k 2 = 4

✷✳✶ ❚r✐❛• ❞✐✈✐s✐♦♥ ❛•❣♦r✐t❤♠ •✐✈❡♥ ♠❛♥② ❝♦♠♣♦s✐t❡ ✐♥t❡❣❡rs N i ✱ 0 ≤ i < n✱ ✇❡ ✇❛♥t t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ♣r✐♠❡s ❢r♦♠ s♦♠❡ s❡t P = {p j , 0 ≤ j < k} ♦❢ s♠❛•• ♦❞❞ ♣r✐♠❡s ❞✐✈✐❞❡ ❡❛❝❤ N i ✳ ❲❡ ❛ss✉♠❡ n ≫ k✳ ❊❛❝❤ N i ✐s ❛ ♠✉•t✐✲✇♦r❞ ✐♥t❡❣❡r ♦❢ ✉♣ t♦ ℓ + 1 ✇♦r❞s✱ N i = ℓ j=0 n i,j β j ✱ ✇❤❡r❡ β ✐s t❤❡ ♠❛❝❤✐♥❡ ✇♦r❞ ❜❛s❡ ✭❡✳❣✳✱ β = 2 32 ♦r β = 2 64 ✮ ❛♥❞ ℓ ✐s ♦♥ t❤❡ ♦r❞❡r ♦❢ ✑❛ ❢❡✇✱✑ s❛② ℓ ≤ 4✳ ❋♦r ❡❛❝❤ ♣r✐♠❡ p ∈ P ✱ ✇❡ ♣r❡❝♦♠♣✉t❡ w j = β j mod p ❢♦r 1 ≤ j ≤ ℓ✱ p ✐♥✈ = p -1 (mod β) ❛♥❞ p •✐♠ = β-1 p ✳ ❈♦♥
√ βℓ -ℓ + 1)✳ ❲❡ t❤❡♥ ❞❡❝♦♠♣♦s❡ r ✐♥t♦ r = r 1 β + r 0 ✱ ✇❤❡r❡ 0 ≤ r 0 < β✳ ❚❤✐s ✐♠♣•✐❡s r 1 < √ βℓ✱ ❛♥❞ r 1 w 1 ≤ r 1 (p -1) < √ βℓ β ℓ -1 = β - √ βℓ✳ ❚❤❡ ❛•❣♦r✐t❤♠ t❤❡♥ ❞♦❡s ❛♥♦t❤❡r r❡❞✉❝t✐♦♥ st❡♣ ❜② s = r 1 w 1 + r 0 ✳ ❲❡ ✇♦✉•❞ •✐❦❡ s = s 1 β + s 0 < 2β -p✱ s♦ t❤❛t ❛ ✜♥❛• r❡❞✉❝t✐♦♥ st❡♣ t = s 0 + s 1 w 1 < β ♣r♦❞✉❝❡s ❛ ♦♥❡✲✇♦r❞ r❡s✉•t✳ ❙✐♥❝❡ r 1 (p-1) < β- √ βℓ✱ s < 2β- √ βℓ-1 < 2β-p✳ ❙✐♥❝❡ s 1 ✐s ❡✐t❤❡r 0 ♦r 1✱ t❤❡ ♠✉•t✐♣•✐❝❛t✐♦♥ ❛♥❞ ❛❞❞✐t✐♦♥ ✐♥ s 0 + s 1 w 1 ✐s r❡❛••② •✉st ❛ ❝♦♥❞✐t✐♦♥❛• ❛❞❞✐t✐♦♥✳ ◆♦✇ ✇❡ ❤❛✈❡ ❛ ♦♥❡✲✇♦r❞ ✐♥t❡❣❡r t ✇❤✐❝❤ ✐s ❞✐✈✐s✐❜•❡ ❜② p ✐❢ ❛♥❞ ♦♥•② ✐❢ N ✐s✳ ❚♦ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r p | t✱ ✇❡ ✉s❡ t❤❡ ✐❞❡❛ ❢r♦♠ ❬✶✸✱ ➓✾( t♦ ❝♦♠♣✉t❡ u = tp -1 mod β✱ ✉s✐♥❣ t❤❡ ♣r❡❝♦♠♣✉t❡❞ p ✐♥✈ = p -1 (mod β)✳ ■❢ p | t✱ t/p ✐s ❛♥ ✐♥t❡❣❡r < β ❛♥❞ s♦ t❤❡ ♠♦❞✉•❛r ❛r✐t❤♠❡t✐❝ mod β ♠✉st ♣r♦❞✉❝❡ t❤❡ ❝♦rr❡❝t u = t/p✳ ❚❤❡r❡ ❛r❡ β-1 p + 1 ♠✉•t✐♣•❡s ♦❢ p ✭✐♥❝•✉❞✐♥❣ ✵✮ •❡ss t❤❛♥ β✱ ✉♥❞❡r ❞✐✈✐s✐♦♥ ❜② p t❤❡s❡ ♠❛♣ t♦ t❤❡ ✐♥t❡❣❡rs 0, . . . , β-1 p ✳ ❙✐♥❝❡ p ✐s ❝♦♣r✐♠❡ t♦ β✱ ♠✉•t✐♣•✐❝❛t✐♦♥ ❜② p -1 (mod β) ✐s ❛ ❜✐•❡❝t✐✈❡ ♠❛♣✱ s♦ ❛•• ♥♦♥✲♠✉•t✐♣•❡s ♦❢ p ♠✉st ♠❛♣ t♦ t❤❡ r❡♠❛✐♥✐♥❣ ✐♥t❡❣❡rs β-1 p + 1, β -1 ✳ ❍❡♥❝❡ t❤❡ t❡st ❢♦r ❞✐✈✐s✐❜✐•✐t② ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✼ ❝❛♥ ❜❡ ❞♦♥❡ ❜② ❛ ♦♥❡✲✇♦r❞ ♠✉•t✐♣•✐❝❛t✐♦♥ ❜② t❤❡ ♣r❡❝♦♠♣✉t❡❞ ❝♦♥st❛♥t p ✐♥✈ ✱ ❛♥❞ ♦♥❡ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ♣r❡❝♦♠♣✉t❡❞ ❝♦♥st❛♥t p •✐♠ = β-1 p ✳ ✷✳✷ ■♠♣•❡♠❡♥t❛t✐♦♥ ❚❤❡ ❛•❣♦r✐t❤♠ ✐s q✉✐t❡ s✐♠♣•❡ t♦ ✐♠♣•❡♠❡♥t ♦♥ ❛♥ ①✽✻ ❈P❯✱ ✇❤✐❝❤ ♦✛❡rs t❤❡ t✇♦✲✇♦r❞ ♣r♦❞✉❝t ♦❢ t✇♦ ♦♥❡✲✇♦r❞ ❛r❣✉♠❡♥ts ❜② ❛ s✐♥❣•❡ ▼❯▲ ✐♥str✉❝t✐♦♥✳ ■t ♠✐❣❤t r✉♥ ❛s s❤♦✇♥ ✐♥ ❆•❣♦r✐t❤♠ ✶✱ ✇❤❡r❡ x 1 ✱ x 0 ❛r❡ r❡❣✐st❡rs t❤❛t t❡♠♣♦r❛r✐•② ❤♦•❞ t✇♦✲✇♦r❞ ♣r♦❞✉❝ts✳ ❆ ♣❛✐r ♦❢ r❡❣✐st❡rs ❤♦•❞✐♥❣ ❛ t✇♦✲✇♦r❞ ✈❛•✉❡ r 1 β + r 0 ✐s ✇r✐tt❡♥ ❛s r 1 : r 0 . ❚❤❡ ✈❛•✉❡s r 0,1 ✱ s 0,1 ✱ ❛♥❞ t 0 ❝❛♥ ❛•• ✉s❡ t❤❡ s❛♠❡ r❡❣✐st❡rs✱ ✇r✐tt❡♥ r 0,1 ❤❡r❡✳ ❚❤❡ •♦♦♣ ♦✈❡r j s❤♦✉•❞ ❜❡ ✉♥r♦••❡❞✳ ■♥♣✉t✿ ▲❡♥❣t❤ ℓ N = ℓ i=0 n i β i , 0 ≤ n i < β ❖❞❞ ♣r✐♠❡ p < β ℓ w j = β j mod p ❢♦r 1 ≤ j ≤ ℓ p ✐♥✈ = p -1 mod β p •✐♠ = β-1 p ❖✉t♣✉t✿ 1 ✐❢ p | N ✱ 0 ♦t❤❡r✇✐s❡ r 0 := n 0 ❀ r 1 := 0❀ ❢♦r 1 ≤ j ≤ ℓ ❞♦
x 1 : x 0 = n j • w j ❀ r 1 : r 0 = r 1 : r 0 + x 1 : x 0 ❀ x 0 = r 1 • w 1 ❀ r 0 = (r 0 + x 0 ) mod β❀ ✐❢ •❛st ❛❞❞✐t✐♦♥ s❡t ❝❛rr② ✢❛❣ t❤❡♥ r 0 = (r 0 + w 1 ) mod β❀ 2,3,5,7,12,19,24,43,67,110,177,287,464,751 2,3,5,8,13,21,34,55,68,123,191,314,437,751. 2,3,4,7,11,18,29,47,76,123,170,293,463,756,1219 2,3,5,6,7,13,20,33,53 10, 3, 3, 0, 3, 3, 0, 5, 3, 3, 3, 11 103 : 10, 3, 0, 3, 3, 0, 3, 3, 0, 4, 3, 11 107 : 10, 3, 0, 3, 3, 0, 3, 0, 4, 3, 3, 3, 11 109 : 10, 3, 0, 3, 0, 1, 1, 3, 11 2, 3, 5, 6, 7, 11, 13, 17, 19, 23, . . . 

r 0 = r 0 • p ✐♥✈ ❀ ✐❢ r 0 ≤ p •✐♠ t❤❡♥ r❡t✉r♥ 1❀ ❡•s❡ r❡t✉r♥ 0❀ ❆•❣♦r✐t❤♠ ✶✿ Ps❡✉❞♦✲❝♦❞❡ ❢♦r tr✐❛• ❞✐✈✐s✐♦♥ ♦❢ ♥✉♠❜❡rs ♦❢ ✉♣ t♦ ℓ + 1 ✇♦r❞s✳ ❚❤✐s ❝♦❞❡ ✉s❡s ℓ ♠✉•t✐♣•✐❝❛t✐♦♥s ♦❢ t✇♦ ✇♦r❞s t♦ ❛ t✇♦✲✇♦r❞ ♣r♦❞✉❝t✳ ❚❤❡s❡ ♠✉•t✐♣•✐❝❛t✐♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♦♥❡ ❛♥♦t❤❡r✱ s♦ t❤❡② ❝❛♥ ♦✈❡r•❛♣ ♦♥ ❛ ❈P❯ ✇✐t❤ ♣✐♣❡•✐♥❡❞ ♠✉•t✐♣•✐❡r✳ ❖♥ ❛♥ ❆t❤•♦♥✻✹✱ ❖♣t❡r♦♥✱ ❛♥❞ P❤❡♥♦♠ ❈P❯s✱ ❛ ♠✉•t✐♣•✐❝❛t✐♦♥ ❝❛♥ st❛rt ❡✈❡r② ✷ ❝•♦❝❦ ❝②❝•❡s✱ t❤❡ •♦✇ ✇♦r❞ ♦❢ t❤❡ ♣r♦❞✉❝t ✐s ❛✈❛✐•❛❜•❡ ❛❢t❡r ✹ ❝•♦❝❦ ❝②❝•❡s✱ t❤❡ ❤✐❣❤ ✇♦r❞ ❛❢t❡r ✺ ❝•♦❝❦ ❝②❝•❡s✳ ❚❤✉s ✐♥ ❝❛s❡ ♦❢ ℓ = 4✱ t❤❡ •❛t❡♥❝② ❢♦r t❤❡ ✜rst 4 ♣r♦❞✉❝ts ❛♥❞ ❜✉✐•❞✐♥❣ t❤❡✐r s✉♠ s❤♦✉•❞ ❜❡ ✶✷ ❝②❝•❡s✳ ❚❤❡ t✇♦ r❡♠❛✐♥✐♥❣ ♠✉•t✐♣•✐❡s✱ t❤❡ ❛❞❞✐t✐♦♥s ❛♥❞ ❝♦♥❞✐t✐♦♥❛• ♠♦✈❡s s❤♦✉•❞ ❜❡ ♣♦ss✐❜•❡ ✐♥ ❛❜♦✉t ✶✶ ❝②❝•❡s✱ ❣✐✈✐♥❣ ❛ t❤❡♦r❡t✐❝❛• t♦t❛• ❝♦✉♥t ♦❢ ❛❜♦✉t ✷✸ ❝•♦❝❦ ❝②❝•❡s ❢♦r tr✐❛• ❞✐✈✐❞✐♥❣ ❛ ✺ ✇♦r❞ ✐♥t❡❣❡r ❜② ❛ s♠❛•• ♣r✐♠❡✳ ❉❛t❛ ♠♦✈❡♠❡♥t
w i ❢♦r •❛r❣❡r i ❞♦ ♥♦t ♦❝❝✉♣② ❝❛❝❤❡ ✇❤✐•❡ t❤❡ N ♣r♦❝❡ss❡❞ ❛r❡ s♠❛••✳ ❙✐♥❝❡ t❤❡ ✈❛•✉❡ ♦❢ p ✐s ♥♦t ❛❝t✉❛••② ♥❡❡❞❡❞ ❞✉r✐♥❣ t❤❡ tr✐❛• ❞✐✈✐s✐♦♥✱ ✐t ✐s ♣♦ss✐❜•❡ t♦ ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✾ ◆✉♠❜❡r ♦❢ ✇♦r❞s ✐♥ N n ✶ ✷ ✸ ✹
n ♦❞❞ ♣r✐♠❡s ♦♥ ❛ 2 •❍③ ❆▼❉ P❤❡♥♦♠ ❈P❯✳ ❚✐♠❡ ♣❡r tr✐❛• ❞✐✈✐s♦♥ ✐♥ ♥❛♥♦s❡❝♦♥❞s ✐♥ ♣❛r❡♥t❤❡s❡s✳ ❛✈♦✐❞ st♦r✐♥❣ ✐t ❛♥❞ r❡❝♦♠♣✉t✐♥❣ ✐t✱ ❡✳❣✳✱ ❢r♦♠ p ✐♥✈ ✇❤❡♥ ✐t ♥❡❡❞s t♦ ❜❡ r❡♣♦rt❡❞ ❛s ❛ ❞✐✈✐s♦r✳ ✸ ▼♦❞✉•❛r ❛r✐t❤♠❡t✐❝ ❚❤❡ ♠♦❞✉•❛r ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ❛r❡ r❡•❛t✐✈❡•② ✐♥❡①♣❡♥s✐✈❡ ✇❤❡♥ ♠♦❞✉•✐ ❛♥❞ r❡s✐❞✉❡s ♦❢ ♦♥•② ❛ ❢❡✇ ♠❛❝❤✐♥❡ ✇♦r❞s ❛r❡ ❝♦♥s✐❞❡r❡❞✱ ❛♥❞ s❤♦✉•❞ ❜❡ ✐♠♣•❡♠❡♥t❡❞ ✐♥ ❛ ✇❛② t❤❛t •❡ts t❤❡ ❝♦♠♣✐•❡r ♣❡r❢♦r♠ ✐♥✲•✐♥✐♥❣ ♦❢ s✐♠♣•❡ ❛r✐t❤♠❡t✐❝ ❢✉♥❝t✐♦♥s t♦ ❛✈♦✐❞ ✉♥♥❡❝❡ss❛r② ❢✉♥❝t✐♦♥ ❝❛•• ♦✈❡r❤❡❛❞ ❛♥❞ ❞❛t❛ ♠♦✈❡♠❡♥t ❜❡t✇❡❡♥ r❡❣✲ ✐st❡rs✱ ♠❡♠♦r② ❛♥❞ st❛❝❦ ❞✉❡ t♦ t❤❡ ❝❛••✐♥❣ ❝♦♥✈❡♥t✐♦♥s ♦❢ t❤❡ •❛♥❣✉❛❣❡ ❛♥❞ ❛r❝❤✐t❡❝t✉r❡✳ ▼❛♥② s✐♠♣•❡ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ❝❛♥ ❜❡ ✐♠♣•❡♠❡♥t❡❞ ❡❛s✐•② ❛♥❞ ❡✣❝✐❡♥t•② ✉s✐♥❣ ❛ss❡♠❜•② •❛♥❣✉❛❣❡✱ ❜✉t ❛r❡ ❝✉♠❜❡rs♦♠❡ t♦ ✇r✐t❡ ✐♥ ♣✉r❡ ❈ ❝♦❞❡✱ ❡s♣❡❝✐❛••② ✐❢ ♠✉•t✐✲✇♦r❞ ♣r♦❞✉❝ts ♦r ❝❛rr② ♣r♦♣❛❣❛t✐♦♥ ❛r❡ ✐♥✈♦•✈❡❞✳ ❚❤❡ •◆❯ ❈ ❝♦♠♣✐•❡r ♦✛❡rs ❛ ✈❡r② ✢❡①✐❜•❡ ♠❡t❤♦❞ ♦❢ ✐♥•❡❝t✐♥❣ ❛ss❡♠❜•② ❝♦❞❡ ✐♥t♦ ❈ ♣r♦❣r❛♠s✱ ✇✐t❤ ❛♥ ✐♥t❡r❢❛❝❡ t❤❛t t❡••s t❤❡ ❝♦♠♣✐•❡r ❛•• ❝♦♥str❛✐♥ts ♦♥ ✐♥♣✉t ❛♥❞ ♦✉t♣✉t ❞❛t❛ ♦❢ t❤❡ ❛ss❡♠❜•② ❜•♦❝❦ s♦ t❤❛t ✐t ❝❛♥ ♣❡r❢♦r♠ ♦♣t✐♠✐③❛t✐♦♥ ♦♥ t❤❡ ❝♦❞❡ s✉rr♦✉♥❞✐♥❣ t❤❡ ❛ss❡♠❜•② st❛t❡♠❡♥ts✳ ❇② ❞❡✜♥✐♥❣ s♦♠❡ ❝♦♠♠♦♥•② ✉s❡❞ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ✐♥ ❛ss❡♠❜•②✱ ♠✉❝❤ ♦❢ t❤❡ ♠♦❞✉•❛r ❛r✐t❤♠❡t✐❝ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ ❈✱ •❡tt✐♥❣ t❤❡ ❝♦♠♣✐•❡r ❤❛♥❞•❡ r❡❣✐st❡r ❛••♦❝❛t✐♦♥ ❛♥❞ ❞❛t❛ ♠♦✈❡♠❡♥t✳ ❚❤❡ r❡s✉•t✐♥❣ ❝♦❞❡ ✐s ✉s✉❛••② ♥♦t ♦♣t✐♠❛•✱ ❜✉t q✉✐t❡ ✉s❡❛❜•❡✳ ❋♦r t❤❡ ♠♦st t✐♠❡✲ ❝r✐t✐❝❛• ♦♣❡r❛t✐♦♥s✱ ✇r✐t✐♥❣ ❤❛♥❞✲♦♣t✐♠✐③❡❞ ❛ss❡♠❜•② ❝♦❞❡ ♦✛❡rs ❛♥ ❛❞❞✐t✐♦♥❛• s♣❡❡❞ ✐♠♣r♦✈❡♠❡♥t✳ ❋♦r t❤❡ ♣r❡s❡♥t ✇♦r❦✱ ♠♦❞✉•❛r ❛r✐t❤♠❡t✐❝ ❢♦r ♠♦❞✉•✐ ♦❢ 1 ♠❛❝❤✐♥❡ ✇♦r❞ ❛♥❞ ♦❢ 2 ♠❛❝❤✐♥❡ ✇♦r❞s ✇✐t❤ t❤❡ t✇♦ ♠♦st s✐❣♥✐✜❝❛♥t ❜✐ts ③❡r♦ ✐s ✐♠♣•❡♠❡♥t❡❞✳ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❛r✐t❤♠❡t✐❝ ❢♦r ♠♦❞✉•✐ ♦❢ 3 ♠❛❝❤✐♥❡ ✇♦r❞s ✐s ✐♥ ♣r♦❣r❡ss✳ ✸✳✶ ❆ss❡♠❜•② s✉♣♣♦rt ❚♦ ❣✐✈❡ ❛♥ ❡①❛♠♣•❡ ♦❢ ❛♥ ❡•❡♠❡♥t❛r② ❢✉♥❝t✐♦♥ t❤❛t ✐s ✐♠♣•❡♠❡♥t❡❞ ✇✐t❤ t❤❡ ❤❡•♣ ♦❢ s♦♠❡ ❛ss❡♠❜•② ❝♦❞❡✱ ✇❡ ❡①❛♠✐♥❡ ♠♦❞✉•❛r ❛❞❞✐t✐♦♥ ✇✐t❤ ❛ ♠♦❞✉•✉s ♦❢ 1 ♠❛❝❤✐♥❡ ✇♦r❞✳ ❚❤✐s ✐s ❛♠♦♥❣ t❤❡ ♠♦st s✐♠♣•❡ ♦♣❡r❛t✐♦♥s ♣♦ss✐❜•❡✱ ❜✉t ✉s❡❢✉• ❛s ❛♥ ❡①❛♠♣•❡✳ ▲❡t ❛ ✏r❡❞✉❝❡❞ r❡s✐❞✉❡✑ ✇✐t❤ r❡s♣❡❝t t♦ ❛ ♣♦s✐t✐✈❡ ♠♦❞✉•✉s m ♠❡❛♥ ❛♥ ✐♥t❡❣❡r r❡♣r❡s❡♥t❛t✐✈❡ 0 ≤ r < m ♦❢ t❤❡ r❡s✐❞✉❡ ❝•❛ss r (mod m). ▼♦❞✉•❛r ❛❞❞✐t✐♦♥ ♦❢ || ♥➦ ✼✵✹✶ ✶✵ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ t✇♦ r❡❞✉❝❡❞ r❡s✐❞✉❡s ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s (a + b) mod m = a + b -m ✐❢ a + b -m ≥ 0 a + b ♦t❤❡r✇✐s❡✳ ■❢ ❛♥② ♠♦❞✉•✉s m < β ✐s ♣❡r♠✐tt❡❞✱ ✇❤❡r❡ β ✐s t❤❡ ♠❛❝❤✐♥❡ ✇♦r❞ ❜❛s❡✱ t❤❡♥ t❤❡ ♣r♦❜•❡♠ t❤❛t a + b ♠✐❣❤t ♦✈❡r✢♦✇ t❤❡ ♠❛❝❤✐♥❡ ✇♦r❞ ❛r✐s❡s✳ ❖♥❡ ❝♦✉•❞ t❡st ❢♦r t❤✐s ❝❛s❡✱ t❤❡♥ t❡st ✐❢ a + b ≥ m✱ ❛♥❞ s✉❜tr❛❝t m ✐❢ ❡✐t❤❡r ✐s tr✉❡✱ ❜✉t t❤✐s ♥❡❝❡ss✐t❛t❡s t✇♦ t❡sts✳ ❲✐t❤ ❛ s•✐❣❤t r❡❛rr❛♥❣❡♠❡♥t✱ ✇❡ ❝❛♥ ❞♦ ✇✐t❤ ♦♥❡✿ r := a + b❀ ✶ s := a -m❀ ✷ t := s + b❀ ✸ ✐❢ •❛st ❛❞❞✐t✐♦♥ s❡t ❝❛rr② ✢❛❣ t❤❡♥ ✹ r ✿❂ t❀ ✺ ❆•• ❛r✐t❤♠❡t✐❝ ✐♥ t❤✐s ❝♦❞❡ ✐s ❛ss✉♠❡❞ ♠♦❞✉•♦ t❤❡ ✇♦r❞ ❜❛s❡ β, ✐✳❡✳✱ t❤❡ ✐♥t❡❣❡rs ✐♥ r, s, ❛♥❞ t ❛r❡ r❡❞✉❝❡❞ r❡s✐❞✉❡s ♠♦❞✉•♦ β✳ ■♥ •✐♥❡ ✷✱ s✐♥❝❡ a ✐s r❡❞✉❝❡❞ ♠♦❞✉•♦ m✱ t❤❡ s✉❜tr❛❝t✐♦♥ a -m ♥❡❝❡ss❛r✐•② ♣r♦❞✉❝❡s ❛ ❜♦rr♦✇✱ s♦ t❤❛t s = a -m + β✳ ■♥ •✐♥❡ ✸✱ ✐❢ s + b < β✱ t❤❡♥ t❤✐s ❛❞❞✐t✐♦♥ ❞♦❡s ♥♦t ♣r♦❞✉❝❡ ❛ ❝❛rr②✱ ❛♥❞ t = a + b -m + β < β✱ ✐✳❡✳✱ a + b -m < 0✳ ■❢ s + b ≥ β✱ t❤❡ ❛❞❞✐t✐♦♥ ❞♦❡s ♣r♦❞✉❝❡ ❛ ❝❛rr②✱ ❛♥❞ 0 ≤ t = s + b -β = a + b -m. ❍❡♥❝❡ t ✐s t❤❡ ♣r♦♣❡r r❡s✉•t ✐❢ ❛♥❞ ♦♥•② ✐❢ ❛ ❝❛rr② ♦❝❝✉rs ✐♥ •✐♥❡ ✸✱ t♦ ♠❛❦❡ ✉♣ ❢♦r t❤❡ ❜♦rr♦✇ ♦❢ •✐♥❡ ✷✳ ▲✐♥❡s ✶ ❛♥❞ ✷ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ❝❛♥ ❜❡ ❡①❡❝✉t❡❞ ✐♥ ♣❛r❛••❡•✱ •❡❛❞✐♥❣ t♦ ❛ ❞❡♣❡♥❞❡♥t ❝❤❛✐♥ ♦❢ •❡♥❣t❤ ✸✳ ❲❡ r❡q✉✐r❡ a < m ❢♦r ❝♦rr❡❝t♥❡ss✱ ✐❢ b ≥ m, t❤❡ r❡s✉•t st✐•• s❛t✐s✜❡s r ≡ a + b (mod m) ❛♥❞ r < b, ❜✉t ♥♦t ♥❡❝❡ss❛r✐•② r < m. ❚❤❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ✐♥ ❈ ✇✐t❤ ❛ •❈❈ ①✽✻ ❛ss❡♠❜•② ❜•♦❝❦ s❤♦✇♥ ❜❡•♦✇✳ ❚❤❡ ✈❛•✉❡ ♦❢ s✱ s❤♦✇♥ s❡♣❛r❛t❡•② ❢♦r ❝•❛r✐t② ❛❜♦✈❡✱ ✐s st♦r❡❞ ✐♥ t ❤❡r❡✳ r ❂ ❛ ✰ ❜❀ t ❂ ❛ ✲ ♠❀ ❴❴❛s♠❴❴ ✭ ✧❛❞❞ ✪✷✱ ✪✶❭♥❭t✧ ✴✯ t ✿❂ t ✰ ❜ ✯✴ ✧❝♠♦✈❝ ✪✶✱ ✪✵❭♥❭t✧ ✴✯ ✐❢ ✭❝❛rr②✮ r ✿❂ t ✯✴ ✿ ✧✰r✧ ✭r✮✱ ✧✰✫r✧ ✭t✮ ✿ ✧❣✧ ✭❜✮ ✿ ✧❝❝✧ ✮❀ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛• t ❛♥❞ r ❛r❡ ❞♦♥❡ ✐♥ ❈✱ t♦ ❣✐✈❡ t❤❡ ❝♦♠♣✐•❡r s♦♠❡ s❝❤❡❞✉•✐♥❣ ❢r❡❡❞♦♠✳ ❙✐♥❝❡ ❈ ❞♦❡s ♥♦t ♣r♦✈✐❞❡ ❞✐r❡❝t ❛❝❝❡ss t♦ t❤❡ ❝❛rr② ✢❛❣✱ t❤❡ ❛❞❞✐t✐♦♥ t := t+b ❛♥❞ t❤❡ ❢♦••♦✇✐♥❣ ❝♦♥❞✐t✐♦♥❛• ❛ss✐❣♥♠❡♥t ❛r❡ ❞♦♥❡ ✐♥ ❛ss❡♠❜•②✳ ❚❤❡ ❝♦♥str❛✐♥ts ♦♥ t❤❡ ❞❛t❛ ♣❛ss❡❞ t♦ t❤❡ ❛ss❡♠❜•② ❜•♦❝❦ st❛t❡ t❤❛t t❤❡ ✈❛•✉❡s ♦❢ r ❛♥❞ t ♠✉st r❡s✐❞❡ ✐♥ r❡❣✐st❡rs ✭✧r✧✮ s✐♥❝❡ t❤❡ t❛r❣❡t ♦❢ t❤❡ ❝♦♥❞✐t✐♦♥❛• ♠♦✈❡ ✐♥str✉❝t✐♦♥ ❝♠♦✈❝ ♠✉st ❜❡ ❛ r❡❣✐st❡r✱ ❛♥❞ ❛t •❡❛st ♦♥❡ ♦❢ s♦✉r❝❡ ♦r t❛r❣❡t ♦❢ t❤❡ ❛❞❞✐t✐♦♥ ✐♥str✉❝t✐♦♥ ❛❞❞ ♠✉st ❜❡ ❛ r❡❣✐st❡r✳ ❲❡ ❛••♦✇ t❤❡ ✈❛r✐❛❜•❡ ❜ t♦ ❜❡ ♣❛ss❡❞ ✐♥ ❛ r❡❣✐st❡r✱ ✐♥ ♠❡♠♦r② ♦r ❛s ❛♥ ✐♠♠❡❞✐❛t❡ ♦♣❡r❛♥❞ ✭✧❣✧✱ ✏❣❡♥❡r❛•✑ ❝♦♥str❛✐♥t✱ ❢♦r ①✽✻❴✻✹ t❤❡ ❝♦rr❡❝t ❝♦♥str❛✐♥t ✐s ✧r♠❡✧ s✐♥❝❡ ✐♠♠❡❞✐❛t❡ ❝♦♥st❛♥ts ❛r❡ ♦♥•② ✸✷ ❜✐t ✇✐❞❡✮✱ ✇❤✐❝❤ ✐s t❤❡ s♦✉r❝❡ ♦♣❡r❛♥❞ t♦ t❤❡ ❛❞❞ ✐♥str✉❝t✐♦♥✳ ❚❤❡ ✧✰✧ ♠♦❞✐✜❡r t❡••s t❤❛t t❤❡ ✈❛•✉❡s ✐♥ r ❛♥❞ t ✇✐•• ❜❡ ♠♦❞✐✜❡❞✱ ❛♥❞ t❤❡ ✧✫✧ ♠♦❞✐✜❡r t❡••s t❤❛t t ♠❛② ❜❡ ♠♦❞✐✜❡❞ ❜❡❢♦r❡ t❤❡ ❡♥❞ ♦❢ t❤❡ ❛ss❡♠❜•② ❜•♦❝❦ ❛♥❞ t❤✉s ♥♦ ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✶✶ ♦t❤❡r ✐♥♣✉t ✈❛r✐❛❜•❡ s❤♦✉•❞ ❜❡ ♣❛ss❡❞ ✐♥ t❤❡ r❡❣✐st❡r ❛ss✐❣♥❡❞ t♦ t✱ ❡✈❡♥ ✐❢ t❤❡✐r ✈❛•✉❡s ❛r❡ ❦♥♦✇♥ t♦ ❜❡ ✐❞❡♥t✐❝❛•✳ ❋✐♥❛••②✱ ✧❝❝✧ t❡••s t❤❡ ❝♦♠♣✐•❡r t❤❛t t❤❡ ✈❛•✉❡s ♦❢ t❤❡ ✢❛❣s r❡❣✐st❡r ♠❛② ❝❤❛♥❣❡✳ ❚❤❡s❡ ❝♦♥str❛✐♥ts ♣r♦✈✐❞❡ t❤❡ ✐♥❢♦r♠❛t✐♦♥ t❤❡ ❝♦♠♣✐•❡r ♥❡❡❞s t♦ ❜❡ ❛❜•❡ t♦ ✉s❡ t❤❡ ❛ss❡♠❜•② ❜•♦❝❦ ❝♦rr❡❝t•②✱ ✇❤✐•❡ •❡❛✈✐♥❣ ❡♥♦✉❣❤ ✢❡①✐❜✐•✐t② t❤❛t ✐t ❝❛♥ ♦♣t✐♠✐③❡ r❡❣✐st❡r ❛••♦❝❛t✐♦♥ ❛♥❞ ❞❛t❛ ♠♦✈❡♠❡♥t✱ ❝♦♠♣❛r❡❞ t♦✱ ❡✳❣✳✱ ❝♦♠♣✐•❡rs t❤❛t r❡q✉✐r❡ ❛•• ♣❛r❛♠❡t❡rs t♦ ❛ss❡♠❜•② ❜•♦❝❦s ✐♥ ❛ ✜①❡❞ s❡t ♦❢ r❡❣✐st❡rs✳ ❆♥ ❛•t❡r♥❛t✐✈❡ s♦•✉t✐♦♥ ✐s t♦ ❝♦♠♣✉t❡ r := b -(m -a) ❛♥❞ ❛❞❞✐♥❣ m ✐❢ t❤❡ ♦✉t❡r s✉❜tr❛❝t✐♦♥ ♣r♦❞✉❝❡❞ ❛ ❜♦rr♦✇✳ ❍♦✇❡✈❡r✱ t❤✐s r❡q✉✐r❡s ❛ ❝♦♥❞✐t✐♦♥❛• ❛❞❞✐t✐♦♥ r❛t❤❡r t❤❛♥ ❛ ❝♦♥❞✐t✐♦♥❛• ♠♦✈❡✳ ❙✐♠✐•❛r t♦ t❤❡ ♠♦❞✉•❛r ❛❞❞✐t✐♦♥✱ ✈❛r✐♦✉s ❢✉♥❝t✐♦♥s s✉❝❤ ❛s ♠♦❞✉•❛r s✉❜✲ tr❛❝t✐♦♥ ❛♥❞ ♠✉•t✐♣•✐❝❛t✐♦♥ ❢♦r ♦♥❡ ❛♥❞ t✇♦✲✇♦r❞ ♠♦❞✉•✐✱ t✇♦✲✇♦r❞ ❛❞❞✐t✐♦♥✱ s✉❜tr❛❝t✐♦♥✱ ♠✉•t✐♣•✐❝❛t✐♦♥ ❛♥❞ ❜✐♥❛r② s❤✐❢t✱ ❛♥❞ ❞✐✈✐s✐♦♥ ✇✐t❤ ❛ t✇♦✲✇♦r❞ ❞✐✈✲ ✐❞❡♥❞ ✭✉s❡❞✱ ❢♦r ❡①❛♠♣•❡✱ ❢♦r ♣r❡♣❛r✐♥❣ ❛ r❡s✐❞✉❡ ❢♦r ✉s❡ ✇✐t❤ |❊❉❈ ♠♦❞✉•❛r r❡❞✉❝t✐♦♥ ✇✐t❤ ❛ t✇♦✲✇♦r❞ ♠♦❞✉•✉s✱ s❡❡ ✸✳✷✮ ❛r❡ ✇r✐tt❡♥ ❛s ❢✉♥❝t✐♦♥s ✇✐t❤ ❛s✲ s❡♠❜•② s✉♣♣♦rt✳ ❆s ♦♣t✐♠✐③❛t✐♦♥ ❡✛♦rt ♣r♦❣r❡ss❡s✱ ♠♦r❡ t✐♠❡✲❝r✐t✐❝❛• ❢✉♥❝t✐♦♥s ❝✉rr❡♥t•② ✇r✐tt❡♥ ✐♥ ❈ ✇✐t❤ ❛ss❡♠❜•② ♠❛❝r♦s ✇✐•• ❜❡ r❡♣•❛❝❡❞ ❜② ❞❡❞✐❝❛t❡❞ ❛s✲ s❡♠❜•② ❝♦❞❡✳ ✸✳✷ ▼♦❞✉•❛r r❡❞✉❝t✐♦♥ ✇✐t❤ |❊❉❈ ▼♦♥t❣♦♠❡r② ♣r❡s❡♥t❡❞ ✐♥ ❬✶✼( ❛ ♠❡t❤♦❞ ❢♦r ❢❛st ♠♦❞✉•❛r r❡❞✉❝t✐♦♥✳ •✐✈❡♥ ❛♥ ✐♥t❡❣❡r 0 ≤ a < βm✱ ❢♦r ♦❞❞ ♠♦❞✉•✉s m ♦❢ ♦♥❡ ♠❛❝❤✐♥❡ ✇♦r❞ ❛♥❞ ♠❛✲ ❝❤✐♥❡ ✇♦r❞ ❜❛s❡ β ✭❤❡r❡ ❛ss✉♠❡❞ ❛ ♣♦✇❡r ♦❢ 2✮✱ ❛♥❞ ❛ ♣r❡❝♦♠♣✉t❡❞ ❝♦♥st❛♥t m ✐♥✈ = -m -1 mod β✱ ✐t ❝♦♠♣✉t❡s ❛♥ ✐♥t❡❣❡r 0 ≤ r < m ✇❤✐❝❤ s❛t✐s✜❡s rβ ≡ a (mod m). ■t ❞♦❡s s♦ ❜② ❝♦♠♣✉t✐♥❣ t❤❡ ♠✐♥✐♠❛• ♥♦♥✲♥❡❣❛t✐✈❡ tm s✉❝❤ t❤❛t a + tm ≡ 0 (mod β), t♦ ♠❛❦❡ ✉s❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ❞✐✈✐s✐♦♥ ❜② β ✐s ✈❡r② ✐♥✲ ❡①♣❡♥s✐✈❡✳ ❙✐♥❝❡ t < β✱ (a + tm)/β < 2m, ❛♥❞ ❛t ♠♦st ♦♥❡ ✜♥❛• s✉❜tr❛❝t✐♦♥ ♦❢ m ❡♥s✉r❡s r < m. ❍❡ ❝❛••s t❤❡ ❛•❣♦r✐t❤♠ t❤❛t ❝❛rr✐❡s ♦✉t t❤✐s r❡❞✉❝t✐♦♥ ✏|❊❉❈✱✑ s❤♦✇♥ ✐♥ ❆•❣♦r✐t❤♠ ✷✳ ■♥♣✉t✿ m✱ t❤❡ ♠♦❞✉•✉s β✱ t❤❡ ✇♦r❞ ❜❛s❡ a < βm✱ ✐♥t❡❣❡r t♦ r❡❞✉❝❡ m ✐♥✈ < β s✉❝❤ t❤❛t mm ✐♥✈ ≡ -1 (mod β) ❖✉t♣✉t✿ r < m ✇✐t❤ rβ ≡ a (mod m) t := a • m ✐♥✈ mod β❀ r := (a + t • m)/β❀ ✐❢ r ≥ m t❤❡♥ r := r -m❀ ❆•❣♦r✐t❤♠ ✷✿ ❆•❣♦r✐t❤♠ |❊❉❈ ❢♦r ♠♦❞✉•❛r r❡❞✉❝t✐♦♥ ✇✐t❤ ♦♥❡✲✇♦r❞ ♠♦❞✲ ✉•✉s✳ ❆•• ✈❛r✐❛❜•❡s t❛❦❡ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡r ✈❛•✉❡s✳ ❚❤❡ r❡❞✉❝❡❞ r❡s✐❞✉❡ ♦✉t♣✉t ❜② t❤✐s ❛•❣♦r✐t❤♠ ✐s ♥♦t ✐♥ t❤❡ s❛♠❡ r❡s✐❞✉❡ ❝•❛ss mod m ❛s t❤❡ ✐♥♣✉t✱ ❜✉t t❤❡ r❡s✐❞✉❡ ❝•❛ss ❣❡ts ♠✉•t✐♣•✐❡❞ ❜② β -1 (mod m) ✐♥ t❤❡ ♣r♦❝❡ss✳ ❚♦ ♣r❡✈❡♥t ❛❝❝✉♠✉•❛t✐♥❣ ♣♦✇❡rs ♦❢ β -1 (mod m) ❛♥❞ ❤❛✈✐♥❣ ✉♥✲ ❡q✉❛• ♣♦✇❡rs ♦❢ β ✇❤❡♥✱ ❡✳❣✳✱ ❛❞❞✐♥❣ ♦r ❝♦♠♣❛r✐♥❣ r❡s✐❞✉❡s✱ ❛♥② r❡s✐❞✉❡ ♠♦❞✉•♦ m ✐s ❝♦♥✈❡rt❡❞ t♦ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ ✜rst✱ ❜② ♠✉•t✐♣•②✐♥❣ ✐t ❜② β ❛♥❞ r❡❞✉❝✐♥❣ ✭✇✐t❤♦✉t |❊❉❈✮ ♠♦❞✉•♦ m✱ ✐✳❡✳✱ t❤❡ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ || ♥➦ ✼✵✹✶ ✶✷ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ r❡s✐❞✉❡ a (mod m) ✐s aβ (mod m)✳ ❚❤✐s ✇❛②✱ ✐❢ t✇♦ r❡s✐❞✉❡s ✐♥ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ aβ (mod m) ❛♥❞ bβ (mod m) ❛r❡ ♠✉•t✐♣•✐❡❞ ❛♥❞ r❡❞✉❝❡❞ ✈✐❛ |❊❉❈✱ t❤❡♥ |❊❉❈(aβbβ) ≡ abβ (mod m) ✐s t❤❡ ♣r♦❞✉❝t ✐♥ ▼♦♥t❣♦♠❡r② r❡♣✲ r❡s❡♥t❛t✐♦♥✳ ❚❤✐s ❡♥s✉r❡s t❤❡ ❡①♣♦♥❡♥t ♦❢ β ✐♥ t❤❡ r❡s✐❞✉❡s ❛•✇❛②s st❛②s 1✱ ❛♥❞ s♦ ❛••♦✇s ❛❞❞✐t✐♦♥✱ s✉❜tr❛❝t✐♦♥✱ ❛♥❞ ❡q✉❛•✐t② t❡sts ♦❢ r❡s✐❞✉❡s ✐♥ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥✳ ❙✐♥❝❡ β ⊥ m✱ ✇❡ ❛•s♦ ❤❛✈❡ aβ ≡ 0 (mod m) ✐❢ ❛♥❞ ♦♥•② ✐❢ a ≡ 0 (mod m)✱ ❛♥❞ gcd(aβ, m) = gcd(a, m). ❙✐♥❝❡ β = 2 32 ♦r 2 64 ✐s ❛♥ ✐♥t❡❣❡r sq✉❛r❡✱ t❤❡ ❏❛❝♦❜✐ s②♠❜♦• s❛t✐s✜❡s aβ m = a m . ❋♦r ♠♦❞✉•✐ m ♦❢ ♠♦r❡ t❤❛♥ ♦♥❡ ♠❛❝❤✐♥❡ ✇♦r❞✱ s❛② m < β k ✱ ❛ ♣r♦❞✉❝t ♦❢ t✇♦ r❡❞✉❝❡❞ r❡s✐❞✉❡s ♠❛② ❡①❝❡❡❞ β✱ ❜✉t ✐s ❜❡•♦✇ mβ k ✳ ❚❤❡ r❡❞✉❝t✐♦♥ ❝❛♥ ❜❡ ❝❛rr✐❡❞ ♦✉t ✐♥ t✇♦ ✇❛②s✿ ♦♥❡ ❡ss❡♥t✐❛••② ♣❡r❢♦r♠s t❤❡ ♦♥❡✲✇♦r❞ |❊❉❈ r❡❞✉❝t✐♦♥ k t✐♠❡s✱ ♣❡r❢♦r♠✐♥❣ O k 2 ♦♥❡✲✇♦r❞ ♠✉•t✐♣•✐❡s✱ t❤❡ ♦t❤❡r r❡♣•❛❝❡s ❛r✐t❤♠❡t✐❝ ♠♦❞✉•♦ β ✐♥ |❊❉❈ ❜② ❛r✐t❤♠❡t✐❝ ♠♦❞✉•♦ β k ✱ ♣❡r❢♦r♠✐♥❣ O(1) k✲✇♦r❞ ♠✉•t✐✲ ♣•✐❝❛t✐♦♥s✳ ■♥ ❡✐t❤❡r ❝❛s❡✱ ❛ ❢✉•• r❡❞✉❝t✐♦♥ ✇✐t❤ ✭r❡♣❡❛t❡❞ ♦♥❡✲✇♦r❞ ♦r ❛ s✐♥❣•❡ ♠✉•t✐✲✇♦r❞✮ |❊❉❈ ❞✐✈✐❞❡s t❤❡ r❡s✐❞✉❡ ❝•❛ss ♦❢ t❤❡ ♦✉t♣✉t ❜② β k , ❛♥❞ t❤❡ ❝♦♥✲ ✈❡rs✐♦♥ t♦ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ ♠✉st ♠✉•t✐♣•② ❜② β k ❛❝❝♦r❞✐♥❣•②✳ ❚❤❡ ❢♦r♠❡r ♠❡t❤♦❞ ❤❛s •♦✇❡r ♦✈❡r❤❡❛❞ ❛♥❞ ✐s ♣r❡❢❡r❛❜•❡ ❢♦r s♠❛•• ♠♦❞✉•✐✱ t❤❡ •❛tt❡r ❝❛♥ ✉s❡ ❛s②♠♣t♦t✐❝❛••② ❢❛st ♠✉•t✐♣•✐❝❛t✐♦♥ ❛•❣♦r✐t❤♠s ✐❢ t❤❡ ♠♦❞✉•✉s ✐s •❛r❣❡✳ ❆s ✐♥ ♦✉r ❛♣♣•✐❝❛t✐♦♥ t❤❡ ♠♦❞✉•✐ ❛r❡ q✉✐t❡ s♠❛••✱ ♥♦ ♠♦r❡ t❤❛♥ t✇♦ ♠❛❝❤✐♥❡ ✇♦r❞s✱ ✇❡ ✉s❡ t❤❡ ❢♦r♠❡r ♠❡t❤♦❞✳ ❇❡❢♦r❡ ♠♦❞✉•❛r ❛r✐t❤♠❡t✐❝ ✇✐t❤ |❊❉❈ ❢♦r ❛ ♣❛rt✐❝✉•❛r m ❝❛♥ ❜❡❣✐♥✱ t❤❡ ❝♦♥st❛♥t m ✐♥✈ ♥❡❡❞s t♦ ❜❡ ❝♦♠♣✉t❡❞✳ ■❢ β ✐s ❛ ♣♦✇❡r ♦❢ 2✱ ❍❡♥s❡• •✐❢t✐♥❣ ♠❛❦❡s t❤✐s ❝♦♠♣✉t❛t✐♦♥ ✈❡r② ❢❛st✳ ❚♦ s♣❡❡❞ ✐t ✉♣ ❢✉rt❤❡r✱ ✇❡ tr② t♦ ❣✉❡ss ❛♥ ❛♣♣r♦①✐✲ ♠❛t✐♦♥ t♦ m ✐♥✈ s♦ t❤❛t ❛ ❢❡✇ •❡❛st s✐❣♥✐✜❝❛♥t ❜✐ts ❛r❡ ❝♦rr❡❝t✱ t❤✉s s❛✈✐♥❣ ❛ ❢❡✇ ◆❡✇t♦♥ ✐t❡r❛t✐♦♥s✳ ❚❤❡ sq✉❛r❡ ♦❢ ❛♥② ♦❞❞ ✐♥t❡❣❡r ✐s ❝♦♥❣r✉❡♥t t♦ 1 (mod 8)✱ s♦ m ✐♥✈ ≡ m (mod 8). ❚❤❡ ❢♦✉rt❤ ❜✐t ♦❢ m ✐♥✈ ✐s ❡q✉❛• t♦ t❤❡ ❜✐♥❛r② ❡①❝•✉s✐✈❡✲♦r ♦❢ t❤❡ s❡❝♦♥❞✱ t❤✐r❞✱ ❛♥❞ ❢♦✉rt❤ ❜✐t ♦❢ m✱ ❜✉t ♦♥ ♠❛♥② ♠✐❝r♦♣r♦❝❡ss♦rs ❛♥ ❛•✲ t❡r♥❛t✐✈❡ s✉❣❣❡st✐♦♥ ❢r♦♠ ▼♦♥t❣♦♠❡r② ❬✷✷( ✐s s•✐❣❤t•② ❢❛st❡r✿ (3m) ❳❖| 2 ❣✐✈❡s t❤❡ •♦✇ 5 ❜✐ts ♦❢ m ✐♥✈ ❝♦rr❡❝t•②✳ ❊❛❝❤ ◆❡✇t♦♥ ✐t❡r❛t✐♦♥ x → 2x -x 2 m ❞♦✉❜•❡s t❤❡ ♥✉♠❜❡r ♦❢ ❝♦rr❡❝t ❜✐ts✱ s♦ t❤❛t ✇✐t❤ ❡✐t❤❡r ❛♣♣r♦①✐♠❛t✐♦♥✱ 3 ✐t❡r❛t✐♦♥s ❢♦r β = 2 32 ♦r 4 ❢♦r β = 2 64 s✉✣❝❡✳ ❈♦♥✈❡rt✐♥❣ r❡s✐❞✉❡s ♦✉t ♦❢ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ ❝❛♥ ❜❡ ♣❡r❢♦r♠❡❞ q✉✐❝❦•② ✇✐t❤ |❊❉❈✱ ❜✉t ❝♦♥✈❡rt✐♥❣ t❤❡♠ t♦ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ r❡✲ q✉✐r❡s ❛♥♦t❤❡r ♠♦❞✉•❛r r❡❞✉❝t✐♦♥ ❛•❣♦r✐t❤♠✳ ■❢ s✉❝❤ ❝♦♥✈❡rs✐♦♥s ❛r❡ t♦ ❜❡ ❞♦♥❡ ❢r❡q✉❡♥t•②✱ ✐t ♣❛②s t♦ ♣r❡❝♦♠♣✉t❡ ℓ = β 2 mod m✱ s♦ t❤❛t |❊❉❈(aℓ) = aβ mod m ❛••♦✇s ✉s✐♥❣ |❊❉❈ ❢♦r t❤❡ ♣✉r♣♦s❡✳ ■♥ s♦♠❡ ❝❛s❡s✱ t❤❡ ✜♥❛• ❝♦♥❞✐t✐♦♥❛• s✉❜tr❛❝t✐♦♥ ♦❢ m ✐♥ |❊❉❈ ❝❛♥ ❜❡ ♦♠✐t✲ t❡❞✳ ■❢ a < m✱ t❤❡♥ a + tm < mβ s✐♥❝❡ t < β, s♦ r = (a + tm)/β < m ✇❤✐❝❤ ❝❛♥ ❜❡ ✉s❡❞ ✇❤❡♥ ❝♦♥✈❡rt✐♥❣ r❡s✐❞✉❡s ♦✉t ♦❢ ▼♦♥t❣♦♠❡r② ❢♦r♠✱ ♦r ✇❤❡♥ ❞✐✈✐s✐♦♥ ❜② ❛ ♣♦✇❡r ♦❢ 2 ♠♦❞✉•♦ m ✐s ❞❡s✐r❡❞✳ ✸✳✸ ▼♦❞✉•❛r ✐♥✈❡rs❡ ❚♦ ❝♦♠♣✉t❡ ❛ ♠♦❞✉•❛r ✐♥✈❡rs❡ r ≡ a -1 (mod m) ❢♦r ❛ ❣✐✈❡♥ r❡❞✉❝❡❞ r❡s✐❞✉❡ a ❛♥❞ ♦❞❞ ♠♦❞✉•✉s m ✇✐t❤ a ⊥ m✱ ✇❡ ✉s❡ ❛ ❜✐♥❛r② ❡①t❡♥❞❡❞ ❊✉❝•✐❞❡❛♥ ❛•❣♦r✐t❤♠✳ ▼♦❞✉•❛r ✐♥✈❡rs❡s ❛r❡ ✉s❡❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ st❛❣❡ ✷ ❢♦r t❤❡ P✕✶ ❛•❣♦r✐t❤♠✱ ❛♥❞ ❢♦r ✐♥✐t✐❛•✐s❛t✐♦♥ ♦❢ st❛❣❡ ✶ ♦❢ ❊❈▼ ✭❡①❝❡♣t ❢♦r ❛ s❡•❡❝t ❢❡✇ ❝✉r✈❡s ✇❤✐❝❤ ❤❛✈❡ s✐♠♣•❡ ❡♥♦✉❣❤ ♣❛r❛♠❡t❡rs t❤❛t t❤❡② ❝❛♥ ❜❡ ✐♥✐t✐❛•✐s❡❞ ✉s✐♥❣ ♦♥•② ❞✐✈✐s✐♦♥ ❜② s♠❛•• ❝♦♥st❛♥ts✮✳ ❖✉r ❝♦❞❡ ❢♦r ❛ ♠♦❞✉•❛r ✐♥✈❡rs❡ t❛❦❡s ❛❜♦✉t 0.5µs ❢♦r ♦♥❡✲ ✇♦r❞ ♠♦❞✉•✐✱ ✇❤✐❝❤ ✐♥ ❝❛s❡ ♦❢ P✕✶ ✇✐t❤ s♠❛•• B 1 ❛♥❞ B 2 ♣❛r❛♠❡t❡rs ❛❝❝♦✉♥ts ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✶✸ ❢♦r s❡✈❡r❛• ♣❡r❝❡♥t ♦❢ t❤❡ t♦t❛• r✉♥✲t✐♠❡✱ s❤♦✇✐♥❣ t❤❛t s♦♠❡ ♦♣t✐♠✐③❛t✐♦♥ ❡✛♦rt ✐s ✇❛rr❛♥t❡❞ ❢♦r t❤✐s ❢✉♥❝t✐♦♥✳ ❚❤❡ ❡①t❡♥❞❡❞ ❊✉❝•✐❞❡❛♥ ❛•❣♦r✐t❤♠ s♦•✈❡s ar + ms = gcd(a, m) ❢♦r ❣✐✈❡♥ a, m ❜② ✐♥✐t✐❛•✐s✐♥❣ e 0 = 0, f 0 = 1, g 0 = m ❛♥❞ e 1 = 1, f 1 = 0, g 1 = a, ❛♥❞ ❝♦♠♣✉t✐♥❣ s❡q✉❡♥❝❡s e i , f i ❛♥❞ g i t❤❛t ♠❛✐♥t❛✐♥ ae i + mf i = g i ✭✶✮ ✇❤❡r❡ gcd(a, m) | g i ❛♥❞ t❤❡ g i ❛r❡ str✐❝t•② ❞❡❝r❡❛s✐♥❣ ✉♥t✐• g i = 0✳ ❚❤❡ ♦r✐❣✐♥❛• ❊✉❝•✐❞❡❛♥ ❛•❣♦r✐t❤♠ ✉s❡s g i = g i-2 mod g i-1 , t❤❛t ✐s✱ ✐♥ ❡❛❝❤ st❡♣ ✇❡ ✇r✐t❡ g i = g i-2 -g i-1 ⌊ gi-2 gi-1 ⌋ ❛♥❞ •✐❦❡✇✐s❡ e i = e i-2 -e i-1 ⌊ gi-2 gi-1 ⌋ ❛♥❞ f i = f i-2 -f i-1 ⌊ gi-2 gi-1 ⌋, s♦ t❤❛t ❡q✉❛t✐♦♥ ✭✶✮ ❤♦•❞s ❢♦r ❡❛❝❤ i. ■❢ n ✐s t❤❡ s♠❛••❡st i s✉❝❤ t❤❛t g i = 0, t❤❡♥ g n-1 = gcd(a, m)✱ s = f n-1 , ❛♥❞ r = e n-1 . ❙✐♥❝❡ ✇❡ ♦♥•② ✇❛♥t