥❞✐♥❣ p ❛♥❞ q ❢r♦♠ n ✐s ❛❧♠♦st ✐♠♣♦ss✐❜❧❡✳ ◆❡①t✱ ✇❡ ❝♦♥s✐❞❡r ❤♦✇ t♦ ❛tt❛❝❦ ❘❙❆✳ ✷✶ ✷✳✷✳✷ ❆tt❛❝❦ ❘❙❆ ■♥ ❘❙❆ s②st❡♠✱ p ❛♥❞ q ♣❧❛② ❛♥ ✐♠♣♦rt❛♥t r♦❧❡✳ ❚❤❡② ❛r❡ t✇♦ ✉♥✐q✉❡ ❢❛❝✲ t♦rs ♦❢ n ❛♥❞ ❢♦r ❝❛❧❝✉❧❛t✐♥❣ d ✇❤❡♥ ✇❡ ❦♥♦✇ e✳ ❘❙❆ ❙②st❡♠✬s ❝♦♠♣❧✐❝❛❝② ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐✣❝✉❧t② ♦❢ ✜♥❞✐♥❣ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ ❧❛r❣❡ ✐♥t❡❣❡r n✳ ❲✐❡♥❡r ♣r♦♣♦s❡❞ ❛♥ ❛❧❣♦r✐t❤♠ t♦ ❛tt❛❝❦ ❘❙❆ ✐♥ ❝❛s❡ d ✐s s♠❛❧❧❡r t❤❛♥ n ✭s❡❡ ❬✸❪✮✳ ❚♦ ❜❡ ♠♦r❡ s♣❡❝✐✜❝✿ p < q < 2p, d < 1√ n ✭✷✳✷✮ ❙✐♥❝❡ de ≡ (mod φ(n))✱ t❤❡♥ de − kφ(n) = ❢♦r s♦♠❡ ✐♥t❡❣❡r k ✳ 1√ k n✱ t❤❡♥ ✐s ❛ ❝♦♥✈❡r❣❡♥t ♦❢ ❚❤❡♦r❡♠ ✷✳✷✳ ■❢ p < q < 2p ❛♥❞ d < d e ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ ✳ n Pr♦♦❢✳ ❋r♦♠ de − kφ(n) = 1✱ ✇❡ ✐♠♣❧② t❤❛t k ≥ ❛♥❞ ♠❛①{e, d} s✐♥❝❡ ❜② t❤❡ ✇❛② ✇❡ ❝♦♥str✉❝t ❘❙❆ ❤❛s e < φ(n) ❛♥❞ d < φ(n)✳ ◆♦t❡ t❤❛t n = pq ✱ p < q < 2p ❛♥❞ φ(n) = (p − 1)(q − 1) = n − (p + q) + 1✱ ✇❤✐❝❤ ✐♠♣❧② √ n − n < φ(n) < n✳ √ ❙✐♥❝❡ de − kφ(n) = ❛♥❞ n > 9d2 ✱ ✇❡ ❤❛✈❡ k e kn − de − = d n dn kn − kφ(n) − = dn kn − kφ(n) k(n − φ(n)) < = dn dn √ 3k n 3k < = √ dn d n 3 3d < √ = √ < < 2d d n n 9d ✷✷ ❇② ❚❤❡♦r❡♠ ✶✳✶✶✱ ♦❢ e ✳ n k ✐s ❛ ❝♦♥✈❡r❣❡♥t ♦❢ t❤❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ d ▲❡t [a0 , a1 , a2 , ] ✐s t❤❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ ❛ r❡❛❧ ♥✉♠❜❡r ξ ✳ ❚❤❡ ❝♦♥✈❡r❣❡♥ts ❢♦r i ≥ hj s❛t✐s❢② h0 = a0 ✱ k0 = 1✱ h1 = a0 a1 + 1, k1 = a1 ✱ ❛♥❞ kj hi = hi−1 + ki−2 ki = ki−1 + ki−2 ❚❤❡r❡❢♦r❡✱ t❤❡ ❞❡♥♦♠✐♥❛t♦rs ❣r♦✇ ❡①♣♦♥❡♥t✐❛❧❧②✳ ❚❤✐s ♠❡❛♥s t❤❛t t♦t❛❧ e ✐s ♦❢ ♦r❞❡r O(❧♦❣n)✳ ■❢ ❛ ❝♦♥✈❡r❣❡♥t ❝❛♥ ❜❡ n t❡st❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✱ t❤✐s ✇✐❧❧ ❣✐✈❡ ✉s ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ t♦ ♥✉♠❜❡r ♦❢ ❝♦♥✈❡r❣❡♥ts ♦❢ ❞❡t❡r♠✐♥❡ d✳ a ❜❡ ❛ b e k ❝♦♥✈❡r❣❡♥t ♦❢ ✳ ■❢ ✐t ✐s t❤❡ ❝♦rr❡❝t ❣✉❡ss ❢♦r ✱ t❤❡♥ φ(n) ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ n d (p + q) ❢r♦♠ φ(n) = (p − 1)(q − 1) = (be − 1)/a✳ ❲❡ ❝❛♥ ♥♦✇ ❝♦♠♣✉t❡ q−p ❛♥❞ r❡s♣❡❝t✐✈❡❧② ❢r♦♠ t❤❡ ✐❞❡♥t✐t②✿ ❲✐❡♥❡r ♣r♦♣♦s❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❡t❤♦❞ ❢♦r t❡st✐♥❣ ❝♦♥✈❡r❣❡♥ts✳ ▲❡t p+q pq − (p − 1)(q − 1) n − φ(n) + = = ; 2 q−p 2 = p+q 2 − pq = p+q 2 − n p+q q−p ❛♥❞ ✱ ♦❜t❛✐♥❡❞ ❜② t❤❡s❡ ✐❞❡♥t✐t✐❡s✱ ❛r❡ ♣♦s✐t✐✈❡ 2 a k ✐♥t❡❣❡rs✱ t❤❡♥ t❤❡ ❝♦♥✈❡r❣❡♥t ✐s ❝♦rr❡❝t ❣✉❡ss ❢♦r ✳ ❲❡ ❝❛♥ ❛❧s♦ r❡❝♦✈❡r b d p+q q−p ❡❛s✐❧② p ❛♥❞ q ❢r♦♠ ❛♥❞ ✳ 2 ■❢ t❤❡ ♥✉♠❜❡rs ✷✸ ❊①❛♠♣❧❡ ✷✳✷✳✷✳ ❈♦♥s✐❞❡r ❘❙❆ ❙②st❡♠ ❤❛s n = 205320043521075746592613, e = 70760135995620281241019, ❛♥❞ ❛ss✉♠❡ t❤❛t d < 224382✳ e ✐s n [0, 2, 1, 9, 6, 54, 5911, 1, 5, 1, 1, 3, 1, 2, 5, 3, 1, 3, 1, 1, 2, 13, 1, 4, 5, ] ❚❤❡ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ♦❢ ❆♥❞ t❤❡ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ✐s 1 10 61 3304 19530005 , , , , , , , 29 177 9587 56668934 3304 ❇② ❲✐❡♥♥❡r ♠❡t❤♦❞✱ ✇❡ ❦❡❡♣ tr②✐♥❣ t♦ ❝♦♥✈❡r❣❡♥t ✳ ❚❤❡♥ ✇❡ ❤❛✈❡ 9587 p+q q−p = 548218898963 ❛♥❞ = 308583728766✳ 2 ❚❤✐s ✐♠♣❧✐❡s p = 239635170197 ❛♥❞ q = 856802627729✳ ❍❡♥❝❡✱ n = 239635170197 · 856802627729✳ ✷✹ ❈♦♥❝❧✉s✐♦♥ ■♥ t❤❡ t❤❡s✐s✱ ✇❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ s♦♠❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧ts • ❲❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ t❤❡ ♠♦st ❜❛s✐❝ ♣r♦❜❧❡♠s ❛♥❞ t❤❡♦r✐❡s ❛❜♦✉t ❝♦♥✲ t✐♥✉❡❞ ❢r❛❝t✐♦♥s✳ • ❲❡ ❛♣♣❧✐❡❞ t❤❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s t♦ r❡♣r❡s❡♥t r❡❛❧ ♥✉♠❜❡rs✱ s♦❧✈❡ t❤❡ ❧✐♥❡❛r ❝♦♥❣r✉❡♥❝❡ ❡q✉❛t✐♦♥✱ ❛tt❛❝❦ ❘❙❆ s②st❡♠✳ ✷✺ ❇✐❜❧✐♦❣r❛♣❤② ❬✶❪ ❍✳ ❈♦❤♥✱ ❆ s❤♦rt ♣r♦♦❢ ♦❢ t❤❡ s✐♠♣❧❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ e✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦♥t❤❧②✱ ❱♦❧✳ ✶✶✻ ✭✷✵✵✻✮✱✺✼✕✻✷✳ ❬✷❪ ■✳ ◆✐✈❡♥✱ ❍✳❙✳ ❩✉❝❦❡r♠❛♥ ❛♥❞ ❍✳ ▲✳ ▼♦♥t❣♦♠❡r②✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ❚❤❡♦r② ♦❢ ◆✉♠❜❡rs✱ ✜❢t❤ ❡❞✐t✐♦♥✱ ❏♦❤♥ ❲✐❧❡② ✫ ❙♦♥s✱ ■♥❝✳ ❬✸❪ ▼✳ ❏✳ ❲✐❡♥❡r✱ ❈r②♣t❛♥❛❧②s✐s ♦❢ s❤♦rt ❘❙❆ s❡❝r❡t ❡①♣♦♥❡♥ts✱ ■❊❊❊ ❚r❛♥s✳ ■♥❢♦r♠✳ ❚❤❡♦r② ✸✻ ✭✶✾✾✵✮✱ ✺✺✸✕✺✺✽✳ ✷✻