t❤❡ ✈❛•✉❡ ♦❢ r = e n-1 , ✇❡ ❞♦♥✬t ♥❡❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ f i ✈❛•✉❡s✳ ❲❡ ❝❛♥ ✇r✐t❡ u = e i-1 , v = e i , x = g i-1 , y = g i ❛♥❞ ❢♦r i = 1 ✐♥✐t✐❛•✐s❡ u = 0, v = 1, x = m, ❛♥❞ y = a. ❚❤❡♥ ❡❛❝❤ ✐t❡r❛t✐♦♥ i → i + 1 ✐s ❝♦♠♣✉t❡❞ ❜② (u, v, x, y) := (v, u -⌊x/y⌋v, y, x -⌊x/y⌋y). ❆t t❤❡ ✜rst ✐t❡r❛t✐♦♥ ✇❤❡r❡ y = 0✱ ✇❡ ❤❛✈❡ r = u ❛♥❞ x = 1 ✐❢ a ❛♥❞ m ✇❡r❡ ✐♥❞❡❡❞ ❝♦♣r✐♠❡✳ ❆ ♣r♦❜•❡♠ ✇✐t❤ t❤✐s ❛•❣♦r✐t❤♠ ✐s t❤❡ ❝♦st•② ❝♦♠♣✉t❛t✐♦♥ ♦❢ ⌊x/y⌋ ❛s ✐♥t❡❣❡r ❞✐✈✐s✐♦♥ ✐s ✉s✉❛••② s•♦✇✳ ❚❤❡ ❜✐♥❛r② ❡①t❡♥❞❡❞ ❊✉❝•✐❞❡❛♥ ❛•❣♦r✐t❤♠ ❛✈♦✐❞s t❤✐s ♣r♦❜•❡♠ ❜② ✉s✐♥❣ ♦♥•② s✉❜tr❛❝t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ❜② ♣♦✇❡rs ♦❢ 2. ❖✉r ✐♠♣•❡♠❡♥✲ t❛t✐♦♥ ✐s ✐♥s♣✐r❡❞ ❜② ❝♦❞❡ ✇r✐tt❡♥ ❜② |♦❜❡rt ❍❛r•❡② ❢♦r t❤❡ ❊❈❈♣✲✾✼ ❝❤❛••❡♥❣❡ ❛♥❞ ✐s s❤♦✇♥ ✐♥ ❆•❣♦r✐t❤♠ ✸✳ ❚❤❡ ✉♣❞❛t❡s ♠❛✐♥t❛✐♥ ua ≡ -x2 t (mod m) ❛♥❞ va ≡ y2 t (mod m) s♦ t❤❛t ✇❤❡♥ y = 1, ✇❡ ❤❛✈❡ r = v2 -t = a -1 (mod m). ■♥♣✉t✿ ❖❞❞ ♠♦❞✉•✉s m |❡❞✉❝❡❞ r❡s✐❞✉❡ a (mod m) ❖✉t♣✉t✿ |❡❞✉❝❡❞ r❡s✐❞✉❡ r (mod m) ✇✐t❤ ar ≡ 1 (mod m)✱ ♦r ❢❛✐•✉r❡ ✐❢ gcd(a, m) > 1 ✐❢ a = 0 t❤❡♥ r❡t✉r♥ ❢❛✐•✉r❡❀ t := Val 2 (a)❀ ✴✯ 2 t || a ✯✴ u := 0; v := 1; x := m; y := a/2 t ❀ ✇❤✐•❡ x = y ❞♦ ℓ := Val 2 (x -y)❀ ✴✯ 2 ℓ || x -y ✯✴ ✐❢ x < y t❤❡♥ (u, v, x, y, t) := (u2 ℓ , u + v, x, (y -x)/2 ℓ , t + ℓ)❀ ❡•s❡ (u, v, x, y, t) := (u + v, v2 ℓ , (x -y)/2 ℓ , y, t + ℓ)❀ ✐❢ y = 1 t❤❡♥ r❡t✉r♥ ❢❛✐•✉r❡❀ r := v2 -t mod m❀ ❆•❣♦r✐t❤♠ ✸✿ ❇✐♥❛r② ❡①t❡♥❞❡❞ •❈❉ ❛•❣♦r✐t❤♠✳ ■♥ ❡❛❝❤ st❡♣ ✇❡ s✉❜tr❛❝t t❤❡ s♠❛••❡r ♦❢ x, y ❢r♦♠ t❤❡ •❛r❣❡r✱ s♦ t❤❡② ❛r❡ ❞❡❝r❡❛s✐♥❣ ❛♥❞ ♥♦♥✲♥❡❣❛t✐✈❡✳ ◆❡✐t❤❡r ❝❛♥ ❜❡❝♦♠❡ ③❡r♦ ❛s t❤❛t ✐♠♣•✐❡s x = y || ♥➦ ✼✵✹✶ ✶✹ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ ✐♥ t❤❡ ♣r❡✈✐♦✉s ✐t❡r❛t✐♦♥✱ ✇❤✐❝❤ t❡r♠✐♥❛t❡s t❤❡ •♦♦♣✳ ❙✐♥❝❡ ❜♦t❤ ❛r❡ ♦❞❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❡❛❝❤ ✐t❡r❛t✐♦♥✱ t❤❡✐r ❞✐✛❡r❡♥❝❡ ✐s ❡✈❡♥✱ s♦ ♦♥❡ ✈❛•✉❡ ❞❡❝r❡❛s❡s ❜② ❛t •❡❛st ❛ ❢❛❝t♦r ♦❢ 2, ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s ✐s ❛t ♠♦st log 2 (am). ■♥ ❡❛❝❤ ✐t❡r❛t✐♦♥✱ uy + vx = m, ❛♥❞ s✐♥❝❡ x ❛♥❞ y ❛r❡ ♣♦s✐t✐✈❡✱ u, v ≤ m s♦ t❤❛t ♥♦ ♦✈❡r✢♦✇ ♦❝❝✉rs ✇✐t❤ ✜①❡❞✲♣r❡❝✐s✐♦♥ ❛r✐t❤♠❡t✐❝✳ ❚♦ ♣❡r❢♦r♠ t❤❡ ♠♦❞✉•❛r ❞✐✈✐s✐♦♥ r = v/2 ti , ✇❡ ❝❛♥ ✉s❡ |❊❉❈✳ ❲❤✐•❡ t ≥ log 2 (β), ✇❡ r❡♣•❛❝❡ v := |❊❉❈(v) ❛♥❞ t := t -log 2 (β). ❚❤❡♥✱ ✐❢ t > 0✱ ✇❡ ♣❡r❢♦r♠ ❛ ✈❛r✐❛❜•❡✲✇✐❞t❤ |❊❉❈ t♦ ❞✐✈✐❞❡ ❜② 2 t r❛t❤❡r t❤❛♥ ❜② β ❜② ❝♦♠♣✉t✐♥❣ r = (v + ((vm ✐♥✈ ) mod 2 t ) m) /2 t ✇✐t❤ mm ✐♥✈ ≡ -1 (mod β). ❙✐♥❝❡ v < m, ✇❡ ❞♦♥✬t ♥❡❡❞ ❛ ✜♥❛• s✉❜tr❛❝t✐♦♥ ✐♥ t❤❡s❡ |❊❉❈✳ ■❢ t❤❡ r❡s✐❞✉❡ a ✇❤♦s❡ ✐♥✈❡rs❡ ✇❡ ✇❛♥t ✐s ❣✐✈❡♥ ✐♥ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ aβ k mod m ✇✐t❤ k✲✇♦r❞ ♠♦❞✉•✉s m✱ ✇❡ ❝❛♥ ✉s❡ |❊❉❈ 2k t✐♠❡s t♦ ❝♦♠♣✉t❡ aβ -k mod m, t❤❡♥ ❝♦♠♣✉t❡ t❤❡ ♠♦❞✉•❛r ✐♥✈❡rs❡ t♦ ♦❜t❛✐♥ t❤❡ ✐♥✈❡rs❡ ♦❢ a ✐♥ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥✿ a -1 β k ≡ aβ -k -1 (mod m). ❚❤✐s ❝❛♥ ❜❡ s✐♠♣•✐✲ ✜❡❞ ❜② ✉s✐♥❣ t❤❡ ❢❛❝t t❤❛t t❤❡ ❜✐♥❛r② ❡①t❡♥❞❡❞ •❈❉ ❝♦♠♣✉t❡s v = a -1 2 t . ■❢ ✇❡ ❦♥♦✇ ❜❡❢♦r❡❤❛♥❞ t❤❛t t ≥ log 2 β, ✇❡ ❝❛♥ s❦✐♣ ❞✐✈✐s✐♦♥s ❜② β ✈✐❛ |❊❉❈ ❜♦t❤ ❜❡❢♦r❡ ❛♥❞ ❛❢t❡r t❤❡ ❜✐♥❛r② ❡①t❡♥❞❡❞ •❈❉✳ ▲❡t t❤❡ ❢✉♥❝t✐♦♥ t(x, y) ❣✐✈❡ t❤❡ ✈❛•✉❡ ♦❢ t ❛t t❤❡ ❡♥❞ ♦❢ ❆•❣♦r✐t❤♠ ✸ ❢♦r ❝♦♣r✐♠❡ ✐♥♣✉ts x, y✳ ■t s❛t✐s✜❡s t(x, y) =          0 ✐❢ x = y (✐♠♣•✐❡s x = y = 1), t(x/2, y) + 1 ✐❢ x = y, 2 | x, t(x -y, y) ✐❢ x > y, 2 ∤ x, t(y, x) ✐❢ x < y, 2 ∤ x. ❆ss✉♠✐♥❣ y ♦❞❞✱ ❝❛s❡ ✸ ✐s ❛•✇❛②s ❢♦••♦✇❡❞ ❜② ❝❛s❡ ✷✱ ❛♥❞ ✇❡ ❝❛♥ s✉❜st✐t✉t❡ ❝❛s❡ ✸ ❜② t(x, y) = t((x -y)/2, y) + 1. ❲❡ ❝♦♠♣❛r❡ t❤❡ ❞❡❝r❡❛s❡ ♦❢ t❤❡ s✉♠ x + y ❛♥❞ t❤❡ ✐♥❝r❡❛s❡ ♦❢ t✳ ■♥ ❝❛s❡ ✷✱ (x + y) → x/2 + y > (x + y)/2, ❛♥❞ t ✐♥❝r❡❛s❡s ❜② 1. ■♥ t❤❡ s✉❜st✐t✉t❡❞ ❝❛s❡ ✸✱ (x + y) → (x + y)/2, ❛♥❞ t ✐♥❝r❡❛s❡s ❜② 1. ❲❡ s❡❡ t❤❛t ✇❤❡♥❡✈❡r x + y ❞❡❝r❡❛s❡s✱ t ✐♥❝r❡❛s❡s✱ ❛♥❞ ✇❤❡♥❡✈❡r t ✐♥❝r❡❛s❡s ❜② 1✱ x + y ❞r♦♣s ❜② ❛t ♠♦st ❤❛•❢✱ ✉♥t✐• x + y = 2. ❍❡♥❝❡ t(x, y) ≥ log 2 (x + y) -1, ❛♥❞ t❤❡r❡❢♦r❡ t(x, y) ≥ log 2 (y), s✐♥❝❡ x > 0. ❚❤✉s ✐♥ ❝❛s❡ ♦❢ k✲✇♦r❞ ♠♦❞✉•✐ β k-1 < m < β k ✱ ✇❡ ❤❛✈❡ t(x, m) ≥ (k - 1) log 2 (β) ❢♦r ❛♥② ♣♦s✐t✐✈❡ x✱ s♦ ✉s✐♥❣ aβ -1 (mod m) ❛s ✐♥♣✉t t♦ t❤❡ ❜✐♥❛r② ❡①t❡♥❞❡❞ •❈❉ ✐s s✉✣❝✐❡♥t t♦ ❡♥s✉r❡ t❤❛t ❛t t❤❡ ❡♥❞ ✇❡ ❣❡t a -1 β ≡ v2 -t (mod m)✱ ♦r a -1 β k ≡ v2 -t+(k-1) log 2 (β) (mod m) ❛♥❞ t❤❡ ❞❡s✐r❡❞ r❡s✉•t a -1 β k ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ v2 -t ✇✐t❤ ❛ ❞✐✈✐s✐♦♥ ❜② 2 t-(k-1) log 2 (β) ✈✐❛ |❊❉❈✳ ✸✳✹ ▼♦❞✉•❛r ❞✐✈✐s✐♦♥ ❜② s♠❛•• ✐♥t❡❣❡rs ■♥✐t✐❛•✐s❛t✐♦♥ ♦❢ P✰✶ ❛♥❞ ❊❈▼ ✐♥✈♦•✈❡s ❞✐✈✐s✐♦♥ ♦❢ r❡s✐❞✉❡s ❜② s♠❛•• ✐♥t❡❣❡rs s✉❝❤ ❛s 3, 5, 7, 11, 13 ♦r 37✳ ❚❤❡s❡ ❝❛♥ ❜❡ ❝❛rr✐❡❞ ♦✉t q✉✐❝❦•② ❜② ✉s❡ ♦❢ ❞❡❞✐✲ ❝❛t❡❞ ❢✉♥❝t✐♦♥s✳ ❚♦ ❝♦♠♣✉t❡ r ≡ ad -1 (mod m) ❢♦r ❛ r❡❞✉❝❡❞ r❡s✐❞✉❡ a ✇✐t❤ d ⊥ m✱ ✇❡ ✜rst ❝♦♠♣✉t❡ t = a + km✱ ✇✐t❤ k s✉❝❤ t❤❛t t ≡ 0 (mod d), ✐✳❡✳✱ k = a -m -1 mod d, ✇❤❡r❡ -m -1 mod d ✐s ❞❡t❡r♠✐♥❡❞ ❜② •♦♦❦✲✉♣ ✐♥ ❛ ♣r❡✲ ❝♦♠♣✉t❡❞ t❛❜•❡ ❢♦r t❤❡ d -1 ♣♦ss✐❜•❡ ✈❛•✉❡s ♦❢ m mod d. ❋♦r ♦♥❡✲✇♦r❞ ♠♦❞✉•✐✱ t❤❡ r❡s✉•t✐♥❣ ✐♥t❡❣❡r t ❝❛♥ ❜❡ ❞✐✈✐❞❡❞ ❜② d ✈✐❛ ♠✉•t✐✲ ♣•✐❝❛t✐♦♥ ❜② t❤❡ ♣r❡❝♦♠♣✉t❡❞ ❝♦♥st❛♥t d ✐♥✈ ≡ d -1 (mod β)✳ ❙✐♥❝❡ t/d < m < β ✐s ❛♥ ✐♥t❡❣❡r✱ t❤❡ r❡s✉•t r = td ✐♥✈ mod β ♣r♦❞✉❝❡s t❤❡ ❝♦rr❡❝t r❡❞✉❝❡❞ r❡s✐❞✉❡ r. ❚❤✐s ✐♠♣•✐❡s t❤❛t ❝♦♠♣✉t✐♥❣ t ♠♦❞✉•♦ β ✐s s✉✣❝✐❡♥t✳ ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✶✺ ❋♦r t✇♦✲✇♦r❞ ♠♦❞✉•✐✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ ❛♥ ❛•❣♦r✐t❤♠ ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡r m ❛♥❞ d ❛r❡ •❛r❣❡ ❡♥♦✉❣❤ t❤❛t t ♠❛② ♦✈❡r✢♦✇ t✇♦ ♠❛❝❤✐♥❡ ✇♦r❞s ♦r ♥♦t✳ ■♥ ❡✐t❤❡r ❝❛s❡✱ ✇❡ ♠❛② ✇r✐t❡ t = t 1 β + t 0 ✇✐t❤ 0 ≤ t 0 < β✱ 0 ≤ t 1 < dβ ❛♥❞ r = r 1 β + r 0 ✇✐t❤ 0 ≤ r 0 , r 1 < β, ❛♥❞ ❝❛♥ ❝♦♠♣✉t❡ r 0 = t 0 d ✐♥✈ mod β✳ ■❢ t ❞♦❡s ♥♦t ♦✈❡r✢♦✇✱ ✇❡ ♠❛② ✇r✐t❡ t = t ′′ + t ′ dβ, 0 ≤ t ′′ < dβ✱ ✇❤❡r❡ d | t ′′ . ❚❤❡♥ r = t/d = t ′ β + t ′′ /d ✇✐t❤ t ′′ /d < β, s♦ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ r 1 = ⌊t 1 /d⌋. ❚❤❡ tr✉♥❝❛t✐♥❣ ❞✐✈✐s✐♦♥ ❜② t❤❡ ✐♥✈❛r✐❛♥t d ❝❛♥ ❜❡ ✐♠♣•❡♠❡♥t❡❞ ❜② t❤❡ ♠❡t❤♦❞s ♦❢ ❬✶✸(✳ ❆♥ ❛❞✈❛♥t❛❣❡ ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ✐s t❤❛t t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ •♦✇ ✇♦r❞ r 0 ❢r♦♠ t 0 ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❤✐❣❤ ✇♦r❞ r 1 ❢r♦♠ t 1 ✳ ■❢ t ♠❛② ♦✈❡r✢♦✇ t✇♦ ♠❛❝❤✐♥❡ ✇♦r❞s✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ r 0 ❛s ❜❡❢♦r❡✱ ❛♥❞ ✉s❡ t❤❛t t -dr 0 ✐s ❞✐✈✐s✐❜•❡ ❜② dβ✱ s♦ ✇❡ ♠❛② ✇r✐t❡ r 1 β + r 0 ≡ t/d (mod β 2 ) ❛s r 1 ≡ (t -dr 0 )/β • d ✐♥✈ (mod β). ✹ P✕✶ ❛•❣♦r✐t❤♠ ❚❤❡ P✕✶ ❛•❣♦r✐t❤♠ ✐s ❞❡s❝r✐❜❡❞✱ ❢♦r ❡①❛♠♣•❡✱ ✐♥ ❬✷✸(✳ ❲❡ r❡❝❛♣✐t✉•❛t❡ s♦♠❡ ❡•❡♠❡♥t❛r② ❢❛❝ts ❤❡r❡✳ ❚❤❡ ✜rst st❛❣❡ ♦❢ P✕✶ ❝♦♠♣✉t❡s x 1 = x e 0 mod N ❢♦r s♦♠❡ st❛rt✐♥❣ ✈❛•✉❡ x 0 ≡ 0, ±1 (mod N ) ❛♥❞ ❛ ❤✐❣❤•② ❝♦♠♣♦s✐t❡ ✐♥t❡❣❡r ❡①♣♦♥❡♥t e✳ ❇② ❋❡r♠❛t✬s •✐tt•❡ t❤❡♦r❡♠✱ ✐❢ p -1 | e ❢♦r ❛♥② p | N ✱ t❤❡♥ x 1 ≡ 1 (mod p) ❛♥❞ p | gcd(x 1 -1, N ). ❚❤✐s ❝♦♥❞✐t✐♦♥ ✐s s✉✣❝✐❡♥t ❜✉t ♥♦t ♥❡❝❡ss❛r②✿ ✐t ✐s ❡♥♦✉❣❤ ✭❛♥❞ ♥❡❝❡ss❛r②✮ t❤❛t ord p (x 0 ) | e✱ ✇❤❡r❡ ord p (x 0 ) ✐s t❤❡ ♦r❞❡r ♦❢ x 0 ✐♥ F * p ✳ ❚♦ ♠❛①✐♠✐s❡ t❤❡ ♣r♦❜❛❜✐•✐t② t❤❛t ord p (x 0 ) | e ❢♦r ❛ ❣✐✈❡♥ s✐③❡ ♦❢ e✱ ✇❡ ❝♦✉•❞ ❝❤♦♦s❡ e t♦ ❝♦♥t❛✐♥ ❛•• ♣r✐♠❡s ❛♥❞ ♣r✐♠❡ ♣♦✇❡rs t❤❛t ❞✐✈✐❞❡ ord p (x 0 ) ✇✐t❤ ♣r♦❜❛❜✐•✐t② ❜❡tt❡r t❤❛♥ s♦♠❡ ❜♦✉♥❞ 1/B 1 ✳ ❖♥❡ t②♣✐❝❛••② ❛ss✉♠❡s t❤❛t ❛ ♣r✐♠❡ ♣♦✇❡r q k ❞✐✈✐❞❡s ord p (x 0 ) ✇✐t❤ ♣r♦❜❛❜✐•✐t② q -k ✱ s♦ t❤❛t e ✐s t❛❦❡♥ ❛s t❤❡ ♣r♦❞✉❝t ♦❢ ❛•• ♣r✐♠❡s ❛♥❞ ♣r✐♠❡ ♣♦✇❡rs ♥♦t ❡①❝❡❡❞✐♥❣ B 1 , ♦r e = lcm(1, 2, 3, 4, . . . , B 1 ). ❚❤❡ ✈❛•✉❡ ♦❢ e ✐s ♣r❡❝♦♠♣✉t❡❞ ❛♥❞ ♣❛ss❡❞ t♦ t❤❡ P✕✶ st❛❣❡ ✶ r♦✉t✐♥❡✱ ✇❤✐❝❤ ❜❛s✐❝❛••② ❝♦♥s✐sts ♦♥•② ♦❢ ❛ ♠♦❞✉•❛r ❡①♣♦♥❡♥t✐❛t✐♦♥✱ ❛ s✉❜tr❛❝t✐♦♥ ❛♥❞ ❛ ❣❝❞✳ ❚❤❡ ❜❛s❡ x 0 ❢♦r t❤❡ ❡①♣♦♥❡♥t✐❛t✐♦♥ ✐s ❝❤♦s❡♥ ❛s 2❀ ✐♥ ❛ •❡❢t✲t♦✲r✐❣❤t ❜✐♥❛r② ♣♦✇❡r✐♥❣ •❛❞❞❡r✱ t❤✐s r❡q✉✐r❡s ♦♥•② sq✉❛r✐♥❣s ❛♥❞ ❞♦✉❜•✐♥❣s✱ ✇❤❡r❡ t❤❡ •❛tt❡r ❝❛♥ ❜❡ ♣❡r❢♦r♠❡❞ q✉✐❝❦•② ✇✐t❤ ❛♥ ❛❞❞✐t✐♦♥ ✐♥st❡❛❞ ♦❢ ❛ ♠✉•t✐♣•✐❝❛t✐♦♥ ❜② x 0 ✳ ❚♦ r❡❞✉❝❡ t❤❡ ♣r♦❜❛❜✐•✐t② t❤❛t ❛•• ♣r✐♠❡ ❢❛❝t♦rs ♦❢ N ✭✐✳❡✳✱ N ✐ts❡•❢✮ ❛r❡ ❢♦✉♥❞ s✐♠✉•t❛♥❡♦✉s•② ❛♥❞ r❡♣♦rt❡❞ ❛s ❛ ❞✐✈✐s♦r ❛t t❤❡ ❡♥❞ ♦❢ st❛❣❡ ✶✱ ♦♥•② t❤❡ ♦❞❞ ♣❛rt ♦❢ e ✐s ♣r♦❝❡ss❡❞ ❛t ✜rst✱ ❛♥❞ t❤❡♥ t❤❡ ❢❛❝t♦rs ♦❢ 2 ✐♥ e ♦♥❡ ❛t ❛ t✐♠❡ ❜② s✉❝❝❡ss✐✈❡ sq✉❛r✐♥❣s✳ ❆❢t❡r ❡❛❝❤ ♦♥❡ ✇❡ ❝❤❡❝❦ ✐❢ t❤❡ ♥❡✇ r❡s✐❞✉❡ ✐s 1 (mod N )✱ ✐♥❞✐❝❛t✐♥❣ t❤❛t ❛•• ♣r✐♠❡ ❢❛❝t♦rs ♦❢ N ❤❛✈❡ ❜❡❡♥ ❢♦✉♥❞ ♥♦✇✱ ❛♥❞ ✐❢ s♦✱ r❡✈❡rt t♦ t❤❡ ♣r❡✈✐♦✉s ✈❛•✉❡✱ t♦ ✉s❡ ✐t ❢♦r t❤❡ ❣❝❞✳ ■❢ ♥♦t t❤❡ s❛♠❡ ♣♦✇❡r ♦❢ 2 ❞✐✈✐❞❡s ord p (x 0 ) ❡①❛❝t•② ❢♦r ❛•• ♣r✐♠❡s p | N ✱ t❤❡♥ t❤✐s ✇✐•• ❞✐s❝♦✈❡r ❛ ♣r♦♣❡r ❢❛❝t♦r✳ ❚❤✐s ❜❛❝❦tr❛❝❦✐♥❣ s❝❤❡♠❡ ✐s s✐♠♣•❡ ❜✉t s❛t✐s❢❛❝t♦r✐•② ❡✛❡❝t✐✈❡✿ ❆♠♦♥❣ 10 6 ❝♦♠♣♦s✐t❡ ♥✉♠❜❡rs t❤❛t ♦❝❝✉rr❡❞ ❞✉r✐♥❣ ❛♥ s✐❡✈✐♥❣ ❡①♣❡r✐♠❡♥t ♦❢ t❤❡ |❙❆✶✺✺ ♥✉♠❜❡r✱ ❡❛❝❤ ❝♦♠♣♦s✐t❡ ❜❡✐♥❣ ♦❢ ✉♣ t♦ 86 ❜✐ts ❛♥❞ ✇✐t❤ ♣r✐♠❡ ❢❛❝t♦rs •❛r❣❡r t❤❛♥ 2 24 , ♦♥•② 48 ❤❛❞ t❤❡ ✐♥♣✉t ♥✉♠❜❡r r❡♣♦rt❡❞ ❛s t❤❡ ❢❛❝t♦r ✐♥ P✕✶ st❛❣❡ ✶ ✇✐t❤ B 1 = 500. ❲✐t❤♦✉t t❤❡ ❜❛❝❦tr❛❝❦✐♥❣ s❝❤❡♠❡ ✭✐✳❡✳✱ ♣r♦❝❡ss✐♥❣ t❤❡ ❢✉•• ❡①♣♦♥❡♥t✐❛t✐♦♥ ❜② e✱ t❤❡♥ t❛❦✐♥❣ ❛ •❈❉✮✱ 879 ✐♥♣✉t ♥✉♠❜❡rs ❛r❡ r❡♣♦rt❡❞ ❛s ❢❛❝t♦rs ✐♥st❡❛❞✳ ❚❤❡ s❡❝♦♥❞ st❛❣❡ ♦❢ P✕✶ ❝❛♥ ✉s❡ ❡①❛❝t•② t❤❡ s❛♠❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ❛s t❤❡ s❡❝♦♥❞ st❛❣❡ ♦❢ P✰✶✱ ❜② ♣❛ss✐♥❣ X 1 = x 1 + x -1 1 t♦ t❤❡ st❛❣❡ ✷ ❛•❣♦r✐t❤♠✳ || ♥➦ ✼✵✹✶ ✶✻ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ ❚❤❡ st❛❣❡ ✷ ❛•❣♦r✐t❤♠ ❢♦r ❊❈▼ ✐s ✈❡r② s✐♠✐•❛r ❛s ✇❡••✱ ❛♥❞ t❤❡② ❛r❡ ❞❡s❝r✐❜❡❞ t♦❣❡t❤❡r ✐♥ ❙❡❝t✐♦♥ ✼✳ ✹✳✶ P✕✶ st❛❣❡ ✶ ♣❡r❢♦r♠❛♥❝❡ ❚❛❜•❡ ✶ ❝♦♠♣❛r❡s t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ P✕✶ st❛❣❡ ✶ ✐♠♣•❡♠❡♥t❛t✐♦♥ ❢♦r ❞✐✛❡r✲ ❡♥t B 1 ✈❛•✉❡s ❛♥❞ ♠♦❞✉•✉s s✐③❡s ♦♥ ❆▼❉ P❤❡♥♦♠ ❛♥❞ ■♥t❡• ❈♦r❡ ✷ ❈P❯s✳ ❈♦r❡ ✷ P❤❡♥♦♠ B 1 1 ✇♦r❞ 2 ✇♦r❞s -2 ❜✐ts 1 ✇♦r❞ 2 ✇♦
❚❛❜•❡ ✶✿ ❚✐♠❡ ✐♥ ♠✐❝r♦s❡❝♦♥❞s ❢♦r P✕✶ st❛❣❡ ✶ ✇✐t❤ ❞✐✛❡r❡♥t B 1 ✈❛•✉❡s ♦♥ 2.146 •❍③ ■♥t❡• ❈♦r❡ ✷ ❛♥❞ 2 •❍③ ❆▼❉ P❤❡♥♦♠ ❈P❯s✳ ✺ P✰✶ ❛•❣♦r✐t❤♠ ❚❤❡ P✰✶ ❛•❣♦r✐t❤♠ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❞❡t❛✐• ✐♥ ❬✷✸(✳ ❲❡ r❡❝❛♣✐t✉•❛t❡ t❤❡ ❜❛s✐❝ ❛•❣♦r✐t❤♠ ❤❡r❡✳ ❚❤❡ ✜rst st❛❣❡ ♦❢ P✰✶ ❝♦♠♣✉t❡s x 1 = V e (x 0 ) mod N, ✇❤❡r❡ x 0 ∈ Z/N Z ✐s ❛ ♣❛r❛♠❡t❡r✱ V n (x) ✐s ❛ ❞❡❣r❡❡✲n ❈❤❡❜②s❤❡✈ ♣♦•②♥♦♠✐❛• ❞❡✜♥❡❞ ❜② V n (x + x -1 ) = x n + x -n , ❛♥❞ e ✐s ❛ ❤✐❣❤•② ❝♦♠♣♦s✐t❡ ✐♥t❡❣❡r ❝❤♦s❡♥ ❛s ❢♦r t❤❡ P✕✶ ♠❡t❤♦❞✳ ❚❤❡s❡ ❈❤❡❜②s❤❡✈ ♣♦•②♥♦♠✐❛•s s❛t✐s❢② V 0 (x) = 2, V 1 (x) = x, V -n (x) = V n (x), V mn (x) = V m (V n (x)), ❛♥❞ V m+n (x) = V m (x)V n (x) -V m-n (x). ❲❡ t❡st ❢♦r ❛ ❢❛❝t♦r ❜② t❛❦✐♥❣ gcd(x 1 -2, N ). ■❢ t❤❡r❡ ✐s ❛ ♣r✐♠❡ p s✉❝❤ t❤❛t p | N ❛♥❞ p -∆ p | e✱ ✇❤❡r❡ ∆ = x 2 0 -4 ❛♥❞ ∆ p ✐s t❤❡ ▲❡❣❡♥❞r❡ s②♠❜♦•✱ t❤❡♥ p | gcd(x 1 -2, N ). ❙✐♥❝❡ V n-m ✐s r❡q✉✐r❡❞ ❢♦r ❝♦♠♣✉t✐♥❣ V n+m , t❤❡s❡ ♣♦•②♥♦♠✐❛•s ❝❛♥♥♦t ❜❡ ❡✈❛•✉❛t❡❞ ✇✐t❤ ❛ s✐♠♣•❡ ❜✐♥❛r② ❛❞❞✐t✐♦♥ ❝❤❛✐♥ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛✲ t✐♦♥ ✐♥ st❛❣❡ ✶ ♦❢ P✕✶✳ ■♥st❡❛❞✱ ❛♥ ❛❞❞✐t✐♦♥ ❝❤❛✐♥ ♥❡❡❞s t♦ ❜❡ ✉s❡❞ t❤❛t ❝♦♥t❛✐♥s n -m ✇❤❡♥❡✈❡r t❤❡ s✉♠ n + m ✐s ❢♦r♠❡❞ ❢r♦♠ n ❛♥❞ m. ❚❤❡s❡ ❝❤❛✐♥s ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✺✳✶✳ ❚❤❡ r❡q✉✐r❡❞ ❛❞❞✐t✐♦♥ ❝❤❛✐♥ ❢♦r t❤❡ st❛❣❡ ✶ ♠✉•t✐♣•✐❡r e ✐s ♣r❡❝♦♠♣✉t❡❞ ❛♥❞ st♦r❡❞ ❛s ❝♦♠♣r❡ss❡❞ ❜②t❡ ❝♦❞❡✱ s❡❡ ❙❡❝t✐♦♥ ✺✳✷✳ ❆s ❢♦r P✕✶✱ ❛ ❜❛❝❦tr❛❝❦✐♥❣ s❝❤❡♠❡ ✐s ✉s❡❞ t♦ ❛✈♦✐❞ ✜♥❞✐♥❣ ❛•• ❢❛❝t♦rs ♦❢ N ❛♥❞ t❤✉s r❡♣♦rt✐♥❣ t❤❡ ✐♥♣✉t ♥✉♠❜❡r ❛s t❤❡ ❢❛❝t♦r ❢♦✉♥❞✳ ❙✐♥❝❡ ❢❛❝t♦rs ♦❢ 2 ✐♥ e ❝❛♥ ❡❛s✐•② ❜❡ ❤❛♥❞•❡❞ ❜② V 2n (x) = V 2 (V n (x)) = V n (x) 2 -2, t❤❡② ♥❡❡❞ ♥♦t ❜❡ st♦r❡❞ ✐♥ t❤❡ ♣r❡✲❝♦♠♣✉t❡❞ ❛❞❞✐t✐♦♥ ❝❤❛✐♥✱ ❛♥❞ ❝❛♥ ❜❡ ♣r♦❝❡ss❡❞ ♦♥❡ ❛t ❛ t✐♠❡✳ ❙✐♠✐•❛r•② ❛s ✐♥ st❛❣❡ ✶ ♦❢ P✕✶✱ ✇❡ r❡♠❡♠❜❡r t❤❡ ♣r❡✈✐♦✉s r❡s✐❞✉❡✱ ♣r♦❝❡ss ♦♥❡ ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✶✼ ❢❛❝t♦r ♦❢ 2 ♦❢ e, ❛♥❞ ✐❢ t❤❡ r❡s✉•t ✐s 2 (mod N )✱ ♠❡❛♥✐♥❣ t❤❛t ❛•• ❢❛❝t♦rs ♦❢ N ❤❛✈❡ ❜❡❡♥ ❢♦✉♥❞✱ ✇❡ r❡✈❡rt t♦ t❤❡ ♣r❡✈✐♦✉s r❡s✐❞✉❡ t♦ t❛❦❡ t❤❡ •❈❉ ✇✐t❤ N. ❯s✐♥❣ t❤❡ s❛♠❡ 10 6 ❝♦♠♣♦s✐t❡ ✐♥♣✉ts ❛s ❢♦r P✕✶✱ P✰✶ ✇✐t❤ B 1 = 500 r❡♣♦rts 117 ✐♥♣✉t ♥✉♠❜❡rs ❛s ❢❛❝t♦rs ✇✐t❤ ❜❛❝❦tr❛❝❦✐♥❣✱ ❛♥❞ 1527 ✇✐t❤♦✉t✳ ■❢ st❛❣❡ ✶ ♦❢ P✰✶ ✐s ✉♥s✉❝❝❡ss❢✉•✱ ✇❡ ❝❛♥ tr② t♦ ✜♥❞ ❛ ❢❛❝t♦r ②❡t ❜② r✉♥♥✐♥❣ st❛❣❡ ✷✱ ✉s✐♥❣ ❛s ✐♥♣✉t t❤❡ ♦✉t♣✉t x 1 ♦❢ st❛❣❡ ✶✳ ❖✉r st❛❣❡ ✷ ✐s ✐❞❡♥t✐❝❛• ❢♦r P✕✶ ❛♥❞ P✰✶✱ ❛♥❞ ✈❡r② s✐♠✐•❛r ❢♦r ❊❈▼✱ ❛♥❞ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✼✳ ✺✳✶ ▲✉❝❛s ❝❤❛✐♥s ▼♦♥t❣♦♠❡r② s❤♦✇s ✐♥ ❬✶✾( ❤♦✇ t♦ ❣❡♥❡r❛t❡ ❛❞❞✐t✐♦♥ ❝❤❛✐♥s a 0 , a 1 , . . . , a ℓ ✇✐t❤ a 0 = 1 ❛♥❞ •❡♥❣t❤ ℓ s✉❝❤ t❤❛t ❢♦r ❛♥② 0 < i ≤ ℓ, t❤❡r❡ ❡①✐st 0 ≤ s, t < i s✉❝❤ t❤❛t a i = a s + a t ❛♥❞ a s -a t ✐s ❡✐t❤❡r ③❡r♦✱ ♦r ✐s ❛•s♦ ♣r❡s❡♥t ✐♥ t❤❡ ❝❤❛✐♥✳ ❍❡ ❝❛••s s✉❝❤ ❝❤❛✐♥s ✏▲✉❝❛s ❝❤❛✐♥s✳✑ ❋♦r ❡①❛♠♣•❡✱ t❤❡ ❛❞❞✐t✐♦♥ ❝❤❛✐♥ 1, 2, 4, 5 ✐s ♥♦t ❛ ▲✉❝❛s ❝❤❛✐♥ s✐♥❝❡ t❤❡ •❛st t❡r♠ 5 ❝❛♥ ❜❡ ❣❡♥❡r❛t❡❞ ♦♥•② ❢r♦♠ 4 + 1✱ ❜✉t 4 -1 = 3 ✐s ♥♦t ✐♥ t❤❡ ❝❤❛✐♥✳ ❚❤❡ ❛❞❞✐t✐♦♥ ❝❤❛✐♥ 1, 2, 3, 5, ❤♦✇❡✈❡r✱ ✐s ❛ ▲✉❝❛s ❝❤❛✐♥✳ ❋♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✱ L(n) ❞❡♥♦t❡s t❤❡ •❡♥❣t❤ ♦❢ ❛♥ ♦♣t✐♠❛• ✭✐✳❡✳✱ s❤♦rt❡st ♣♦ss✐❜•❡✮ ▲✉❝❛s ❝❤❛✐♥ t❤❛t ❡♥❞s ✐♥ n. ❆ s✐♠♣•❡ ❜✉t ❣❡♥❡r❛••② ♥♦♥✲♦♣t✐♠❛• ✇❛② ♦❢ ❣❡♥❡r❛t✐♥❣ s✉❝❤ ❝❤❛✐♥s ✉s❡s t❤❡ r❡❞✉❝t✐♦♥ (n, n -1) → (⌈n/2⌉, ⌈n/2⌉ -1). ❲❡ ❝❛♥ ❝♦♠♣✉t❡ V n (x) ❛♥❞ V n-1 (x) ❢r♦♠ V ⌈n/2⌉ (x)✱ V ⌈n/2⌉-1 (x), V 1 (x) = x, ❛♥❞ V 0 (x) = 2. ■♥ t❤❡ ❝❛s❡ ♦❢ n ❡✈❡♥✱ ✇❡ ✉s❡ V n (x) = V ⌈n/2⌉ (x) 2 -V 0 (x), ❛♥❞ V n-1 (x) = V ⌈n/2⌉ (x)V ⌈n/2⌉-1 (x)-V 1 (x) ❛♥❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ n ♦❞❞✱ ✇❡ ✉s❡ V n = V ⌈n/2⌉ (x)V ⌈n/2⌉-1 (x) -V 1 (x). ❛♥❞ V n-1 (x) = V ⌈n/2⌉-1 (x) 2 -V 0 (x) ❚❤❡ r❡s✉•t✐♥❣ ❝❤❛✐♥ ❛••♦✇s ♣r♦❝❡ss✐♥❣ t❤❡ ♠✉•t✐♣•✐❡r •❡❢t✲ t♦✲r✐❣❤t ♦♥❡ ❜✐t ❛t ❛ t✐♠❡✱ ❛♥❞ t❤✉s ✐s ❝❛••❡❞ ❜✐♥❛r② ❝❤❛✐♥ ❜② ▼♦♥t❣♦♠❡r②✳ ❊❛❝❤ ❜✐t ✐♥ t❤❡ ♠✉•t✐♣•✐❡r ❛❞❞s t✇♦ t❡r♠s t♦ t❤❡ ❛❞❞✐t✐♦♥ ❝❤❛✐♥✱ ❡①❝❡♣t t❤❛t ✇❤❡♥ ♣r♦❝❡ss✐♥❣ t❤❡ ✜♥❛• ❜✐t✱ ♦♥•② ♦♥❡ ♦❢ t❤❡ t✇♦ ✈❛•✉❡s ♥❡❡❞s t♦ ❜❡ ❝♦♠♣✉t❡❞✱ ❛♥❞ ✐❢ t❤❡ t✇♦ ♠♦st s✐❣♥✐✜❝❛♥t ❜✐ts ✭▼❙❇✮ ❛r❡ 10 b ✱ t❤❡ ❛❜♦✈❡ r✉•❡ ✇♦✉•❞ ❝♦♠♣✉t❡ V 2 (x) t✇✐❝❡ ♦❢ ✇❤✐❝❤ ♦♥❡ s❤♦✉•❞ ❜❡ s❦✐♣♣❡❞✳ ❆♥② tr❛✐•✐♥❣ ③❡r♦ ❜✐ts ❝❛♥ ❜❡ ❤❛♥❞•❡❞ ❜② V 2n (x) = V n (x) 2 -V 0 (x) ❛t t❤❡ ❝♦st ♦❢ 1 ♠✉•t✐♣•✐❝❛t✐♦♥ ❡❛❝❤✳ ❚❤❡ •❡♥❣t❤ L b (n2 k ) ❢♦r t❤❡ ❜✐♥❛r② ▲✉❝❛s ❝❤❛✐♥ ❢♦r ❛ ♥✉♠❜❡r n2 k ✇✐t❤ n ♦❞❞ ✐s t❤❡r❡❢♦r❡ 2⌊log 2 (n)⌋ -1 + k ✐❢ t❤❡ t✇♦ ▼❙❇ ❛r❡ 10 b ✱ ♦r 2⌊log 2 (n)⌋ + k ✐❢ n = 1 ♦r t❤❡ t✇♦ ▼❙❇ ❛r❡ 11 b . ❊①❛♠♣•❡s ❛r❡ ✐♥ ❚❛❜•❡ ✷✳ ■t •✐sts t❤❡ ❜✐♥❛r② ❝❤❛✐♥✱ t❤❡ •❡♥❣t❤ L b (n) ♦❢ t❤❡ ❜✐♥❛r② ❝❤❛✐♥✱ ❛♥ ♦♣t✐♠❛• ❝❤❛✐♥✱ ❛♥❞ t❤❡ •❡♥❣t❤ L(n) ♦❢ ❛♥ ♦♣t✐♠❛• ❝❤❛✐♥✱ ❢♦r ♦❞❞ n ✉♣ t♦ 15. n ❇✐♥❛r② ❝❤❛✐♥ L b (n) ❖♣t✐♠❛• ❝❤❛✐♥ ▲✭♥✮ 3 = 11 b 1, 2, 3 ✷ 1, 2, 3 ✷ 5 = 101 b 1, 2, 3, 5 ✸ 1, 2, 3, 5 ✸ 7 = 111 b 1, 2, 3, 4, 7 ✹ 1, 2, 3, 4, 7 ✹ 9 = 1001 b 1, 2, 3, 4, 5, 9 ✺ 1, 2, 3, 6, 9 ✹ 11 = 1011 b 1, 2,
✺ ❚❛❜•❡ ✷✿ ❇✐♥❛r② ❛♥❞ ♦♣t✐♠❛• ▲✉❝❛s ❝❤❛✐♥s ❢♦r s♠❛•• ♦❞❞ ✈❛•✉❡s n ❚❤❡ ❜✐♥❛r② ❝❤❛✐♥ ✐s ✈❡r② ❡❛s② t♦ ✐♠♣•❡♠❡♥t✱ ❜✉t ♣r♦❞✉❝❡s ♥♦♥✲♦♣t✐♠❛• ▲✉❝❛s ❝❤❛✐♥s ❡①❝❡♣t ❢♦r ✈❡r② s♠❛•• ♠✉•t✐♣•✐❡rs✳ ❚❤❡ s♠❛••❡st ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ✇❤❡r❡ t❤❡ ❜✐♥❛r② ♠❡t❤♦❞ ❞♦❡s ♥♦t ♣r♦❞✉❝❡ ❛♥ ♦♣t✐♠❛• ❝❤❛✐♥ ✐s 9✱ ❛♥❞ t❤❡ s♠❛••❡st s✉❝❤ || ♥➦ ✼✵✹✶ ✶✽ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ ♣r✐♠❡ ✐s 13. ▼♦♥t❣♦♠❡r② s❤♦✇s t❤❛t ✐❢ n ✐s ❛ ♣r✐♠❡ ❜✉t ♥♦t ❛ ❋✐❜♦♥❛❝❝✐ ♣r✐♠❡✱ ❛♥ ♦♣t✐♠❛• ▲✉❝❛s ❝❤❛✐♥ ❢♦r n ❤❛s •❡♥❣t❤ L(n) ≥ r ✇✐t❤ r ♠✐♥✐♠❛• s✉❝❤ t❤❛t n ≤ F r+2 -F r-3 , ✇❤❡r❡ F k ✐s t❤❡ k✲t❤ ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡r✳ ❙✐♥❝❡ F k = (φ k -φ -k )/ √ 5 ✇❤❡r❡ φ = (1 + √ 5)/2 ✐s t❤❡ •♦•❞❡♥ |❛t✐♦✱ t❤✐s s✉❣❣❡sts t❤❛t ✐❢ t❤✐s ❜♦✉♥❞ ✐s t✐❣❤t✱ ❢♦r •❛r❣❡ n ❛♥ ♦♣t✐♠❛• ❝❤❛✐♥ ❢♦r n s❤♦✉•❞ ❜❡ ❛❜♦✉t 28% s❤♦rt❡r t❤❛♥ t❤❡ ❜✐♥❛r② ❝❤❛✐♥✳ ■♥ ❛ ▲✉❝❛s ❝❤❛✐♥ a 0 , a 1 , . . . , a ℓ ♦❢ •❡♥❣t❤ ℓ✱ ❛ ❞♦✉❜•✐♥❣ st❡♣ a k+1 = 2a k ❝❛✉s❡s ❛•• a i ✇✐t❤ k ≤ i ≤ ℓ t♦ ❜❡ ♠✉•t✐♣•❡s ♦❢ a k , ❛♥❞ ❛•• t❤❡s❡ t❡r♠s a i ❛r❡ ❢♦r♠❡❞ ✉s✐♥❣ s✉♠s ❛♥❞ ❞✐✛❡r❡♥❝❡s ♦♥•② ♦❢ t❡r♠s a j , k < j ≤ ℓ, s❡❡ ❬✶✾(✳ ❙✉❝❤ ❛ ❞♦✉❜•✐♥❣ st❡♣ ❝♦rr❡s♣♦♥❞s t♦ ❛ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ▲✉❝❛s ❝❤❛✐♥s✳ ❋♦r ❝♦♠♣♦s✐t❡ n = n 1 •n 2 , ❛ ▲✉❝❛s ❝❤❛✐♥ ❝❛♥ ❜❡ ♠❛❞❡ ❜② ❝♦♥❝❛t❡♥❛t✐♥❣ t❤❡ ❝❤❛✐♥s ♦❢ ✐ts ❢❛❝t♦rs✳ ❊✳❣✳✱ ❢♦r n = 15, ✇❡ ❝♦✉•❞ ♠✉•t✐♣•② ❡✈❡r② ❡♥tr② ✐♥ t❤❡ ❝❤❛✐♥ 1, 2, 3, 5 ❜② 3 ❛♥❞ ❛♣♣❡♥❞ ✐t ❞♦ t❤❡ ❝❤❛✐♥ 1, 2, 3 ✭♦♠✐tt✐♥❣ t❤❡ r❡♣❡❛t❡❞ ❡♥tr② 3✮ t♦ ❢♦r♠ t❤❡ ▲✉❝❛s ❝❤❛✐♥ 1, 2, 3, 6, 9, 15. ❙✐♥❝❡ ❛♥② ▲✉❝❛s ❝❤❛✐♥ st❛rts ✇✐t❤ 1, 2✱ ❡✈❡r② ❝♦♥❝❛t❡♥❛t✐♦♥ ✐♥tr♦❞✉❝❡s ♦♥❡ ❞♦✉❜•✐♥❣ st❡♣✱ ❛♥❞ ❡✈❡r② ❞♦✉❜•✐♥❣ st❡♣ •❡❛❞s t♦ ❛ ❝❤❛✐♥ t❤❛t ✐s t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ t✇♦ ▲✉❝❛s ❝❤❛✐♥s✳ ❈❤❛✐♥s t❤❛t ❛r❡ ♥♦t t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ♦t❤❡r ❝❤❛✐♥s ✭✐✳❡✳✱ t❤❛t ❝♦♥t❛✐♥ ♥♦ ❞♦✉❜•✐♥❣ st❡♣ ♦t❤❡r t❤❛♥ 1, 2✮ ❛r❡ ❝❛••❡❞ s✐♠♣•❡ ❝❤❛✐♥s✳ ❋♦r ♣r✐♠❡ n, ♦♥•② s✐♠♣•❡ ❝❤❛✐♥s ❡①✐st✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ❜✐♥❛r② ▲✉❝❛s ❝❤❛✐♥s✱ t❤❡ ❝♦♥❝❛t❡♥❛t❡❞ ❝❤❛✐♥ ✐s ♥❡✈❡r •♦♥❣❡r t❤❛♥ t❤❡ ❝❤❛✐♥ ❢♦r t❤❡ ❝♦♠♣♦s✐t❡ ✈❛•✉❡ ❛♥❞ ✉s✉❛••② s❤♦rt❡r✱ s♦ t❤❛t ❢♦r♠✐♥❣ ❛ ❝♦♥❝❛t❡♥❛t❡❞ ▲✉❝❛s ❝❤❛✐♥ ❢r♦♠ ❝❤❛✐♥s ♦❢ t❤❡ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ n ✭✐❢ ❦♥♦✇♥✮ ✐s ❛•✇❛②s ❛❞✈✐s❛❜•❡✳ ❚❤❡ s❛♠❡ ✐s ♥♦t tr✉❡ ❢♦r ♦♣t✐♠❛• ❝❤❛✐♥s✱ ❛s s❤♦✇♥ ❜❡•♦✇✳ ❖♣t✐♠❛• ❝❤❛✐♥s ❝❛♥ ❜❡ ❢♦✉♥❞ ❜② ❡①❤❛✉st✐✈❡ s❡❛r❝❤ ❢♦r ❛ ❝❤♦s❡♥ ♠❛①✐♠❛• •❡♥❣t❤ l ♠❛① ❛♥❞ ♠❛①✐♠❛• ❡♥❞✲✈❛•✉❡ n ♠❛① ✳ ❋♦r ♦❞❞ n ≥ 3✱ ❛ ▲✉❝❛s ❝❤❛✐♥ ❢♦r n ❛•✇❛②s st❛rts ✇✐t❤ 1, 2, 3 s✐♥❝❡ ❛ ❞♦✉❜•✐♥❣ st❡♣ 2, 4 ✇♦✉•❞ ♣r♦❞✉❝❡ ♦♥•② ❡✈❡♥ ✈❛•✉❡s ✐♥ t❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤❡ ❝❤❛✐♥✳ ■♥ t❤❡ ❡①❤❛✉st✐✈❡ s❡❛r❝❤✱ t❤❡ ▲✉❝❛s ❝❤❛✐♥ a 0 , . . . , a k ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ r❡❝✉rs✐✈❡•② ✐❢ k < ℓ ♠❛① ❛♥❞ a k < n ♠❛① ❜② ❛❞❞✐♥❣ ❛♥ ❡•❡♠❡♥t a k+1 > a k s✉❝❤ t❤❛t t❤❡ r❡s✉•t✐♥❣ s❡q✉❡♥❝❡ ✐s st✐•• ❛ ▲✉❝❛s ❝❤❛✐♥✱ ✐✳❡✳✱ s❛t✐s❢②✐♥❣ t❤❛t t❤❡r❡ ❛r❡ 0 ≤ i, j ≤ k s✉❝❤ t❤❛t ❡✐t❤❡r a k+1 = 2a i ✱ ♦r a k+1 = a i + a j ❛♥❞ a i -a j ✐s ♣r❡s❡♥t ✐♥ t❤❡ ❝❤❛✐♥✳ ❋♦r ❡❛❝❤ ❝❤❛✐♥ s♦ ❝r❡❛t❡❞✱ ✇❡ ❝❤❡❝❦ ✐♥ ❛ t❛❜•❡ ♦❢ ❜❡st ❦♥♦✇♥ •❡♥❣t❤s ✇❤❡t❤❡r t❤❡ •❡♥❣t❤ k + 1 ✐s s♠❛••❡r t❤❛♥ t❤❡ ♣r❡✈✐♦✉s•② ❦♥♦✇♥ s❤♦rt❡st •❡♥❣t❤ ❢♦r r❡❛❝❤✐♥❣ a k+1 ✱ ❛♥❞ ✐❢ s♦✱ ✉♣❞❛t❡ ✐t t♦ k + 1 ❛♥❞ st♦r❡ t❤❡ ❝✉rr❡♥t ❝❤❛✐♥ ❛s t❤❡ ❜❡st ❦♥♦✇♥ ❢♦r r❡❛❝❤✐♥❣ a k+1 . ❇② tr②✐♥❣ ❛•• ♣♦ss✐❜•❡ ❝❤❛✐♥ ❡①♣❛♥s✐♦♥s✱ ✇❡ ❛r❡ ❝❡rt❛✐♥ t♦ ✜♥❞ ❛♥ ♦♣t✐♠❛• ❝❤❛✐♥ ❢♦r ❡✈❡r② n ≤ n ♠❛① . ❚❤✐s r❡❝✉rs✐✈❡ s❡❛r❝❤ ✐s ✈❡r② t✐♠❡ ❝♦♥s✉♠✐♥❣ ❞✉❡ t♦ ❛ •❛r❣❡ ♥✉♠❜❡r ♦❢ ❝♦♠❜✐♥❛t✐♦♥s t♦ tr②✳ ❚♦ r❡❛❝❤ ❛ ✇♦rt❤✇❤✐•❡ s❡❛r❝❤ ❞❡♣t❤✱ t❤❡ ♣♦ss✐❜•❡ ❝❤❛✐♥ ❡①t❡♥s✐♦♥s ❝❛♥ ❜❡ r❡str✐❝t❡❞✳ ❚❤❡ •❛st st❡♣ ♦❢ ❛♥ ♦♣t✐♠❛• ❝❤❛✐♥ ✐s ❛•✇❛②s a l = a ℓ-1 + a ℓ-2 ❛s ♦t❤❡r✇✐s❡ ♦♥❡ ♦r ❜♦t❤ ♦❢ a ℓ-1 , a ℓ-2 ❛r❡ ♦❜s♦•❡t❡✱ s♦ t❤❡ t❛❜•❡ ♦❢ ❜❡st ❦♥♦✇♥ •❡♥❣t❤s ♥❡❡❞s t♦ ❜❡ ❝❤❡❝❦❡❞ ❛♥❞ ✉♣❞❛t❡❞ ♦♥•② ❛❢t❡r s✉❝❤ ❛♥ ❛❞❞✐t✐♦♥ st❡♣✱ ❛♥❞ t❤❡ ✜♥❛• r❡❝✉rs✐♦♥ •❡✈❡• ♦❢ t❤❡ s❡❛r❝❤ ♥❡❡❞s t♦ ❝♦♥s✐❞❡r ♦♥•② t❤✐s ❛❞❞✐t✐♦♥ st❡♣✳ ❆♥② ❞♦✉❜•✐♥❣ st❡♣ a k+1 = 2a k ❝❛✉s❡s t❤❡ ❝❤❛✐♥ t♦ ❜❡❝♦♠❡ t❤❡ ❡q✉✐✈❛•❡♥t ♦❢ ❛ ❝♦♥❝❛t❡♥❛t❡❞ ❝❤❛✐♥✱ s♦ ❞✉r✐♥❣ t❤❡ r❡❝✉rs✐✈❡ ❝❤❛✐♥ ❡①♣❛♥s✐♦♥✱ ❞♦✉❜•✐♥❣ st❡♣s ♥❡❡❞ ♥♦t ❜❡ ❝♦♥s✐❞❡r❡❞✳ ■♥st❡❛❞ t❤❡ r❡❝✉rs✐✈❡ s❡❛r❝❤ ♣r♦❞✉❝❡s ♦♥•② t❤❡ ♦♣t✐♠❛• •❡♥❣t❤s ♦❢ s✐♠♣•❡ ❝❤❛✐♥s✳ ❚❤❡♥ ❢♦r ❛•• ♣♦ss✐❜•❡ ♣❛✐rs 3 ≤ m ≤ n ≤ √ n ♠❛① ✱ t❤❡ •❡♥❣t❤ ♦❢ t❤❡ ❝❤❛✐♥ ❢♦r mn ✐s ✉♣❞❛t❡❞ ✇✐t❤ t❤❡ s✉♠ ♦❢ t❤❡ •❡♥❣t❤s ♦❢ ❝❤❛✐♥s ❢♦r m ❛♥❞ n✱ ✐❢ t❤❡ •❛tt❡r ✐s s❤♦rt❡r✳ ❚❤✐s ✐s r❡♣❡❛t❡❞ ✉♥t✐• ♥♦ ♠♦r❡ ✐♠♣r♦✈❡♠❡♥ts ♦❝❝✉r✳ ❆❢t❡r t❤❡ ✜rst ♣❛ss✱ t❤❡ ♦♣t✐♠❛• •❡♥❣t❤s ♦❢ ❝❤❛✐♥s ❢♦r ❛•• n ✇❤❡r❡ n ❤❛s ❛t ♠♦st t✇♦ ♣r✐♠❡ ❢❛❝t♦rs ❛r❡ ❦♥♦✇♥✳ ❆❢t❡r t❤❡ s❡❝♦♥❞ ♣❛ss✱ ❢♦r ❛•• n t❤❛t ❝♦♥t❛✐♥ ❛t ♠♦st t❤r❡❡ ♣r✐♠❡s✱ ❡t❝✳✱ ✉♥t✐• ❛❢t❡r ❛t ♠♦st O(log(n ♠❛① )) ♣❛ss❡s ♦♣t✐♠❛• •❡♥❣t❤s ❢♦r ❛•• ✈❛•✉❡s ❛r❡ ❦♥♦✇♥✳ ❯s✐♥❣ t❤✐s ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦
❋✐❣✉r❡ ✸✿ ▲❡♥❣t❤ ♦❢ ❜✐♥❛r② ❛♥❞ ♦♣t✐♠❛• ▲✉❝❛s ❝❤❛✐♥s ❢♦r ♦❞❞ ♣r✐♠❡s p ✐♥ [100, 15000]✱ ❛♥❞ ❛ •♦✇❡r ❜♦✉♥❞ ♦♥ t❤❡ •❡♥❣t❤ ❢♦r ♣r✐♠❡s t❤❛t ❛r❡ ♥♦t ❋✐❜♦♥❛❝❝✐ ♣r✐♠❡s✳ ❚❤❡ ❣r❛♣❤ ❢♦r t❤❡ ❜♦✉♥❞ ✐s s❡t 0.5 •♦✇❡r t♦ ♠❛❦❡ ✐t ✈✐s✐❜•❡✳ ❚❤❡ ❋✐✲ ❜♦♥❛❝❝✐ ♣r✐♠❡ 1597 ✐s s❡❡♥ t♦ ✉♥❞❡rs❤♦♦t t❤✐s •♦✇❡r ❜♦✉♥❞✳ s❡❛r❝❤ ♠❡t❤♦❞✱ t❤❡ ♠✐♥✐♠❛• •❡♥❣t❤s ♦❢ ▲✉❝❛s ❝❤❛✐♥s ❢♦r ♣r✐♠❡s 100 < n < 10000 ❤❛✈❡ ❜❡❡♥ ❞❡t❡r♠✐♥❡❞✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✸✳ ■t ❝♦♠♣❛r❡s t❤❡ •❡♥❣t❤ ♦❢ t❤❡ ❜✐♥❛r② ▲✉❝❛s ❝❤❛✐♥✱ t❤❡ ♦♣t✐♠❛• ▲✉❝❛s ❝❤❛✐♥ ❛♥❞ t❤❡ •♦✇❡r ❜♦✉♥❞ ♦♥ t❤❡ •❡♥❣t❤ ♦❢ ▲✉❝❛s ❝❤❛✐♥s ❢♦r ♣r✐♠❡s t❤❛t ❛r❡♥✬t ❋✐❜♦♥❛❝❝✐ ♣r✐♠❡s✳ ❚❤✐s •♦✇❡r ❜♦✉♥❞ ✐s q✉✐t❡ t✐❣❤t✱ ✐♥ t❤❡ ❡①❛♠✐♥❡❞ ❞❛t❛ L(n) ❞♦❡s ♥♦t ❡①❝❡❡❞ ✐t ❜② ♠♦r❡ t❤❛♥ 1. ❚❤❡ ❋✐❜♦♥❛❝❝✐ ♣r✐♠❡ 1597 ✈✐s✐❜•② ✉♥❞❡rs❤♦♦ts t❤✐s •♦✇❡r ❜♦✉♥❞ ✭❛s ❞♦❡s t❤❡ s♠❛••❡r ❋✐❜♦♥❛❝❝✐ ♣r✐♠❡ 233, ❜✉t ✐t ✐s ❞✐✣❝✉•t t♦ s❡❡ ✐♥ t❤❡ ❣r❛♣❤✮✳ ❚❤❡ ❡①❤❛✉st✐✈❡ s❡❛r❝❤ ♠❡t❤♦❞ ✐s ❡①tr❡♠❡•② s•♦✇ ❛♥❞ ✉s❡•❡ss ❢♦r ♣r♦❞✉❝✐♥❣ ❛❞❞✐t✐♦♥ ❝❤❛✐♥s ❢♦r P✰✶ ♦r ❊❈▼ ✐❢ •❛r❣❡ B 1 ✈❛•✉❡s ❛r❡ ❞❡s✐r❡❞✳ ▼♦♥t❣♦♠❡r② ❬✶✾( s✉❣❣❡sts t❤❡ ❛•❣♦r✐t❤♠ ✏P|❆❈✱✑ ✇❤✐❝❤ ♣r♦❞✉❝❡s ▲✉❝❛s ❝❤❛✐♥s ❜❛s❡❞ ♦♥ •❈❉ ❝❤❛✐♥s✱ ♥♦t✐♥❣ t❤❛t ❛ s✉❜tr❛❝t✐✈❡ •❈❉ ❛•❣♦r✐t❤♠ ❢♦r n, r ✇✐t❤ n > r ❛♥❞ n ⊥ r ❛•✇❛②s ♣r♦❞✉❝❡s ❛ ✈❛•✐❞ ▲✉❝❛s ❝❤❛✐♥ ❢♦r n✳ ❍♦✇❡✈❡r✱ t❤❡ r❡s✉•t✐♥❣ ▲✉❝❛s ❝❤❛✐♥ ❤❛s •❡♥❣t❤ ❡q✉❛• t♦ t❤❡ s✉♠ ♦❢ t❤❡ ♣❛rt✐❛• q✉♦t✐❡♥ts ✐♥ t❤❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ n/(n-r), ❛♥❞ ✐❢ ❛ •❛r❣❡ ♣❛rt✐❛• q✉♦t✐❡♥t ❛♣♣❡❛rs✱ t❤❡ r❡s✉•t✐♥❣ ▲✉❝❛s ❝❤❛✐♥ ✐s ✉♥r❡❛s♦♥❛❜•② •♦♥❣✳ ❍❡ ✜①❡s t❤✐s ♣r♦❜•❡♠ ❜② ✐♥tr♦❞✉❝✐♥❣ ❛❞❞✐t✐♦♥❛• r✉•❡s ❢♦r r❡❞✉❝t✐♦♥ ✐♥ t❤❡ •❈❉ ❝❤❛✐♥ ✭r❛t❤❡r t❤❛♥ •✉st r❡♣•❛❝✐♥❣ t❤❡ •❛r❣❡r ♦❢ t❤❡ t✇♦ ♣❛rt✐❛• r❡♠❛✐♥❞❡rs ❜② t❤❡✐r ❛❜s♦•✉t❡ ❞✐✛❡r❡♥❝❡ ❛s ✐♥ ❛ ♣✉r❡•② s✉❜tr❛❝t✐✈❡ •❈❉ ❝❤❛✐♥✮ t♦ ❛✈♦✐❞ s✐t✉❛t✐♦♥s ✇❤❡r❡ t❤❡ q✉♦t✐❡♥t ♦❢ t❤❡ ♣❛rt✐❛• r❡♠❛✐♥❞❡rs ❞❡✈✐❛t❡s t♦♦ ❢❛r ❢r♦♠ t❤❡ •♦•❞❡♥ |❛t✐♦✱ ②❡t s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s ❢♦r ❛ ▲✉❝❛s ❝❤❛✐♥✳ ❚❤❡ ❣r❡❛t ❛❞✈❛♥t❛❣❡ ✐s t❤❛t P|❆❈ ✉s✉❛••② ♣r♦❞✉❝❡s ✈❡r② ❣♦♦❞ ❝❤❛✐♥s ❛♥❞ ❞♦❡s s♦ r❛♣✐❞•②✳ ❚❤✐s ✇❛② ✐t ✐s ❢❡❛s✐❜•❡ t♦ tr② ❛ ❢❡✇ ❞✐✛❡r❡♥t s✉✐t❛❜•❡ r ❢♦r ❛ ❣✐✈❡♥ n, ❛♥❞ ❢♦r n ✐♥ t❤❡ r❛♥❣❡ ♦❢ ✐♥t❡r❡st ❢♦r P✰✶ ❛♥❞ ❊❈▼✱ ♦♥❡ ✉s✉❛••② ❞✐s❝♦✈❡rs ❛♥ ♦♣t✐♠❛• ❝❤❛✐♥ t❤✐s ✇❛②✳ ■t r❡♠❛✐♥s t❤❡ ♣r♦❜•❡♠ ♦❢ ❝❤♦♦s✐♥❣ ❛ s✉✐t❛❜•❡ r ⊥ n t♦ st❛rt t❤❡ •❈❉ ❝❤❛✐♥✱ ❤♦♣✐♥❣ t♦ ✜♥❞ ❛ ✭♥❡❛r✮ ♦♣t✐♠❛• ❝❤❛✐♥✳ ▼♦♥t❣♦♠❡r② s✉❣❣❡sts tr②✐♥❣ r = n-⌊n/c⌉ ❢♦r s❡✈❡r❛• ✐rr❛t✐♦♥❛• c s✉❝❤ t❤❛t t❤❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ c ❤❛s s♠❛•• || ♥➦ ✼✵✹✶ ✷✵ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ ♣❛rt✐❛• q✉♦t✐❡♥ts✳ ❚❤✐s ✇❛②✱ t❤❡ ♣❛rt✐❛• ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ n/(n -r) st❛rts ✇✐t❤ s♠❛•• ♣❛rt✐❛• q✉♦t✐❡♥ts ❛s ✇❡••✳ •♦♦❞ ❝❤♦✐❝❡s ❛r❡ t❤❡ ❣♦•❞❡♥ |❛t✐♦ c 0 = φ, ✇❤♦s❡ ♣❛rt✐❛• q✉♦t✐❡♥ts ❛•• ❛r❡ 1, ♦r ♥✉♠❜❡rs ✇✐t❤ ♣❛rt✐❛• q✉♦t✐❡♥ts ❛•• 1 ❡①❝❡♣t ❢♦r ♦♥❡ ♦r t✇♦ 2 ❛♠♦♥❣ t❤❡ ✜rst 10 ♣❛rt✐❛• q✉♦t✐❡♥ts✳ ❚❤❡ r❡s✉•t✐♥❣ •❛r❣❡ ♥✉♠❜❡r ♦❢ ♠✉•t✐♣•✐❡rs ✐s ♥♦t ❛ ♣r♦❜•❡♠ ✐❢ t❤❡ ▲✉❝❛s ❝❤❛✐♥s ❛r❡ ♣r❡❝♦♠♣✉t❡❞✱ ❜✉t ✐♥ ❝❛s❡s ✇❤❡r❡ t❤❡② ❛r❡ ❝♦♠♣✉t❡❞ ♦♥✲t❤❡✲✢② ❞✉r✐♥❣ st❛❣❡ ✶ ♦❢ P✰✶ ♦r ❊❈▼✱ ❛ s♠❛••❡r s❡t ♦❢ ♠✉•t✐♣•✐❡rs s❤♦✉•❞ ❜❡ ✉s❡❞✱ s❛②✱ ♦♥•② t❤♦s❡ ✇✐t❤ ❛t ♠♦st ♦♥❡ 2 ❛♠♦♥❣ t❤❡ ✜rst t❡♥ ♣❛rt✐❛• q✉♦t✐❡♥ts✳ ❊✈❡♥ ✇✐t❤ ❛ •❛r❣❡ s❡t ♦❢ c i ✈❛•✉❡s t♦ tr②✱ P|❆❈ ✐♥ t❤❡ ❢♦r♠ ❣✐✈❡♥ ❜② ▼♦♥t✲ ❣♦♠❡r② ❝❛♥♥♦t ❛•✇❛②s ♦❜t❛✐♥ ❛♥ ♦♣t✐♠❛• ❝❤❛✐♥✳ ❚❤❡ s♠❛••❡st ❡①❛♠♣•❡ ✐s n = 751 ✇❤✐❝❤ ❤❛s t✇♦ ▲✉❝❛s ❝❤❛✐♥s ♦❢ ♦♣t✐♠❛• •❡♥❣t❤ L(751) = 14✿ 1,
❛♥❞ 1,
❇♦t❤ ❝❤❛✐♥s ✐♥✈♦•✈❡ ❛♥ ❛❞❞✐t✐♦♥ st❡♣ t❤❛t r❡❢❡r❡♥❝❡s ❛ ❞✐✛❡r❡♥❝❡ t❤❛t ♦❝❝✉rr❡❞ 5 st❡♣s ❜❡❢♦r❡ t❤❡ ♥❡✇ t❡r♠✿ ❢♦r t❤❡ ❢♦r♠❡r s❡q✉❡♥❝❡ ✐♥ t❤❡ st❡♣ a 8 = 43 = a 7 + a 6 = 24 + 19✱ ✇✐t❤ ❞✐✛❡r❡♥❝❡ a 7 -a 6 = 5 = a 3 ✱ ❛♥❞ ❢♦r t❤❡ •❛tt❡r s❡q✉❡♥❝❡ ✐♥ t❤❡ st❡♣ a 10 = 123 = a 9 + a 8 = 68 + 55, ✇✐t❤ ❞✐✛❡r❡♥❝❡ a 9 -a 8 = 13 = a 5 . ❚❤❡ ♦r✐❣✐♥❛• P|❆❈ ❛•❣♦r✐t❤♠ ❞♦❡s ♥♦t ❤❛✈❡ ❛♥② r✉•❡ t❤❛t ❛••♦✇s ✉t✐•✐③✐♥❣ ❛ ❞✐✛❡r❡♥❝❡ t❤❛t ♦❝❝✉rr❡❞ ♠♦r❡ t❤❛♥ 4 st❡♣s ❜❡❢♦r❡ t❤❡ ♥❡✇ t❡r♠ ❛♥❞ s♦ ❝❛♥♥♦t ✜♥❞ ❡✐t❤❡r ♦❢ t❤❡s❡ t✇♦ ❝❤❛✐♥s✳ ❆♥♦t❤❡r✱ s✐♠✐•❛r ❝❛s❡ ✐s n = 1087. ❋♦r ♣r✐♠❡s ❜❡•♦✇ 10000✱ ■ ❢♦✉♥❞ 40 ❝❛s❡s ✇❤❡r❡ P|❆❈ ❞✐❞ ♥♦t ✜♥❞ ❛♥ ♦♣t✐♠❛• ❝❤❛✐♥✳ ❋♦r t❤❡ ♣✉r♣♦s❡ ♦❢ ❣❡♥❡r❛t✐♥❣ ▲✉❝❛s ❝❤❛✐♥s ❢♦r P✰✶ ❛♥❞ ❊❈▼✱ t❤❡s❡ ♠✐ss❡❞ ♦♣♣♦r✲ t✉♥✐t✐❡s ❛t ♦♣t✐♠❛• ❝❤❛✐♥s ❛r❡ ♦❢ ♥♦ ❣r❡❛t ❝♦♥s❡q✉❡♥❝❡✳ ❲❤❡♥ ✉s✐♥❣ P✰✶ ❛♥❞ ❊❈▼ ❛s ❛ ❢❛❝t♦r✐♥❣ s✉❜r♦✉t✐♥❡ ✐♥ ◆❋❙✱ t❤❡ B 1 ✈❛•✉❡ ✐s ♦❢t❡♥ •❡ss t❤❛♥ 751 s♦ t❤❛t s✉❝❤ ❝❛s❡s ❞♦ ♥♦t ♦❝❝✉r ❛t ❛••✱ ❛♥❞ ✐❢ ❛ ❣r❡❛t❡r B 1 s❤♦✉•❞ ❜❡ ✉s❡❞✱ t❤❡② ♦❝❝✉r s♦ r❛r❡•② t❤❛t ❛❞❞✐♥❣ ♠♦r❡ r✉•❡s t♦ P|❆❈ s♦ t❤❛t ♦♣t✐♠❛• ❝❤❛✐♥s ❛r❡ ❢♦✉♥❞ ❢♦r ❛•• ♣r✐♠❡s ❜❡•♦✇ B 1 ✇♦✉•❞ ✐♥❝r❡❛s❡ t❤❡ ❝♦❞❡ ❝♦♠♣•❡①✐t② ♦❢ ♦✉r P✰✶ ♦r ❊❈▼ st❛❣❡ ✶✱ ✇❤✐❝❤ ✐♠♣•❡♠❡♥ts ❡❛❝❤ P|❆❈ r✉•❡ ✭s❡❡ ❙❡❝t✐♦♥ ✺✳✷✮✱ ❢♦r •✐tt•❡ ❣❛✐♥✳ ❋♦r ♦✉r ✐♠♣•❡♠❡♥t❛t✐♦♥✱ t❤✐s ✇❛s ♥♦t ❞❡❡♠❡❞ ✇♦rt❤✇❤✐•❡✳ ❋♦r t❤❡ ♣✉r✲ ♣♦s❡ ♦❢ ✜♥❞✐♥❣ ♦♣t✐♠❛• ▲✉❝❛s ❝❤❛✐♥s r❛♣✐❞•②✱ ✐t ✇♦✉•❞ ❜❡ ✐♥t❡r❡st✐♥❣ t♦ ❛✉❣♠❡♥t P|❆❈ ✇✐t❤ ❛ s✉✐t❛❜•❡ r✉•❡ ❢♦r t❤❡ r❡q✉✐r❡❞ ❛❞❞✐t✐♦♥ st❡♣ a k = a k-1 + a k-2 ✇✐t❤ a k-1 -a k-2 = a k-5 ✱ ❛♥❞ t❡st✐♥❣ ✇❤✐❝❤ ♣r✐♠❡s r❡♠❛✐♥ s✉❝❤ t❤❛t t❤❡ ♠♦❞✐✜❡❞ P|❆❈ ❝❛♥♥♦t ✜♥❞ ♦♣t✐♠❛• ❝❤❛✐♥s✳ ❋♦r ❝♦♠♣♦s✐t❡ n = pq✱ ✇❡ tr✐✈✐❛••② ❤❛✈❡ L(n) ≤ L(p) + L(q), s✐♥❝❡ ✇❡ ❝❛♥ ❝♦♥❝❛t❡♥❛t❡ t❤❡ ❝❤❛✐♥ ❢♦r p ❛♥❞ t❤❡ ❝❤❛✐♥ ❢♦r q. ■♥ s♦♠❡ ❝❛s❡s✱ ❛ s❤♦rt❡r ❝❤❛✐♥ ❢♦r t❤❡ ❝♦♠♣♦s✐t❡ n ❡①✐sts t❤❛♥ ❢♦r t❤❡ ❝♦♥❝❛t❡♥❛t❡❞ ❝❤❛✐♥s ♦❢ ✐ts ❢❛❝t♦rs✳ ❚❤❡ s♠❛••❡st ❡①❛♠♣•❡ ✐s 1219 = 23 • 53 ✇❤✐❝❤ ❤❛s 1,
❛s ❛♥ ♦♣t✐♠❛• ❝❤❛✐♥ ♦❢ •❡♥❣t❤ ✶✺✱ ✇❤✐•❡ ❛♥ ♦♣t✐♠❛• ❝❤❛✐♥ ❢♦r 23 ✐s 1, 2, 3, 4, 5, 9, 14, 23 ♦❢ •❡♥❣t❤ 7✱ ❛♥❞ ❢♦r 53 ✐s 1,
♦❢ •❡♥❣t❤ 9. ❙✐♠✐•❛r•②✱ ❝♦♠♣♦s✐t❡ ♥✉♠❜❡rs n ❡①✐st ✇❤❡r❡ P|❆❈ ✇✐t❤ ❛ ❝❡rt❛✐♥ s❡t ♦❢ ♠✉•t✐♣•✐❡rs ✜♥❞s ❛ ❝❤❛✐♥ ❢♦r n t❤❛t ✐s s❤♦rt❡r t❤❛♥ t❤❡ ❝♦♥❝❛t❡♥❛t❡❞ s✐♠♣•❡ ❝❤❛✐♥s ❢♦r t❤❡ ❞✐✈✐s♦rs ♦❢ n. ❆ ♣r♦❜•❡♠ ✐s t❤❛t t❤❡ st❛rt✐♥❣ ♣❛✐r n, r ❢♦r t❤❡ •❈❉ s❡q✉❡♥❝❡ ♠✉st ❜❡ ❝♦♣r✐♠❡✱ ♣♦ss✐❜•② ♠❛❦✐♥❣ s❡✈❡r❛• c ♠✉•t✐♣•✐❡rs ✐♥❡•✐❣✐❜•❡ ❢♦r ❛♥ n ✇✐t❤ s♠❛•• ♣r✐♠❡ ❢❛❝t♦rs✳ ❙t❛rt✐♥❣ ✇✐t❤ ❛ •❛r❣❡ ❡♥♦✉❣❤ s❡t ♦❢ ♠✉•t✐♣•✐❡rs✱ ✉s✉❛••② ❡♥♦✉❣❤ ♦❢ t❤❡♠ ♣r♦❞✉❝❡ ❝♦♣r✐♠❡ n ❛♥❞ r t❤❛t ❛♥ ♦♣t✐♠❛• ❝❤❛✐♥ ❝❛♥ ❜❡ ❢♦✉♥❞✱ ✐❢ ♦♥❡ ❡①✐sts ♦❢ ❛ ❢♦r♠ s✉✐t❛❜•❡ ❢♦r P|❆❈✳ ❚❤❡ ❡①❛♠♣•❡ n = 1219 ❛❜♦✈❡ ✐s ❢♦✉♥❞✱ ❡✳❣✳✱ ✇✐t❤ r = 882, ✉s✐♥❣ t❤❡ ♠✉•t✐♣•✐❡r 3 -Φ ✇✐t❤ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ //1, 2, 1, 1, 1, 1, . . . //. ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✷✶ ✺✳✷ ❇②t❡ ❝♦❞❡ ❛♥❞ ❝♦♠♣r❡ss✐♦♥ ■♥ ✐♠♣•❡♠❡♥t❛t✐♦♥s ♦❢ P✰✶ ♦r ❊❈▼ s✉❝❤ ❛s ✐♥ •▼P✲❊❈▼ ❬✷✾( t❤❛t t②♣✐❝❛••② ♦♣❡r❛t❡s ♦♥ ♥✉♠❜❡rs ♦❢ ❤✉♥❞r❡❞s t♦ t❡♥✲t❤♦✉s❛♥❞s ♦❢ ❞✐❣✐ts✱ ♦r ✐♥ t❤❡ ❊❈▼ ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ Pr✐♠❡✾✺ ❬✷✽( t❤❛t ♦♣❡r❛t❡s ♦♥ ♥✉♠❜❡r ♦❢ ✉♣ t♦ s❡✈❡r❛• ♠✐••✐♦♥ ❞✐❣✐ts✱ t❤❡ ❝♦st ♦❢ ❣❡♥❡r❛t✐♥❣ ❣♦♦❞ ▲✉❝❛s ❝❤❛✐♥s ♦♥✲t❤❡✲✢② ❞✉r✐♥❣ st❛❣❡ ✶ ✐s ♠♦st•② ♥❡❣•✐❣✐❜•❡✱ ❡①❝❡♣t ❢♦r P✰✶ ♦♥ r❡•❛t✐✈❡•② s♠❛•• ♥✉♠❜❡rs ♦❢ ♦♥•② ❛ ❢❡✇ ❤✉♥❞r❡❞ ❞✐❣✐ts✳ ❍♦✇❡✈❡r✱ ✐♥ ❛♥ ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❛♥❞ ❡s♣❡❝✐❛••② P✰✶ ❞❡s✐❣♥❡❞ ❢♦r ♥✉♠❜❡rs ♦❢ ♦♥•② ❛ ❢❡✇ ♠❛❝❤✐♥❡ ✇♦r❞s✱ t❤❡ ♦♥✲t❤❡✲✢② ❣❡♥❡r❛t✐♦♥ ♦❢ ▲✉❝❛s ❝❤❛✐♥s ✇♦✉•❞ t❛❦❡ ❛♥ ✐♥❛❝❝❡♣t❛❜•❡ ♣❛rt ♦❢ t❤❡ t♦t❛• r✉♥✲t✐♠❡✳ ❙✐♥❝❡ ✐♥ ♦✉r ❛♣♣•✐❝❛t✐♦♥ ♦❢ ✉s✐♥❣ P✰✶ ❛♥❞ ❊❈▼ ❛s ❛ ❢❛❝t♦r✐♥❣ s✉❜✲r♦✉t✐♥❡ ✐♥ ◆❋❙✱ ✐❞❡♥t✐❝❛• st❛❣❡ ✶ ♣❛r❛♠❡t❡rs ❛r❡ ✉s❡❞ ♠❛♥② t✐♠❡s ♦✈❡r ❛❣❛✐♥✱ ✐t ✐s ♣♦ss✐❜•❡ t♦ ♣r❡✲❝♦♠♣✉t❡ ♦♣t✐♠✐③❡❞ ▲✉❝❛s ❝❤❛✐♥s ❛♥❞ ♣r♦❝❡ss t❤❡ st♦r❡❞ ❝❤❛✐♥ ❞✉r✐♥❣ P✰✶ ♦r ❊❈▼ st❛❣❡ ✶✳ ❚❤✐s r❛✐s❡s t❤❡ q✉❡st✐♦♥ ❤♦✇ t❤❡ ❝❤❛✐♥ s❤♦✉•❞ ❜❡ st♦r❡❞✳ ❙✐♥❝❡ t❤❡ P|❆❈ ❛•❣♦r✐t❤♠ r❡♣❡❛t❡❞•② ❛♣♣•✐❡s ♦♥❡ ♦❢ ♥✐♥❡ r✉•❡s t♦ ♣r♦❞✉❝❡ ❛ ▲✉❝❛s ❝❤❛✐♥ ❢♦r ❛ ❣✐✈❡♥ ✐♥♣✉t✱ ❛♥ ♦❜✈✐♦✉s ♠❡t❤♦❞ ✐s t♦ st♦r❡ t❤❡ s❡q✉❡♥❝❡ ♦❢ P|❆❈ r✉•❡s t♦ ❛♣♣•②✳ ❚❤❡ ♣r❡❝♦♠♣✉t❛t✐♦♥ ♦✉t♣✉ts ❛ s❡q✉❡♥❝❡ ♦❢ ❜②t❡s ✇❤❡r❡ ❡❛❝❤ ❜②t❡ st♦r❡s t❤❡ ✐♥❞❡① ♦❢ t❤❡ P|❆❈ r✉•❡ t♦ ✉s❡✱ ♦r ♦♥❡ ♦❢ t✇♦ ❡①tr❛ ✐♥❞✐❝❡s ❢♦r t❤❡ ✐♥✐t✐❛• ❞♦✉❜•✐♥❣ r❡s♣✳ t❤❡ ✜♥❛• ❛❞❞✐t✐♦♥ t❤❛t ✐s ❝♦♠♠♦♥ t♦ ❛•• ✭♥❡❛r✲✮♦♣t✐♠❛• ▲✉❝❛s ❝❤❛✐♥s✳ ❚❤✐s ✇❛②✱ ❛ ❜②t❡ ❝♦❞❡ ✐s ❣❡♥❡r❛t❡❞ t❤❛t ❝❛♥ ❜❡ ♣r♦❝❡ss❡❞ ❜② ❛♥ ✐♥t❡r♣r❡t❡r t♦ ❝❛rr② ♦✉t t❤❡ st❛❣❡ ✶ ❝♦♠♣✉t❛t✐♦♥s ❢♦r P✰✶ ♦r ❊❈▼✳ ❋♦r ❡❛❝❤ ♣r✐♠❡ t♦ ✐♥❝•✉❞❡ ✐♥ st❛❣❡ ✶✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❜②t❡ ❝♦❞❡ ✐s s✐♠♣•② ❛♣♣❡♥❞❡❞ t♦ t❤❡ ❜②t❡ ❝♦❞❡✱ ✇❤✐❝❤ r❡s✉•ts ✐♥ ❛ ✭•♦♥❣✮ ❝♦♥❝❛t❡♥❛t❡❞ ▲✉❝❛s ❝❤❛✐♥ ❢♦r t❤❡ ♣r♦❞✉❝t ♦❢ ❛•• st❛❣❡ ✶ ♣r✐♠❡s✳ ■❢ ♣r✐♠❡s ❛r❡ t♦ ❜❡ ✐♥❝•✉❞❡❞ ✇❤♦s❡ ♣r♦❞✉❝t ✐s ❦♥♦✇♥ t♦ ❤❛✈❡ ❛ ❜❡tt❡r s✐♠♣•❡ ▲✉❝❛s ❝❤❛✐♥ t❤❛♥ t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ t❤❡ ❝❤❛✐♥s ❢♦r t❤❡ ✐♥❞✐✈✐❞✉❛• ♣r✐♠❡s✱ t❤❡♥ t❤❡✐r ♣r♦❞✉❝t s❤♦✉•❞ ❜❡ ♣❛ss❡❞ t♦ t❤❡ ❜②t❡✲❝♦❞❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❜②t❡ ❝♦❞❡ ❣❡♥❡r❛t❡❞ ❜② P|❆❈ ✐s ❤✐❣❤•② r❡♣❡t✐t✐✈❡✳ ❋♦r ❡①❛♠♣•❡✱ ❜②t❡ ❝♦❞❡s ❢♦r t❤❡ P|❆❈ ❝❤❛✐♥s ❢♦r t❤❡ ♣r✐♠❡s ✶✵✶✱ ✶✵✸✱ ✶✵✼✱ ❛♥❞ ✶✵✾ ❛r❡ 101 :
■t ✐s ❜❡♥❡✜❝✐❛• t♦ r❡❞✉❝❡ r❡❞✉♥❞❛♥❝② ✐♥ t❤❡ ❜②t❡ ❝♦❞❡ t♦ s♣❡❡❞ ✉♣ st❛❣❡ ✶✳ ❚❤❡ ❜②t❡ ❝♦❞❡ ✐♥t❡r♣r❡t❡r t❤❛t ❡①❡❝✉t❡s st❛❣❡ ✶ ♠✉st ❢❡t❝❤ ❛ ❝♦❞❡ ❜②t❡✱ t❤❡♥ ❝❛•• t❤❡ ♣r♦❣r❛♠ ❝♦❞❡ t❤❛t ❝❛rr✐❡s ♦✉t t❤❡ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s t❤❛t ✐♠♣•❡♠❡♥t t❤❡ P|❆❈ r✉•❡ ✐♥❞✐❝❛t❡❞ ❜② t❤❡ ❝♦❞❡❀ t❤✉s t❤❡r❡ ✐s ❛ ❝♦st ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡❛❝❤ ❝♦❞❡ ❜②t❡✳ ■❢ t❤❡ ✐♥t❡r♣r❡t❡r ✉s❡s ❛ ❝♦♠♣✉t❡❞ •✉♠♣ t♦ t❤❡ ❝♦❞❡ ✐♠♣•❡♠❡♥t✐♥❣ ❡❛❝❤ P|❆❈ r✉•❡✱ t❤❡r❡ ✐s ❛•s♦ ❛ ❜r❛♥❝❤ ♠✐s♣r❡❞✐❝t✐♦♥ ❡❛❝❤ t✐♠❡ ❛ ❝♦❞❡ ❜②t❡ ✐s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s ♦♥❡✱ ❛s ❝✉rr❡♥t ♠✐❝r♦♣r♦❝❡ss♦rs t②♣✐❝❛••② ♣r❡❞✐❝t ❝♦♠♣✉t❡❞ •✉♠♣s ❛s ❣♦✐♥❣ t♦ t❤❡ s❛♠❡ ❛❞❞r❡ss ❛s t❤❡② ❞✐❞ t❤❡ ♣r❡✈✐♦✉s t✐♠❡✳ ❙♦♠❡ P|❆❈ r✉•❡s ❢r❡q✉❡♥t•② ♦❝❝✉r t♦❣❡t❤❡r✱ s✉❝❤ ❛s r✉•❡ ✸ ❢♦••✇❡❞ ❜② r✉•❡ ✵✱ s♦ t❤❛t ♠❡r❣✐♥❣ t❤❡♠ ♠❛② •❡❛❞ t♦ s✐♠♣•✐✜❝❛t✐♦♥s ✐♥ t❤❡ ❛r✐t❤♠❡t✐❝✳ ■♥ ♣❛rt✐❝✉•❛r✱ r✉•❡s 11 ✭❡♥❞ ♦❢ ❛ s✐♠♣•❡ ❝❤❛✐♥✮ ❛♥❞ 10 ✭st❛rt ♦❢ ❛ ♥❡✇ s✐♠♣•❡ ❝❤❛✐♥✮ ❛•✇❛②s ❛♣♣❡❛r t♦❣❡t❤❡r✱ ❡①❝❡♣t ❛t t❤❡ ✈❡r② st❛rt ❛♥❞ ❛t t❤❡ ✈❡r② ❡♥❞ ♦❢ t❤❡ ❜②t❡ ❝♦❞❡✳ ❚❤❡s❡ ✐ss✉❡s ❛r❡ ❛❞❞r❡ss❡❞ ❜② ❜②t❡ ❝♦❞❡ ❝♦♠♣r❡ss✐♦♥✳ ❆ s✐♠♣•❡ st❛t✐❝ ❞✐❝✲ t✐♦♥❛r② ❝♦❞❡r ❣r❡❡❞✐•② tr❛♥s•❛t❡s ❢r❡q✉❡♥t•② ♦❜s❡r✈❡❞ ♣❛tt❡r♥s ✐♥t♦ ♥❡✇ ❝♦❞❡s✳ ❚❤❡ ❜②t❡ ❝♦❞❡ ✐♥t❡r♣r❡t❡r ✐♠♣•❡♠❡♥ts ♠❡r❣❡❞ r✉•❡s ❛❝❝♦r❞✐♥❣•②✳ ❋♦r ❡①❛♠♣•❡✱ t❤❡ ❜②t❡ ❝♦❞❡ s❡q✉❡♥❝❡ ✧✸✱ ✵✧ ✭❢♦r ❛♥ ❛❞❞✐t✐♦♥ ❢♦••✇❡❞ ❜② ❛ s✇❛♣ ♦❢ ✈❛r✐❛❜•❡ || ♥➦ ✼✵✹✶ ✷✷ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ ❝♦♥t❡♥ts✮ ♦❝❝✉rs ✈❡r② ❢r❡q✉❡♥t•② ❛♥❞ ♠❛② ❜❡ tr❛♥s•❛t❡❞ t♦ ❛ ♥❡✇ ❝♦❞❡✱ s❛② 12✱ ❛♥❞ t❤❡ ✐♥t❡r♣r❡t❡r ♣❡r❢♦r♠s ❛ ♠❡r❣❡❞ ❛❞❞✐t✐♦♥ ❛♥❞ s✇❛♣✳ ❚❤❡ ❝♦❞❡s ✧✶✶✱ ✶✵✧ ❛•✇❛②s ♦❝❝✉r ❛s ❛ ♣❛✐r ❛♥❞ ❝❛♥ ❜❡ s✉❜st✐t✉t❡❞ ❡①❝❡♣t ❛t t❤❡ ✈❡r② st❛rt ❛♥❞ t❤❡ ✈❡r② ❡♥❞ ♦❢ t❤❡ ❜②t❡❝♦❞❡✱ ❜✉t t❤❡s❡ t✇♦ ♦❝❝✉rr❡♥❝❡s ❝❛♥ ❜❡ ❤❛r❞✲❝♦❞❡❞ ✐♥t♦ t❤❡ ✐♥t❡r♣r❡t❡r✱ s♦ t❤❡② ❞♦ ♥♦t ♥❡❡❞ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ✐♥❞✐✈✐❞✉❛••② ❛t ❛••✳ ❙✐♥❝❡ ✇❡ ♦❢t❡♥ ❝❛♥ ❝❤♦♦s❡ ❛♠♦♥❣ s❡✈❡r❛• ❞✐✛❡r❡♥t ▲✉❝❛s ❝❤❛✐♥s ♦❢ ❡q✉❛• •❡♥❣t❤ ❢♦r ❛ ❣✐✈❡♥ st❛❣❡ ✶ ♣r✐♠❡ ❜② ✉s✐♥❣ ❞✐✛❡r❡♥t ♠✉•t✐♣•✐❡rs ✐♥ P|❆❈✱ ✇❡ ❝❛♥ ♣✐❝❦ ♦♥❡ t❤❛t •❡❛❞s t♦ t❤❡ s✐♠♣•❡st ❝♦♠♣r❡ss❡❞ ❝♦❞❡ ❜② ❝♦♠♣r❡ss✐♥❣ ❡❛❝❤ ❝❛♥❞✐❞❛t❡ ❝❤❛✐♥✱ ❛♥❞ ❝❤♦♦s✐♥❣ t❤❡ ♦♥❡ t❤❛t ❤❛s t❤❡ s♠❛••❡st ♥✉♠❜❡r ♦❢ ❝♦❞❡ ❜②t❡s ❛♥❞ ❝♦❞❡ ❜②t❡ ❝❤❛♥❣❡s✳ ❋♦r ❝♦♠♣❛r✐s♦♥✱ ✇✐t❤♦✉t ❛♥② ❝♦♠♣r❡ss✐♦♥ ♦r ❡✛♦rt t♦ r❡❞✉❝❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦❞❡ ❜②t❡s ♦r ❝♦❞❡ ❝❤❛♥❣❡s ✇❤❡♥ ❝❤♦♦s✐♥❣ P|❆❈ ♠✉•t✐♣•✐❡rs✱ t❤❡ ❜②t❡ ❝♦❞❡ ❢♦r ❛ st❛❣❡ ✶ ✇✐t❤ B 1 = 500 ❝♦♥s✐sts ♦❢ 1487 ❝♦❞❡ ❜②t❡s ❛♥❞ 1357 ❝♦❞❡ ❝❤❛♥❣❡s✱ ✇❤❡r❡❛s ❡✈❡♥ ✇✐t❤ t❤❡ s✐♠♣•❡ s✉❜st✐t✉t✐♦♥ r✉•❡s ❞❡s❝r✐❜❡❞ ❛❜♦✈❡ ❛♥❞ ❝❛r❡❢✉• ❝❤♦✐❝❡ ♦❢ P|❆❈ ♠✉•t✐♣•✐❡rs t♦ ♠✐♥✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦❞❡ ❜②t❡s ❛♥❞ ❝♦❞❡ ❝❤❛♥❣❡s✱ ♦♥•② 554 ❝♦❞❡ ❜②t❡s ✇✐t❤ 435 ❝♦❞❡ ❝❤❛♥❣❡s r❡♠❛✐♥✳ ✺✳✸ P✰✶ st❛❣❡ ✶ ♣❡r❢♦r♠❛♥❝❡ ❚❛❜•❡ ✸ ❝♦♠♣❛r❡s t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ P✰✶ st❛❣❡ ✶ ✐♠♣•❡♠❡♥t❛t✐♦♥ ❢♦r ❞✐❢✲ ❢❡r❡♥t B 1 ✈❛•✉❡s ❛♥❞ ♠♦❞✉•✉s s✐③❡s ♦♥ ❆▼❉ P❤❡♥♦♠ ❛♥❞ ■♥t❡• ❈♦r❡ ✷ ❈P❯s✳ ❈♦r❡ ✷ P❤❡♥♦♠ B 1 1 ✇♦r❞ 2 ✇♦r❞s -2 ❜✐ts 1 ✇♦r❞ 2 ✇♦
❚❛❜•❡ ✸✿ ❚✐♠❡ ✐♥ ♠✐❝r♦s❡❝♦♥❞s ❢♦r P✰✶ st❛❣❡ ✶ ✇✐t❤ ❞✐✛❡r❡♥t B 1 ✈❛•✉❡s ♦♥ 2.146 •❍③ ■♥t❡• ❈♦r❡ ✷ ❛♥❞ 2 •❍③ ❆▼❉ P❤❡♥♦♠ ❈P❯s✱ ✉s✐♥❣ ♣r❡❝♦♠♣✉t❡❞ ▲✉❝❛s ❝❤❛✐♥s st♦r❡❞ ❛s ❝♦♠♣r❡ss❡❞ ❜②t❡ ❝♦❞❡✳ ❋♦r ❝♦♠♣❛r✐s♦♥✱ ✇✐t❤♦✉t ✉s✐♥❣ ❜②t❡ ❝♦❞❡ ❝♦♠♣r❡ss✐♦♥ ♦r ❝❤♦♦s✐♥❣ t❤❡ P|❆❈ ♠✉•t✐♣•✐❡rs t♦ ♠✐♥✐♠✐③❡ ❜②t❡ ❝♦❞❡ •❡♥❣t❤ ❛♥❞ ♥✉♠❜❡r ♦❢ ❝♦❞❡ ❝❤❛♥❣❡s✱ ♦♥ ❈♦r❡ ✷✱ P✰✶ st❛❣❡ ✶ ✇✐t❤ 1 ✇♦r❞ ❛♥❞ B 1 = 500 t❛❦❡s 20.4µs ❛♥❞ s♦ ✐s ❛❜♦✉t 22% s•♦✇❡r✱ ❛♥❞ ✇✐t❤ 2 ✇♦r❞s t❛❦❡s 50.4µs ❛♥❞ s♦ ✐s ❛❜♦✉t 24% s•♦✇❡r✳ ✻ ❊❈▼ ❛•❣♦r✐t❤♠ ❚❤❡ ❊••✐♣t✐❝ ❈✉r✈❡ ▼❡t❤♦❞ ♦❢ ❢❛❝t♦r✐③❛t✐♦♥ ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② ❍✳ ❲✳ ▲❡♥str❛ ✐♥ ✶✾✽✼ ❬✶✻(✳ ❲❤❡r❡❛s P✕✶ ✇♦r❦s ✐♥ t❤❡ ❣r♦✉♣ F * p ♦❢ ♦r❞❡r p -1 ❛♥❞ P✰✶ ✐♥ ❛ s✉❜❣r♦✉♣ ♦❢ F * p 2 ♦❢ ♦r❞❡r p -1 ♦r p + 1✱ ❊❈▼ ✇♦r❦s ✐♥ t❤❡ ▼♦r❞❡••✲❲❡✐• ❣r♦✉♣ ♦❢ ♣♦✐♥ts ♦♥ ❛♥ ❡••✐♣t✐❝ ❝✉r✈❡ ❞❡✜♥❡❞ ♦✈❡r F p . ❇② ❍❛ss❡✬s t❤❡♦r❡♠✱ t❤❡ ♥✉♠❜❡r ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✷✸ ♦❢ ♣♦✐♥ts ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ ♦r❞❡r ♦❢ t❤❡ ▼♦r❞❡••✲❲❡✐• ❣r♦✉♣ ♦❢ ❛♥ ❡••✐♣t✐❝ ❝✉r✈❡ ♦✈❡r F p ✐s ✐♥ [p + 1 -2 √ p, p + 1 + 2 √ p]. ❚❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦♥ ❛ ♣❛rt✐❝✉•❛r ❝✉r✈❡ ❞❡♣❡♥❞s ♦♥ ❜♦t❤ t❤❡ ❝✉r✈❡ ♣❛r❛♠❡t❡rs ❛♥❞ t❤❡ ✜❡•❞✳ ❊❈▼ ✜♥❞s ❛ ♣r✐♠❡ ❢❛❝t♦r p ♦❢ N ✐❢ t❤❡ ❝✉r✈❡ ♦✈❡r F p ❤❛s s♠♦♦t❤ ♦r❞❡r❀ t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ❊❈▼ ♦✈❡r ♣r❡✈✐♦✉s ❛•❣♦r✐t❤♠s s✉❝❤ ❛s P✕✶ ❛♥❞ P✰✶ ✭✇❤✐❝❤ ❛•✇❛②s ✇♦r❦ ✐♥ ❛ ❣r♦✉♣ ♦❢ ♦r❞❡r p -1 ♦r p + 1✮ ✐s t❤❛t ♠❛♥② ❞✐✛❡r❡♥t ❝✉r✈❡s ❝❛♥ ❜❡ tr✐❡❞✱ ✉♥t✐• ♦♥❡ ✇✐t❤ s✉✣❝✐❡♥t•② s♠♦♦t❤ ♦r❞❡r ❝♦♠❡s ❛•♦♥❣✳ ❆♥② ❡••✐♣t✐❝ ❝✉r✈❡ E ♦✈❡r ❛ ✜❡•❞ K ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ ♥❡✐t❤❡r 2 ♥♦r 3 ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② t❤❡ ❲❡✐❡rstr❛ÿ ❡q✉❛t✐♦♥ y 2 = x 3 + ax + b. ✭✷✮ ❚❤✐s ❡q✉❛t✐♦♥ ❞❡✜♥❡s ❛♥ ❡••✐♣t✐❝ ❝✉r✈❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ t❤❡ ❞✐s❝r✐♠✐♥❛♥t 4a 3 + 27b 2 ❞♦❡s ♥♦t ✈❛♥✐s❤✳ ❚❤❡ s❡t ♦❢ ♣♦✐♥ts ♦♥ E ❝♦♥s✐sts ♦❢ t❤❡ s♦•✉t✐♦♥s (x, y) ∈ K 2 ♦❢ ✭✷✮✱ ♣•✉s t❤❡ ♣♦✐♥t ❛t ✐♥✜♥✐t② O. ❚❤❡ ❣r♦✉♣ ❛❞❞✐t✐♦♥ •❛✇ ♦❢ t✇♦ ♣♦✐♥ts ♦♥ t❤❡ ❝✉r✈❡ ✐s ❞❡✜♥❡❞ ❣❡♦♠❡tr✐❝❛••② ❜② ♣✉tt✐♥❣ ❛ str❛✐❣❤t •✐♥❡ t❤r♦✉❣❤ t❤❡ t✇♦ ♣♦✐♥ts ✭♦r✱ ✐❢ t❤❡ ♣♦✐♥ts ❛r❡ ✐❞❡♥t✐❝❛•✱ t❤❡ t❛♥❣❡♥t ♦❢ t❤❡ ❝✉r✈❡ ✐♥ t❤❛t ♣♦✐♥t✮✱ t❛❦✐♥❣ t❤❡ •✐♥❡✬s ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥t ✇✐t❤ t❤❡ ❝✉r✈❡ ❛♥❞ ♠✐rr♦r✐♥❣ ✐t ❛t t❤❡ x✲❛①✐s✳ ❙✐♥❝❡ t❤❡ ❝✉r✈❡ ✐s s②♠♠❡tr✐❝ ❛r♦✉♥❞ t❤❡ x✲❛①✐s✱ t❤❡ ♠✐rr♦r❡❞ ♣♦✐♥t ✐s ♦♥ t❤❡ ❝✉r✈❡✱ ❛♥❞ ✐s t❤❡ r❡s✉•t✳ ■❢ t❤❡ str❛✐❣❤t •✐♥❡ ✐s ✈❡rt✐❝❛•✱ ♥♦ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥t ❡①✐sts❀ ✐♥ t❤✐s ❝❛s❡ t❤❡ ♣♦✐♥t ❛t ✐♥✜♥✐t② ✐s t❛❦❡♥ ❛s t❤❡ r❡s✉•t✳ ❚❤❡ ♣♦✐♥t ❛t ✐♥✜♥✐t② ✐s ✐❞❡♥t✐t② ❡•❡♠❡♥t ♦❢ t❤❡ ❣r♦✉♣✱ ❛❞❞✐♥❣ ✐t t♦ ❛♥② ♣♦✐♥t r❡s✉•ts ✐♥ t❤❡ s❛♠❡ ♣♦✐♥t✳ ❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ♣♦✐♥t ✐s t❤❡ ♣♦✐♥t ♠✐rr♦r❡❞ ❛t t❤❡ x✲❛①✐s✳ ❚❤✐s ❛❞❞✐t✐♦♥ •❛✇ ♦♥ t❤❡ ♣♦✐♥ts ♦❢ ❛♥ ❡••✐♣t✐❝ ❝✉r✈❡ ❞❡✜♥❡s ❛♥ ❆❜❡•✐❛♥ ❣r♦✉♣✱ s❡❡ ❢♦r ❡①❛♠♣•❡ ❬✷✻(✳ ❚❤❡ ❲❡✐❡rstr❛ÿ ❢♦r♠ ♦❢ ❡••✐♣t✐❝ ❝✉r✈❡s ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ✐♠♣•❡♠❡♥t✐♥❣ ❊❈▼✱ ❜✉t r❡q✉✐r❡s ❛ ❝♦st•② ♠♦❞✉•❛r ✐♥✈❡rs❡ ✐♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ♣♦✐♥t ❛❞❞✐t✐♦♥s✳ ▼♦♥t❣♦♠❡r② ❬✶✽( ♣r♦♣♦s❡s ❛♥ ❛•t❡r♥❛t✐✈❡ ❢♦r♠ ♦❢ ❡••✐♣t✐❝ ❝✉r✈❡ ❡q✉❛t✐♦♥ ✐♥ ♣r♦✲ •❡❝t✐✈❡ ❝♦♦r❞✐♥❛t❡s s♦ t❤❛t ✐ts ❛❞❞✐t✐♦♥ •❛✇ ❛✈♦✐❞s ♠♦❞✉•❛r ✐♥✈❡rs❡s✱ ✇❤✐•❡ st✐•• ❦❡❡♣✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ r❡q✉✐r❡❞ ♠✉•t✐♣•✐❝❛t✐♦♥s •♦✇✳ ❍✐s ❝✉r✈❡s ❛r❡ ♦❢ ❢♦r♠ BY 2 Z = X(X 2 + AXZ + Z 2 ), ✭✸✮ ✇✐t❤ ♣♦✐♥ts (X : Y : Z) ∈ K 3 s❛t✐s❢②✐♥❣ ✭✸✮✱ ✇❤❡r❡ X, Y, Z ❛r❡ ♥♦t ❛•• ③❡r♦✳ ❚✇♦ ♣♦✐♥ts ❛r❡ ✐❞❡♥t✐❝❛• ✐❢ (X 2 : Y 2 : Z 2 ) = (kX 1 : kY 1 : kZ 1 ) ❢♦r s♦♠❡ k ∈ K✱ k = 0. ❚❤❡ ♣♦✐♥t ❛t ✐♥✜♥✐t② ✐s O = (0 : 1 : 0). ◆♦t ❛•• ❡••✐♣t✐❝ ❝✉r✈❡s ♦✈❡r ✜♥✐t❡ ✜❡•❞s ❝❛♥ ❜❡ ❜r♦✉❣❤t ✐♥t♦ ❢♦r♠ ✭✸✮✱ ❜✉t ✇❡ ♠❛② r❡str✐❝t ♦✉r ❊❈▼ ✐♠♣•❡♠❡♥t❛t✐♦♥ t♦ ✉s❡ ♦♥•② t❤❡s❡ ❝✉r✈❡s✳ ▼♦♥t❣♦♠❡r② ❞❡s❝r✐❜❡s ❛♥ ❛❞❞✐t✐♦♥ •❛✇ ❢♦r ❝✉r✈❡s ♦❢ t❤✐s ❢♦r♠✳ •✐✈❡♥ t✇♦ ❞✐st✐♥❝t ♣♦✐♥ts P 1 ❛♥❞ P 2 ✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ X ❛♥❞ Z✲❝♦♦r❞✐♥❛t❡s ♦❢ P 1 + P 2 ❢r♦♠ t❤❡ X ❛♥❞ Z✲❝♦♦r❞✐♥❛t❡s ♦❢ P 1 ✱ P 2 ❛♥❞ P 1 -P 2 . ❙✐♠✐•❛r•②✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ X ❛♥❞ Z✲ ❝♦♦r❞✐♥❛t❡s ♦❢ 2P ❢r♦♠ t❤❡ X ❛♥❞ Z✲❝♦♦r❞✐♥❛t❡s ♦❢ P ❛♥❞ t❤❡ ❝✉r✈❡ ♣❛r❛♠❡t❡rs✳ ❙✉r♣r✐s✐♥❣•②✱ t❤❡ Y ✲❝♦♦r❞✐♥❛t❡ ✐s ♥♦t ♥❡❡❞❡❞ ✐♥ t❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s✱ ❛♥❞ ❝❛♥ ❜❡ ✐❣♥♦r❡❞ ❡♥t✐r❡•② ✇❤❡♥ ✉s✐♥❣ ❝✉r✈❡s ✐♥ ▼♦♥t❣♦♠❡r② ❢♦r♠ ❢♦r ❊❈▼✱ ❛♥❞ ♣♦✐♥ts ❛r❡ ❝♦♠♠♦♥•② ✇r✐tt❡♥ ❛s ♦♥•② (X :: Z) ✇✐t❤ Y ✲❝♦♦r❞✐♥❛t❡ ♦♠✐tt❡❞✳ ❚❤❡ ❞❡t❛✐•s ♦❢ t❤❡ ❛❞❞✐t✐♦♥ •❛✇ ❛r❡ ❢♦✉♥❞ ✐♥ ❬✶✽✱ ✶✵✳✸✳✶( ♦r ❬✷✵✱ ✷✳✸(✳ ❚❤✐s ❛❞❞✐t✐♦♥ •❛✇ r❡q✉✐r❡s t❤❛t ✐♥ ♦r❞❡r t♦ ❢♦r♠ t❤❡ s✉♠ ♦❢ t✇♦ ♣♦✐♥ts✱ t❤❡✐r ❞✐✛❡r❡♥❝❡ ✐s ❦♥♦✇♥ ♦r ③❡r♦✳ ❚❤✐s ✐s r❡♠✐♥✐s❝❡♥t ♦❢ t❤❡ P✰✶ ♠❡t❤♦❞ ✇❤❡r❡ ✇❡ ♥❡❡❞ V m-n (x) t♦ ❝♦♠♣✉t❡ V m+n (x) ❢r♦♠ V m (x) ❛♥❞ V n (x), ❛♥❞ t❤❡ s❛♠❡ ▲✉❝❛s ❝❤❛✐♥s ✉s❡❞ t♦ ❝♦♠♣✉t❡ V k (x) ❢♦r ✐♥t❡❣❡r k ✐♥ P✰✶ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ♠✉•t✐♣•❡ kP ♦❢ ❛ ♣♦✐♥t P ♦♥ ❛ ❝✉r✈❡ ✐♥ ▼♦♥t❣♦♠❡r② ❢♦r♠ ✐♥ ❊❈▼✳ || ♥➦ ✼✵✹✶ ✷✹ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ ✻✳✶ ❊❈▼ st❛❣❡ ✶ ■♥ st❛❣❡ ✶ ♦❢ ❊❈▼✱ ✇❡ ❝❤♦♦s❡ ❛ s✉✐t❛❜•❡ ❝✉r✈❡ E ♦❢ ❢♦r♠ ✭✸✮ ❞❡✜♥❡❞ ♦✈❡r Z/N Z✱ ✇❤❡r❡ N ✐s t❤❡ ✐♥t❡❣❡r ✇❡ ✇✐s❤ t♦ ❢❛❝t♦r✳ ◆❛t✉r❛••② N ✐s ❝♦♠♣♦s✐t❡✱ s♦ Z/N Z ✐s ❛ r✐♥❣ ❜✉t ♥♦t ❛ ✜❡•❞✱ ❜✉t t❤✐s ❤❛s •✐tt•❡ ❝♦♥s❡q✉❡♥❝❡ ❢♦r t❤❡ ❛r✐t❤♠❡t✐❝ ♦❢ t❤❡ ❝✉r✈❡ ❛s t❤❡ ♦♥•② ♦♣❡r❛t✐♦♥ t❤❛t ❝♦✉•❞ ❢❛✐• ✐s ✐♥✈❡rs✐♦♥ ♦❢ ❛ r✐♥❣ ❡•❡♠❡♥t✱ ❛♥❞ ❛♥ ✉♥s✉❝❝❡ss❢✉• ✐♥✈❡rs✐♦♥ ♦❢ ❛ ♥♦♥✲③❡r♦ ❡•❡♠❡♥t ✐♥ Z/N Z r❡✈❡❛•s ❛ ♣r♦♣❡r ❢❛❝t♦r ♦❢ N ✇❤✐❝❤ ✐s t❤❡ ❡①❛❝t ♣✉r♣♦s❡ ♦❢ ❊❈▼✳ ❲❡ ♦❢t❡♥ ❝♦♥s✐❞❡r t❤❡ ❝✉r✈❡ E p ❢♦r ❛ ♣r✐♠❡ p | N ✱ ✇❤✐❝❤ ✐s t❤❡ ❝✉r✈❡ E r❡❞✉❝❡❞ ♠♦❞✉•♦ p, ✐✳❡✳✱ E ♦✈❡r t❤❡ ✜❡•❞ F p . ❲❡ t❤❡♥ ❝❤♦♦s❡ ❛ ♣♦✐♥t P 0 ♦♥ E ❛♥❞ ❝♦♠♣✉t❡ P 1 = e • P 0 ❢♦r ❛ ❤✐❣❤•② ❝♦♠♣♦s✐t❡ ✐♥t❡❣❡r e✱ ✉s✉❛••② t❛❦❡♥ t♦ ❜❡ ❞✐✈✐s✐❜•❡ ❜② ❛•• ♣r✐♠❡s ❛♥❞ ♣r✐♠❡ ♣♦✇❡rs ✉♣ t♦ ❛ s✉✐t❛❜•② ❝❤♦s❡♥ ✈❛•✉❡ B 1 , ✐✳❡✳✱ e = lcm(1, 2, 3, 4, . . . , B 1 ). ❲❡ ❤♦♣❡ t❤❛t ❢♦r s♦♠❡ ♣r✐♠❡ ❢❛❝t♦r p ♦❢ N ✱ t❤❡ ♦r❞❡r ♦❢ P 0 ♦♥ E p ✐s B 1 ✲s♠♦♦t❤ ✭❛♥❞ t❤✉s ❞✐✈✐❞❡s e✮✱ s✐♥❝❡ t❤❡♥ t❤❡ ♣♦✐♥t P 1 ♦♥ E p ✇✐•• ❜❡ t❤❡ ♣♦✐♥t ❛t ✐♥✜♥✐t② (0 :: 0) s♦ t❤❛t P 1 ❤❛s Z✲❝♦♦r❞✐♥❛t❡ 0 (mod p) ❛♥❞ p | gcd(P Z , N ). ❚♦ ✜♥❞ ❛ ♣♦✐♥t P 0 ♦♥ E ♦✈❡r Z/N Z✱ ✇❡ ❝❤♦♦s❡ ❛ ♣♦✐♥t ♦❢ E ♦✈❡r Q ❛♥❞ ♠❛♣ ✐t t♦ Z/N Z. ❚❤❡ ♣♦✐♥t ♦✈❡r Q ♠✉st ♥♦t ❜❡ ❛ t♦rs✐♦♥ ♣♦✐♥t✱ ♦r P 0 ✇✐•• ❤❛✈❡ ✐❞❡♥t✐❝❛• ♦r❞❡r ♦♥ E p ❢♦r ❛•• p | N s♦ t❤❛t P 1 ✐s t❤❡ ♣♦✐♥t ❛t ✐♥✜♥✐t② ❡✐t❤❡r ❢♦r ❛•• E p ♦r ❢♦r ♥♦♥❡✱ ♣r♦❞✉❝✐♥❣ ♦♥•② t❤❡ tr✐✈✐❛• ❢❛❝t♦r✐③❛t✐♦♥s N ♦r 1✳ ❇② ❝❛r❡❢✉• s❡•❡❝t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡ E ✇❡ ❝❛♥ ❡♥s✉r❡ t❤❛t ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦❢ E p ✐s ❛ ♠✉•t✐♣•❡ ♦❢ 12 ♦r 16, s✐❣♥✐✜❝❛♥t•② ✐♥❝r❡❛s✐♥❣ t❤❡ ♣r♦❜❛❜✐•✐t② t❤❛t t❤❡ ♦r❞❡r ♦❢ P 0 ✐s s♠♦♦t❤✳ ❚❤❡ ❝❤♦✐❝❡ ♦❢ E ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✻✳✷✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ P 1 = e • P 0 ♦♥ E ✐s ❝❛rr✐❡❞ ♦✉t ❜② ✉s❡ ♦❢ ♣r❡✲❝♦♠♣✉t❡❞ ▲✉❝❛s ❝❤❛✐♥s✱ s✐♠✐•❛r•② ❛s ✐♥ t❤❡ P✰✶ ❛•❣♦r✐t❤♠✳ ❚❤❡ s❡•❡❝t✐♦♥ ♦❢ ♥❡❛r✲♦♣t✐♠❛• ▲✉❝❛s ❝❤❛✐♥s ❢♦r ❊❈▼ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✻✳✸✳ ■❢ st❛❣❡ ✶ ♦❢ ❊❈▼ ✐s ✉♥s✉❝❝❡ss❢✉•✱ ✇❡ tr② st❛❣❡ ✷ ✇❤❡r❡ ✇❡ ❤♦♣❡ t♦ ✜♥❞ ❛ ♣r✐♠❡ p s✉❝❤ t❤❛t t❤❡ ♦r❞❡r ♦❢ P 0 ♦♥ E p ❢❛❝t♦rs ✐♥t♦ ♣r✐♠❡s ❛♥❞ ♣r✐♠❡ ♣♦✇❡rs ✉♣ t♦ B 1 , ❡①❝❡♣t ❢♦r ♦♥❡ ❜✐❣❣❡r ✭❜✉t ♥♦t t♦♦ ♠✉❝❤ ❜✐❣❣❡r✮ ♣r✐♠❡ q. ❖✉r st❛❣❡ ✷ ✐s ✈❡r② s✐♠✐•❛r ❢♦r P✕✶✱ P✰✶✱ ❛♥❞ ❊❈▼ ❛♥❞ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✼✳ ✻✳✷ ❈❤♦✐❝❡ ♦❢ ❝✉r✈❡ ■♥ ❛ •❡tt❡r t♦ |✐❝❤❛r❞ ❇r❡♥t✱ ❍✐r♦♠✐ ❙✉②❛♠❛ ❬✷✼( s❤♦✇❡❞ t❤❛t ❝✉r✈❡s ♦❢ ❢♦r♠ ✭✸✮ ♦✈❡r F p ❛•✇❛②s ❤❛✈❡ ❣r♦✉♣ ♦r❞❡r ❞✐✈✐s✐❜•❡ ❜② 4✱ ❛♥❞ ❛•s♦ s❤♦✇❡❞ ❛ ♣❛r❛♠❡tr✐③❛✲ t✐♦♥ t❤❛t ❡♥s✉r❡s t❤❛t t❤❡ ❣r♦✉♣ ♦r❞❡r ✐s ❞✐✈✐s✐❜•❡ ❜② 12✱ ✇❤✐❝❤ ❇r❡♥t ❞❡s❝r✐❜❡s ✐♥ ❬✸(✳ ❚❤✐s ♣❛r❛♠❡tr✐③❛t✐♦♥ ❣❡♥❡r❛t❡s ❛♥ ✐♥✜♥✐t❡ ❢❛♠✐•② ♦❢ ❝✉r✈❡s ♦✈❡r Q ✇❤✐❝❤ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❣❡♥❡r❛t❡ ❛ •❛r❣❡ ♥✉♠❜❡r ♦❢ ❞✐st✐♥❝t ❝✉r✈❡s ♠♦❞✉•♦ N ✳ ❋♦r ❛ ❣✐✈❡♥ ✐♥t❡❣❡r ♣❛r❛♠❡t❡r σ = 0, 1, 3, 5, •❡t u = σ 2 -5, v = 4σ, X 0 = u 3 , Z 0 = v 3 ❛♥❞ A = (v-u) 3 (3u+v) 4u 3 v -2. ✭✹✮ ❚❤❡♥ t❤❡ ♣♦✐♥t (X 0 :: Z 0 ) ✐s ♦♥ t❤❡ ❝✉r✈❡ ✭✸✮ ✇✐t❤ ♣❛r❛♠❡t❡r A. ❚❤❡ s❛♠❡ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✐s ✉s❡❞ ❜② •▼P✲❊❈▼ ❬✷✾✱ ✶( ❛♥❞ Pr✐♠❡✾✺ ❬✷✽(✳ ▼♦♥t❣♦♠❡r② s❤♦✇❡❞ ✐♥ ❤✐s t❤❡s✐s ❬✷✵( ❤♦✇ t♦ ❝❤♦♦s❡ ❝✉r✈❡s ♦❢ ❢♦r♠ ✸ s✉❝❤ t❤❛t t❤❡ ❝✉r✈❡ ♦✈❡r Q ❤❛s ❛ t♦rs✐♦♥ s✉❜❣r♦✉♣ ♦❢ ♦r❞❡r 12 ♦r 16, •❡❛❞✐♥❣ t♦ ❣r♦✉♣ ♦r❞❡r ❞✐✈✐s✐❜•❡ ❜② 12 ♦r 16, r❡s♣❡❝t✐✈❡•②✱ ✇❤❡♥ t❤❡ ❝✉r✈❡ ✐s ♠❛♣♣❡❞ t♦ F p ❢♦r ❛•♠♦st ❛•• p. ❋♦r ❝✉r✈❡s ✇✐t❤ r❛t✐♦♥❛• t♦rs✐♦♥ ❣r♦✉♣ ♦❢ ♦r❞❡r 12 ❤❡ ✉s❡s t 2 = u 2 -12 4u , a = t 2 -1 t 2 +3 X 0 = 3a 2 + 1, Z 0 = 4a ❛♥❞ A = -3a 4 -6a 2 +1 4a 3 , ✭✺✮ ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✷✺ ✇❤❡r❡ u 3 -12u ✐s ❛ r❛t✐♦♥❛• sq✉❛r❡✳ ❚❤❡ s♦•✉t✐♦♥s ♦❢ v 2 = u 3 -12u ❢♦r♠ ❛♥ ❡••✐♣t✐❝ ❝✉r✈❡ ♦❢ r❛♥❦ 1 ❛♥❞ 2✲t♦rs✐♦♥ ♦✈❡r Q✱ ✇✐t❤ ❣❡♥❡r❛t♦r (-2, 4) ❛♥❞ ✷✲ t♦rs✐♦♥ ♣♦✐♥t (0, 0). ❍♦✇❡✈❡r✱ ❛❞❞✐♥❣ t❤❡ t♦rs✐♦♥ ♣♦✐♥t ♦r ♥♦t s❡❡♠s t♦ ♣r♦❞✉❝❡ ✐s♦♠♦r♣❤✐❝ ❝✉r✈❡s ❢♦r ❊❈▼✱ s♦ ✇❡ ✐❣♥♦r❡ ✐t✳ ❍❡♥❝❡ ❢♦r ❛ ❣✐✈❡♥ ✐♥t❡❣❡r ♣❛r❛♠❡t❡r k > 1 ✇❡ ❝❛♥ ❝♦♠♣✉t❡ s✉✐t❛❜•❡ ✈❛•✉❡s ♦❢ u ❛♥❞ v ❜② ❝♦♠♣✉t✐♥❣ k • (-2, 4) ♦♥ v 2 = u 3 -12u. ❲❡ ❝❛♥ t❤❡♥ •❡t t = v/(2u). ❚❤✐s ♣r♦❞✉❝❡s ❛♥ ✐♥✜♥✐t❡ ❢❛♠✐•② ♦❢ ❝✉r✈❡s ♦✈❡r Q. ❈✉r✈❡s ✇✐t❤ t♦rs✐♦♥ 16 ❛♥❞ ♣♦s✐t✐✈❡ r❛♥❦ ♦✈❡r Q ❛r❡ ♠♦r❡ ❞✐✣❝✉•t t♦ ❣❡♥❡r✲ ❛t❡✱ s❡❡ ❬✷✵✱ ✻✳✷( ❢♦r ❞❡t❛✐•s✳ ❲❡ ❝✉rr❡♥t•② ✐♠♣•❡♠❡♥t ♦♥•② ♦♥❡ s✉❝❤ ❝✉r✈❡ ✇✐t❤ X 0 = 8, Z 0 = 15, ❛♥❞ A = 54721/14400. ❚❤❡s❡ ♣❛r❛♠❡tr✐③❛t✐♦♥s ❡♥s✉r❡ t❤❛t t❤❡ ❣r♦✉♣ ♦r❞❡r ✐s ❞✐✈✐s✐❜•❡ ❜② 12 ♦r 16✱ r❡s♣❡❝t✐✈❡•②✱ ❜✉t t❤❡ r❡s✉•t✐♥❣ ❣r♦✉♣ ♦r❞❡r ♦❢ t❤❡ ❝✉r✈❡ ♦✈❡r F p ❞♦❡s ♥♦t ❜❡❤❛✈❡ •✐❦❡ ❛♥ ✐♥t❡❣❡r ❝❤♦s❡♥ ✉♥✐❢♦r♠•② ❛t r❛♥❞♦♠ ❢r♦♠ t❤❡ ✐♥t❡❣❡rs t❤❛t ❛r❡ ♠✉•t✐♣•❡s ♦❢ 12 ♦r 16, r❡s♣❡❝t✐✈❡•②✱ ✐♥ t❤❡ ❍❛ss❡ ✐♥t❡r✈❛• ❛r♦✉♥❞ p. ■♥ ♣❛rt✐❝✉•❛r✱ t❤❡ ❛✈❡r❛❣❡ ✈❛•✉❛t✐♦♥ ♦❢ 2 ✐♥ t❤❡ ❣r♦✉♣ ♦r❞❡r ❢♦r ❝✉r✈❡s ✇✐t❤ r❛t✐♦♥❛• t♦rs✐♦♥ 12 ✐s 11/3, s•✐❣❤t•② ❤✐❣❤❡r t❤❛♥ 10/3 ❢♦r ❝✉r✈❡s ✐♥ ❇r❡♥t✲❙✉②❛♠❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✭✇❤✐❝❤ ❤❛✈❡ r❛t✐♦♥❛• t♦rs✐♦♥ 6✮✱ ♠❛❦✐♥❣ t❤❡♠ s♦♠❡✇❤❛t ♠♦r❡ •✐❦❡•② t♦ ✜♥❞ ❢❛❝t♦rs✳ ❱❡r② s♠❛•• σ✲✈❛•✉❡s ❢♦r t❤❡ ❇r❡♥t✲❙✉②❛♠❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ •❡❛❞ t♦ ❝✉r✈❡s ✇✐t❤ s✐♠♣•❡ r❛t✐♦♥❛•s ❢♦r t❤❡ ♣♦✐♥t ❝♦♦r❞✐♥❛t❡ ❛♥❞ ❝✉r✈❡ ♣❛r❛♠❡t❡r✱ ❛♥❞ ✈❡r② s♠❛•• k✲✈❛•✉❡s ❢♦r ▼♦♥t❣♦♠❡r②✬s ♣❛r❛♠❡tr✐③❛t✐♦♥ ❢♦r ❝✉r✈❡s ✇✐t❤ r❛t✐♦♥❛• t♦rs✐♦♥ 12 •❡❛❞ t♦ s✐♠♣•❡ r❛t✐♦♥❛•s ❢♦r a, s❡❡ ❚❛❜•❡ ✹✳ ❚❤❡s❡ r❛t✐♦♥❛•s ❝❛♥ ❜❡ ♠❛♣♣❡❞ t♦ Z/N Z ❡❛s✐•②✱ ❛s t❤❡ ❞❡♥♦♠✐♥❛t♦rs ❛r❡ ❤✐❣❤•② ❝♦♠♣♦s✐t❡ ✐♥t❡❣❡rs s♦ t❤❛t t❤❡ r❡q✉✐r❡❞ ❞✐✈✐s✐♦♥s ♠♦❞✉•♦ N ❝❛♥ ❜❡ ❞♦♥❡ ❜② t❤❡ ♠❡t❤♦❞s ♦❢ ❙❡❝t✐♦♥ ✸✳✹ ❛♥❞ ❛ ❢❡✇ ♠✉•t✐♣•✐❝❛t✐♦♥s✳ ❲❤❡♥ ❢❛❝t♦r✐♥❣ ❝♦❢❛❝t♦rs ❛❢t❡r t❤❡ s✐❡✈✐♥❣ st❡♣ ♦❢ ◆❋❙ ✐♥t♦ •❛r❣❡ ♣r✐♠❡s✱ ♦♥•② ✈❡r② ❢❡✇ ❝✉r✈❡s ❛r❡ r❡q✉✐r❡❞ ♦♥ ❛✈❡r❛❣❡ s✐♥❝❡ t❤❡ ♣r✐♠❡s t♦ ❜❡ ❢♦✉♥❞ ❛r❡ r❡•❛t✐✈❡•② s♠❛••✱ ❛♥❞ ✇✐t❤ ❛♥ ❡❛r•②✲❛❜♦rt str❛t❡❣②✱ ♦♥•② t❤❡ ✜rst ❢❡✇ ❝✉r✈❡s ✇♦r❦ ♦♥ •❛r❣❡r ❝♦♠♣♦s✐t❡s ✇❤❡r❡ ❛r✐t❤♠❡t✐❝ ✐s ♠♦r❡ ❡①♣❡♥s✐✈❡✳ ■♥ s♣✐t❡ ♦❢ t❤❡ s♠❛•• ♥✉♠❜❡r ♦❢ ❝✉r✈❡s ✇✐t❤ s✉❝❤ s✐♠♣•❡ r❛t✐♦♥❛•s ❛s ❝✉r✈❡ ♣❛r❛♠❡t❡rs✱ ✐t ✐s ✉s❡❢✉• t♦ ✐♠♣•❡♠❡♥t t❤❡♠ ❛s s♣❡❝✐❛• ❝❛s❡s✳ σ X 0 Z 0 A 2 -
❚❛❜•❡ ✹✿ ❙♦♠❡ ❡••✐♣t✐❝ ❝✉r✈❡s ❝❤♦s❡♥ ❜② t❤❡ ❇r❡♥t✲❙✉②❛♠❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✇✐t❤ ❣r♦✉♣ ♦r❞❡r ❞✐✈✐s✐❜•❡ ❜② 12✱ ❛♥❞ ❜② ▼♦♥t❣♦♠❡r②✬s ♣❛r❛♠❡tr✐③❛t✐♦♥ ✇✐t❤ r❛t✐♦♥❛• t♦rs✐♦♥ ❣r♦✉♣ ♦❢ ♦r❞❡r 12✳ ✻✳✸ ▲✉❝❛s ❝❤❛✐♥s ❢♦r ❊❈▼ ■♥ ♣r✐♥❝✐♣•❡✱ ▲✉❝❛s ❝❤❛✐♥s ❢♦r ❊❈▼ ❝❛♥ ❜❡ ❝❤♦s❡♥ ❡①❛❝t•② ❛s ❢♦r P✰✶✳ ❍♦✇❡✈❡r✱ ❛ s✉❜t•❡ ❞✐✛❡r❡♥❝❡ ❡①✐sts✿ ✐♥ P✰✶✱ t❤❡ ❝♦st ♦❢ ❛ ❞♦✉❜•✐♥❣ V 2n (x) = V n (x) 2 -2 ✐s ✐❞❡♥t✐❝❛• t♦ t❤❛t ♦❢ ❛♥ ❛❞❞✐t✐♦♥ V m+n (x) = V m (x)V n (x) -V m-n ✐❢ V m-n ✐s || ♥➦ ✼✵✹✶ ✷✻ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ ❦♥♦✇♥ ❛♥❞ ❛ sq✉❛r✐♥❣ ✐s t❛❦❡♥ t♦ ❤❛✈❡ t❤❡ s❛♠❡ ❝♦st ❛s ❛ ♠✉•t✐♣•✐❝❛t✐♦♥✳ ❚❤✐s ✇❛②✱ t❤❡ ❝♦st ♦❢ ❛ ▲✉❝❛s ❝❤❛✐♥ ❞❡♣❡♥❞s ♦♥•② ♦♥ ✐ts •❡♥❣t❤✳ ■♥ ❊❈▼✱ t❤❡ ❝♦st ♦❢ ❛ ♣♦✐♥t ❞♦✉❜•✐♥❣ ✉s✉❛••② ❞✐✛❡rs ❢r♦♠ t❤❡ ❝♦st ♦❢ ❛♥ ❛❞✲ ❞✐t✐♦♥ ♦❢ ❞✐st✐♥❝t ♣♦✐♥ts✳ ■♥ t❤❡ ❛❞❞✐t✐♦♥ r✉•❡s ❣✐✈❡♥ ❜② ▼♦♥t❣♦♠❡r②✱ ❛ ❞♦✉❜•✐♥❣ t❛❦❡s 5 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s ♦❢ ✇❤✐❝❤ 2 ❛r❡ sq✉❛r✐♥❣s✱ ✇❤❡r❡❛s ❛♥ ❛❞❞✐t✐♦♥ ♦❢ ❞✐st✐♥❝t ♣♦✐♥ts t❛❦❡s 6 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s ♦❢ ✇❤✐❝❤ ❛❣❛✐♥ 2 ❛r❡ sq✉❛r✐♥❣s✳ ❚❤❡s❡ ❞✐✛❡r❡♥t ❝♦sts ❝❛♥ ❜❡ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✇❤❡♥ ❝❤♦♦s✐♥❣ ▲✉❝❛s ❝❤❛✐♥s✳ ❋♦r ❡①❛♠♣•❡✱ t♦ ♠✉•t✐♣•② ❛ ♣♦✐♥t ❜② 7✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ ❜❡t✇❡❡♥ t❤❡ ❝❤❛✐♥s 1, 2, 3, 5, 7 ♦r 1, 2, 3, 4, 7 ♦❢ ❡q✉❛• •❡♥❣t❤✳ ■♥ t❤❡ ❢♦r♠❡r✱ ❛•• ❛❞❞✐t✐♦♥s ❡①❝❡♣t ❢♦r t❤❡ ✐♥✐t✐❛• ❞♦✉❜•✐♥❣ 1, 2 ❛r❡ ❛❞❞✐t✐♦♥s ♦❢ ❞✐st✐♥❝t ✈❛•✉❡s✳ ■♥ t❤❡ •❛tt❡r✱ ✇❡ ❝❛♥ ♣r♦❞✉❝❡ 4 ❜② ❞♦✉❜•✐♥❣ 2, s♦ t❤❛t t❤✐s ▲✉❝❛s ❝❤❛✐♥ ✇♦✉•❞ s❛✈❡ 1 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥ ✐♥ t❤❡ ❡••✐♣t✐❝ ❝✉r✈❡ ❛r✐t❤♠❡t✐❝✳ ❲❤❡♥ ❣❡♥❡r❛t✐♥❣ ▲✉❝❛s ❝❤❛✐♥s ✇✐t❤ P|❆❈ ✉s✐♥❣ s❡✈❡r❛• ♠✉•t✐♣•✐❡rs✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ t❤❡ ❜❡st ❝❤❛✐♥ ♥♦t ❛❝❝♦r❞✐♥❣ t♦ ✐ts •❡♥❣t❤ ❜✉t ❜② t❤❡ ❝♦st ♦❢ t❤❡ ❛r✐t❤♠❡t✐❝ ♣❡r❢♦r♠❡❞ ✐♥ ❡❛❝❤ P|❆❈ r✉•❡ t❤❛t ✐s ✉s❡❞ t♦ ❜✉✐•❞ t❤❡ ❝❤❛✐♥✳ ❚❤❡ s♣❡❡❞✉♣ ✐♥ ♣r❛❝t✐❝❡ ✐s r❡•❛t✐✈❡•② s♠❛••✿ ✇✐t❤ t✇♦✲✇♦r❞ ♠♦❞✉•✉s✱ ❊❈▼ st❛❣❡ ✶ ✇✐t❤ B 1 = 500 ✐s ❛❜♦✉t 1% ❢❛st❡r ✇❤❡♥ ❝♦✉♥t✐♥❣ t❤❡ ❝♦st ♦❢ ❛ ❞♦✉❜•✐♥❣ ❛s 5/6 ♦❢ t❤❡ ❝♦✉♥t ♦❢ ❛♥ ❛❞❞✐t✐♦♥ ✇❤❡♥ ❝❤♦♦s✐♥❣ ▲✉❝❛s ❝❤❛✐♥s✳ ❙t✐••✱ t❤✐s ✐♠♣r♦✈❡♠❡♥t ✐s s♦ s✐♠♣•❡ t♦ ✐♠♣•❡♠❡♥t t❤❛t ✐t ♠❛② ❜❡ ❝♦♥s✐❞❡r❡❞ ✇♦rt❤✇❤✐•❡✳ ❆s ❢♦r P✰✶✱ t❤❡ ♣r❡❝♦♠♣✉t❡❞ ❛❞❞✐t✐♦♥ ❝❤❛✐♥s ❛r❡ st♦r❡❞ ❛s ❜②t❡ ❝♦❞❡ t❤❛t ❞❡s❝r✐❜❡s ❛ s❡q✉❡♥❝❡ ♦❢ P|❆❈ r✉•❡s t♦ ❛♣♣•②✳ ❈♦❞❡ ❝♦♠♣r❡ss✐♦♥ ♠❛② ❜❡ ✉s❡❞ t♦ r❡❞✉❝❡ t❤❡ ♦✈❡r❤❡❛❞ ✐♥ t❤❡ ❜②t❡ ❝♦❞❡ ✐♥t❡r♣r❡t❡r✱ ❜✉t s✐♥❝❡ t❤❡ ❡••✐♣t✐❝ ❝✉r✈❡ ❛r✐t❤♠❡t✐❝ ✐s ♠♦r❡ ❡①♣❡♥s✐✈❡ t❤❛♥ ✐♥ t❤❡ ❝❛s❡ ♦❢ P + 1, t❤❡ r❡•❛t✐✈❡ s♣❡❡❞✉♣ ❣❛✐♥❡❞ ❜② ❝♦♠♣r❡ss✐♦♥ ✐s ♠✉❝❤ s♠❛••❡r✳ ✻✳✹ ❊❈▼ st❛❣❡ ✶ ♣❡r❢♦r♠❛♥❝❡ ❚❛❜•❡ ✺ ❝♦♠♣❛r❡s t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ❊❈▼ st❛❣❡ ✶ ✐♠♣•❡♠❡♥t❛t✐♦♥ ❢♦r ❞✐✛❡r❡♥t B 1 ✈❛•✉❡s ❛♥❞ ♠♦❞✉•✉s s✐③❡s ♦♥ ❆▼❉ P❤❡♥♦♠ ❛♥❞ ■♥t❡• ❈♦r❡ ✷ ❈P❯s✳ ❈♦r❡ ✷ P❤❡♥♦♠ B 1 1 ✇♦r❞ 2 ✇♦r❞s -2 ❜✐ts 1 ✇♦r❞ 2 ✇♦
❚❛❜•❡ ✺✿ ❚✐♠❡ ✐♥ ♠✐❝r♦s❡❝♦♥❞s ❢♦r ❊❈▼ st❛❣❡ ✶ ✇✐t❤ ❞✐✛❡r❡♥t B 1 ✈❛•✉❡s ♦♥ 2.146 •❍③ ❈♦r❡ ✷ ❛♥❞ 2 •❍③ ❆▼❉ P❤❡♥♦♠ ❈P❯s ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✷✼ ✼ ❙t❛❣❡ ✷ ❢♦r P✕✶✱ P✰✶✱ ❛♥❞ ❊❈▼ ❙t❛❣❡ ✶ ♦❢ P✕✶✱ P✰✶✱ ❛♥❞ ❊❈▼ ❛•• ❝♦♠♣✉t❡ ❛♥ ❡•❡♠❡♥t g e 0 ♦❢ s♦♠❡ ✭♠✉•t✐♣•✐❝❛✲ t✐✈❡•② ✇r✐tt❡♥✮ ❣r♦✉♣ G ❢♦r ❛ ❤✐❣❤•② ❝♦♠♣♦s✐t❡ ✐♥t❡❣❡r e✱ t②♣✐❝❛••② ❝❤♦s❡♥ ❛s e = lcm(1, 2, 3, 4, . . . , B 1 ) ❢♦r s♦♠❡ ✐♥t❡❣❡r B 1 ✳ ■❢ t❤❡ ♦r❞❡r ♦❢ g 0 ✐s B 1 ✲s♠♦♦t❤✱ t❤❡♥ g 1 = g e 0 ✐s t❤❡ ✐❞❡♥t✐t② ✐♥ G. ❙✐♥❝❡ G ✐s ❞❡✜♥❡❞ ♦✈❡r F p ✇❤❡r❡ p ❞✐✈✐❞❡s N, t❤❡ ♥✉♠❜❡r t♦ ❢❛❝t♦r✱ ✇❡ ❝❛♥ ❝♦♥str✉❝t ❢r♦♠ t❤❡ ✐❞❡♥t✐t② ✐♥ G ❛ r❡s✐❞✉❡ r (mod N ) s✉❝❤ t❤❛t r ≡ 0 (mod p) ❜✉t ❤♦♣❡❢✉••② ♥♦t r ≡ 0 (mod N ), ❛♥❞ t❤❡♥ gcd(r, N ) ✉s✉❛••② r❡✈❡❛•s p. ■❢ t❤❡ ♦r❞❡r ♦❢ g 0 ✐s ♥♦t B 1 ✲s♠♦♦t❤✱ st❛❣❡ ✶ ❢❛✐•s t♦ ✜♥❞ p✳ ❍♦✇❡✈❡r✱ ✇❡ ♠❛② ❜❡ ❛❜•❡ t♦ ✜♥❞ ✐t ②❡t ✐❢ t❤❡ ♦r❞❡r ♦❢ g 0 ❝♦♥s✐sts ♦❢ ❛ B 1 ✲s♠♦♦t❤ ♣❛rt t✐♠❡s ❛ ♥♦t✲t♦♦✲•❛r❣❡ ♣r✐♠❡ q✳ ❙t❛❣❡ ✷ ♦❢ P✕✶✱ P✰✶✱ ❛♥❞ ❊❈▼ tr✐❡s t♦ ✜♥❞ t❤❡ ✈❛•✉❡ ♦❢ q ❡✣❝✐❡♥t•② ♦♥ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t q ✐s ♣r✐♠❡ ❛♥❞ ♥♦t ✈❡r② •❛r❣❡✱ ❛•t❤♦✉❣❤ •❛r❣❡r t❤❛♥ B 1 ✱ ❜② •♦♦❦✐♥❣ ❢♦r ❛ ♠❛t❝❤ g m 1 = g n 1 ✇❤✐❝❤ ♦❝❝✉rs ✇❤❡♥ q | m -n. ❲❡ ✇✐•• ❞❡s❝r✐❜❡ t❤❡ st❛❣❡ ✷ ❢♦r t❤❡ P✰✶ ❛•❣♦r✐t❤♠❀ P✕✶ ❝❛♥ ✉s❡ t❤❡ s❛♠❡ ❛•❣♦r✐t❤♠ ❜② ❛❞•✉st✐♥❣ ✐ts ♦✉t♣✉t✱ ❛♥❞ t❤❡ st❛❣❡ ✷ ❢♦r ❊❈▼ ✐s str✉❝t✉r❛••② ✈❡r② s✐♠✐•❛r✳ ❉✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ P✰✶ ❛♥❞ ❊❈▼ st❛❣❡ ✷ ❛r❡ ♥♦t❡❞✳ ❖✉r st❛❣❡ ✷ ✐s ♠♦❞❡•❡❞ ❛❢t❡r t❤❡ ❡♥❤❛♥❝❡❞ st❛♥❞❛r❞ ❝♦♥t✐♥✉❛t✐♦♥ ❞❡s❝r✐❜❡❞ ❜② ▼♦♥t❣♦♠❡r② ❬✶✽(✳ ❋♦r ❣✐✈❡♥ s❡❛r❝❤ •✐♠✐ts B 1 ❛♥❞ B 2 ❛♥❞ ✐♥♣✉t X 1 ✐t ❝❤♦♦s❡s ❛ ✈❛•✉❡ d ✇✐t❤ 6 | d ❛♥❞ ❝♦♠♣✉t❡s t✇♦ •✐sts f i = V id (X 1 ) mod N ❢♦r ⌊B 1 /d⌉ ≤ i ≤ ⌊B 2 /d⌉ ❛♥❞ ✭✻✮ g j = V j (X 1 ) mod N ❢♦r 1 ≤ j < d/2 ❛♥❞ j ⊥ d, ✭✼✮ s♦ t❤❛t ❛•• ♣r✐♠❡s q ✐♥ ]B 1 , B 2 ] ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s q = id -j ♦r q = id + j. ▲❡t X 1 ≡ α 1 + 1/α 1 (mod N ), ✇❤❡r❡ α 1 ♠❛② ❜❡ ✐♥ ❛ q✉❛❞r❛t✐❝ ❡①t❡♥s✐♦♥ ♦❢ Z/N Z, ❛♥❞ ❛ss✉♠❡ α q 1 ≡ 1 (mod p) ✭✽✮ ❢♦r s♦♠❡ ✉♥❦♥♦✇♥ ♣r✐♠❡ p, p | N ❛♥❞ ❛ ♣r✐♠❡ q, B 1 < q ≤ B 2 . ▲❡t q = id -j ♦r q = id + j. ❚❤❡♥✱ ✉s✐♥❣ V -n (X) = V n (X), ✇❡ ❤❛✈❡ V id (X 1 ) ≡ V q±j (X 1 ) ≡ α q±j 1 + 1/α q±j 1 ≡ α ±j 1 + 1/α ±j 1 ≡ V ±j (X 1 ) ≡ V j (X 1 ) (mod p) ❛♥❞ s♦ V id (X 1 ) -V j (X 1 ) ≡ 0 (mod p). ✭✾✮ ❆❢t❡r t❤❡ •✐sts f i , g j ❛r❡ ❝♦♠♣✉t❡❞✱ ✇❡ ❝❛♥ ❝♦••❡❝t t❤❡ ♣r♦❞✉❝t A = id±j=q B1<q≤B2 (f i -g j ) mod N. ✭✶✵✮ ■❢ t❤❡r❡ ✐s ❛ ♣r✐♠❡ q ✐♥ ]B 1 , B 2 ] s✉❝❤ t❤❛t ✭✽✮ ❤♦•❞s✱ t❤❡ ♣r♦❞✉❝t ✭✶✵✮ ✇✐•• ✐♥❝•✉❞❡ i, j s✉❝❤ t❤❛t ✭✾✮ ❤♦•❞s✱ ❛♥❞ t❤✉s p | gcd(A, N ). ❙t❛❣❡ ✶ ♦❢ P✕✶ ❝♦♠♣✉t❡s x 1 = x e 0 (mod N ) ❛♥❞ ✇❡ ❝❛♥ s❡t X 1 = x 1 + 1/x 1 t♦ ♠❛❦❡ t❤❡ P✕✶ st❛❣❡ ✶ ♦✉t♣✉t ❝♦♠♣❛t✐❜•❡ ✇✐t❤ ♦✉r st❛❣❡ ✷ ❛t t❤❡ ❝♦st ♦❢ ♦♥❡ ♠♦❞✉•❛r ✐♥✈❡rs❡✳ ❙t❛❣❡ ✶ ♦❢ P✰✶ ❝♦♠♣✉t❡s x 1 = V e (x 0 ) = V e (α 0 + 1/α 0 ) = α e 0 + 1/α e 0 ❛♥❞ ✇❡ ♠❛② s✐♠♣•② s❡t X 1 = x 1 . ❋♦r P✕✶ st❛❣❡ ✷✱ ✇❡ ❝♦✉•❞ ❛•s♦ ✉s❡ f i = x id 1 mod N ❛♥❞ g j = x j 1 mod N ✱ ❢♦r 1 ≤ j < d ❛♥❞ j ⊥ d, ✐♥st❡❛❞ ♦❢ ✭✻✮✳ ❆♥ ❛❞✈❛♥t❛❣❡ ♦❢ ✉s✐♥❣ ✭✻✮ ✐s t❤❛t || ♥➦ ✼✵✹✶ ✷✽ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ V -n (X) = V n (X), s♦ t❤❛t g j = V j (X) mod N ♥❡❡❞s t♦ ❜❡ ❝♦♠♣✉t❡❞ ♦♥•② ❢♦r 1 ≤ j < d/2✱ ❛♥❞ ♦♥❡ (i, j)✲♣❛✐r ❝❛♥ s♦♠❡t✐♠❡s ✐♥❝•✉❞❡ t✇♦ ♣r✐♠❡s ❛t ♦♥❝❡✳ ❚❤❡ s❛♠❡ ❝♦✉•❞ ❜❡ ❛❝❤✐❡✈❡❞ ❜② ✉s✐♥❣ f i = x (id) 2 1 ❛♥❞ g j = x j 2 1 , ❜✉t ❝♦♠♣✉t✐♥❣ t❤❡s❡ ✈❛•✉❡s ❢♦r s✉❝❝❡ss✐✈❡ i ♦r j ✈✐❛ (x (n+1) 2 , x 2(n+1)+1 ) = (x n 2 • x 2n+1 , x 2n+1 • x 2 ) ❝♦sts t✇♦ ♠✉•t✐♣•✐❝❛t✐♦♥s✱ ✇❤❡r❡❛s V n+1 (x) = V n (x)V 1 (x) -V n-1 (x) ❝♦sts ♦♥•② ♦♥❡✳ ❍♦✇❡✈❡r✱ ❛ ♠♦❞✉•❛r ✐♥✈❡rs❡ ✐s r❡q✉✐r❡❞ t♦ ❝♦♥✈❡rt t❤❡ P✕✶ st❛❣❡ ✶ ♦✉t♣✉t ✐♥t♦ t❤❡ r❡q✉✐r❡❞ ❢♦r♠✳ ❲❤✐❝❤ ❛♣♣r♦❛❝❤ ✐s ❜❡tt❡r t❤✉s ❞❡♣❡♥❞s ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ st❛❣❡ ✷ ♣❛r❛♠❡t❡rs✱ ✐✳❡✳✱ ♦♥ ❤♦✇ ♠❛♥② ✈❛•✉❡s ♥❡❡❞ t♦ ❜❡ ♣r❡❝♦♠♣✉t❡❞ ❢♦r t❤❡ f i ❛♥❞ g j •✐sts✳ ❆ss✉♠✐♥❣ ❛ s♠❛•• B 1 , ✇❤❡♥ ✉s✐♥❣ B2 ≈ 5000 ❛♥❞ d = 210✱ ✇❡ ♥❡❡❞ ❛❜♦✉t 24 ✈❛•✉❡s ❢♦r f i ❛♥❞ ❛♥♦t❤❡r 24 ❢♦r g j ✳ ❚❤❡ ❝♦st ♦❢ ❛ ♠♦❞✉•❛r ✐♥✈❡rs❡ ✐s r♦✉❣❤•② 50 t✐♠❡s t❤❡ ❝♦st ♦❢ ❛ ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥ ✐♥ ♦✉r ✐♠♣•❡♠❡♥t❛t✐♦♥✱ s♦ t❤❡ t✇♦ ❛♣♣r♦❛❝❤❡s ❛r❡ ❛❜♦✉t ❡q✉❛••② ❢❛st✳ ❯s✐♥❣ t❤❡ s❛♠❡ st❛❣❡ ✷ ❢♦r P✕✶ ❛♥❞ P✰✶ ❤❛s t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ r❡q✉✐r✐♥❣ ♦♥•② ♦♥❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ❢♦r ❜♦t❤ ♠❡t❤♦❞s✳ ❋♦r ❊❈▼✱ ✇❡ ❛❣❛✐♥ ✇♦✉•❞ •✐❦❡ t✇♦ •✐sts f i ❛♥❞ g j s✉❝❤ t❤❛t f i ≡ g j (mod p) ✐❢ id • P 1 = j • P 1 ♦♥ E p ✱ ✇❤❡r❡ P 1 ✐s t❤❡ ♣♦✐♥t t❤❛t ✇❛s ♦✉t♣✉t ❜② ❊❈▼ st❛❣❡ ✶✳ ❲❡ ❝❛♥ ✉s❡ f i = (id • P 1 ) X ✱ t❤❡ X✲❝♦♦r❞✐♥❛t❡ ♦❢ id • P 1 , ❛♥❞ g j = (jP 1 ) X . ❆ ♣♦✐♥t ❛♥❞ ✐ts ✐♥✈❡rs❡ ❤❛✈❡ t❤❡ s❛♠❡ X✲❝♦♦r❞✐♥❛t❡ ♦♥ ❝✉r✈❡s ✐♥ ❲❡✐❡rstr❛ÿ ❛♥❞ ▼♦♥t❣♦♠❡r② ❢♦r♠✱ s♦ ❛❣❛✐♥ ✇❡ ❤❛✈❡ f i -g j ≡ 0 (mod p) ✐❢ q | id ± j. ❲✐t❤ ♣♦✐♥ts ✐♥ ♣r♦•❡❝t✐✈❡ ❝♦♦r❞✐♥❛t❡s✱ t❤❡ ♣♦✐♥ts ♥❡❡❞ t♦ ❜❡ ❝❛♥♦♥✐❝❛•✐③❡❞ ✜rst t♦ ❡♥s✉r❡ t❤❛t ✐❞❡♥t✐❝❛• ♣♦✐♥ts ❤❛✈❡ ✐❞❡♥t✐❝❛• X✲❝♦♦r❞✐♥❛t❡s✱ ✇❤✐❝❤ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✼✳✷✳ ❍♦✇ t♦ ❝❤♦♦s❡ ♣❛r❛♠❡t❡r d ❛♥❞ t❤❡ s❡t (i, j)✲♣❛✐rs ♥❡❡❞❡❞ ❞✉r✐♥❣ st❛❣❡ ✷ ❢♦r ❣✐✈❡♥ B 1 ❛♥❞ B 2 ✈❛•✉❡s ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✼✳✶✳ ❙❡❝t✐♦♥ ✼✳✷ s❤♦✇s ❤♦✇ t♦ ❝♦♠♣✉t❡ t❤❡ •✐sts f i ❛♥❞ g j ❡✣❝✐❡♥t•②✱ ❛♥❞ ❙❡❝t✐♦♥ ✼✳✸ ❞❡s❝r✐❜❡s ❤♦✇ t♦ ❛❝❝✉♠✉•❛t❡ t❤❡ ♣r♦❞✉❝t ✭✶✵✮✳ ✼✳✶ •❡♥❡r❛t✐♥❣ ♣•❛♥s ❚❤❡ ❝❤♦✐❝❡ ♦❢ d✱ t❤❡ s❡ts ♦❢ i ❛♥❞ j ✈❛•✉❡s t♦ ✉s❡ ❢♦r ❣❡♥❡r❛t✐♥❣ f i ❛♥❞ g j ✱ r❡s♣❡❝t✐✈❡•②✱ ❛♥❞ t❤❡ s❡t ♦❢ (i, j)✲♣❛✐rs ❢♦r ✇❤✐❝❤ t♦ ❛❝❝✉♠✉•❛t❡ t❤❡ ♣r♦❞✉❝t ♦❢ f i -g j ❞❡♣❡♥❞ ♦♥ t❤❡ B 1 ❛♥❞ B 2 ♣❛r❛♠❡t❡rs ❢♦r st❛❣❡ ✷✱ ❜✉t ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ N, t❤❡ ♥✉♠❜❡r t♦ ❢❛❝t♦r✳ ❚❤❡s❡ ❝❤♦✐❝❡s ❛r❡ ♣r❡❝♦♠♣✉t❡❞ ❢♦r ❣✐✈❡♥ B 1 ❛♥❞ B 2 ❛♥❞ ❛r❡ st♦r❡❞ ❛s ❛ ✏st❛❣❡ ✷ ♣•❛♥✳✑ ❚❤❡ st❛❣❡ ✷ ✐♠♣•❡♠❡♥t❛t✐♦♥ t❤❡♥ ❝❛rr✐❡s ♦✉t t❤❡ ♦♣❡r❛t✐♦♥s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ♣•❛♥✱ ✉s✐♥❣ ❛r✐t❤♠❡t✐❝ ♠♦❞✉•♦ N. ❚❤❡ ♣•❛♥ ♣r♦✈✐❞❡s t❤❡ ✈❛•✉❡s d, i 0 , i 1 , ❛ s❡t S ❛♥❞ ❛ s❡t T, ❝❤♦s❡♥ s♦ t❤❛t ❛•• ♣r✐♠❡s q ✐♥ ]B 1 , B 2 ] ❛♣♣❡❛r ❛s q = id ± j ❢♦r s♦♠❡ (i, j) ∈ T ✇✐t❤ i 0 ≤ i ≤ i 1 ❛♥❞ j ∈ S. ❲❡ tr② t♦ ❝❤♦♦s❡ ♣❛r❛♠❡t❡rs t❤❛t ♠✐♥✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ ❣r♦✉♣ ♦♣❡r❛t✐♦♥s r❡q✉✐r❡❞ ❢♦r ❜✉✐•❞✐♥❣ t❤❡ •✐sts f i ❛♥❞ g j ❛♥❞ ♠✐♥✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ (i, j)✲♣❛✐rs r❡q✉✐r❡❞ t♦ ❝♦✈❡r ❛•• ♣r✐♠❡s ✐♥ t❤❡ ]B 1 , B 2 ] ✐♥t❡r✈❛•✳ ❚❤✐s ♠❡❛♥s t❤❛t ✇❡ ✇♦✉•❞ •✐❦❡ t♦ ♠❛①✐♠✐s❡ i 0 ✱ ♠✐♥✐♠✐③❡ i 1 , ❛♥❞ ❝♦✈❡r t✇♦ ♣r✐♠❡s ✐♥ ]B 1 , B 2 ] ✇✐t❤ ❛ s✐♥❣•❡ (i, j)✲♣❛✐r ✇❤❡r❡✈❡r ♣♦ss✐❜•❡✳ ❲❡ ❝❤♦♦s❡ d ❤✐❣❤•② ❝♦♠♣♦s✐t❡ ❛♥❞ S = {1 ≤ j < d/2, j ⊥ d}, s♦ t❤❛t ❛•• ✐♥t❡❣❡rs ❝♦♣r✐♠❡ t♦ d✱ ✐♥ ♣❛rt✐❝✉•❛r ❛•• ♣r✐♠❡s ♥♦t ❞✐✈✐❞✐♥❣ d✱ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s id ± j ❢♦r s♦♠❡ ✐♥t❡❣❡r i ❛♥❞ j ∈ S. ❲❡ ❛ss✉♠❡ B 1 ✐s •❛r❣❡ ❡♥♦✉❣❤ t❤❛t ♥♦ ♣r✐♠❡ ❣r❡❛t❡r t❤❛♥ B 1 ❞✐✈✐❞❡s d✳ ❈❤♦♦s✐♥❣ ✈❛•✉❡s ♦❢ i 0 = ⌊B 1 /d⌉ ❛♥❞ i 1 = ⌊B 2 /d⌉ ✐s s✉✣❝✐❡♥t✱ ❜✉t ♠❛② ❜❡ ✐♠♣r♦✈❡❞ ❛s s❤♦✇♥ ❜❡•♦✇✳ ❈♦♠♣✉t✐♥❣ t❤❡ •✐sts f i ❛♥❞ g j r❡q✉✐r❡s ❛t •❡❛st ♦♥❡ ❣r♦✉♣ ♦♣❡r❛t✐♦♥ ♣❡r •✐st ❡♥tr②✱ ✇❤✐❝❤ ✐s ❡①♣❡♥s✐✈❡ ❡s♣❡❝✐❛••② ✐♥ t❤❡ ❝❛s❡ ♦❢ ❊❈▼✳ ❚❤❡ •✐st f i ❤❛s i 1 -i 0 +1 ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✷✾ ❡♥tr✐❡s ✇❤❡r❡ i 1 -i 0 ≈ (B 2 -B 1 )/d✱ ❛♥❞ g j ❤❛s φ(d)/2 ❡♥tr✐❡s✱ s♦ ✇❡ ❝❤♦♦s❡ d ❤✐❣❤•② ❝♦♠♣♦s✐t❡ t♦ ❛❝❤✐❡✈❡ s♠❛•• φ(d) ❛♥❞ tr② t♦ ♠✐♥✐♠✐③❡ i 1 -i 0 +1+φ(d)/2 ❜② ❡♥s✉r✐♥❣ t❤❛t i 1 -i 0 + 1 ❛♥❞ φ(d)/2 ❛r❡ ❛❜♦✉t ❡q✉❛••② •❛r❣❡✳ ■♥ ♦✉r ❛♣♣•✐❝❛t✐♦♥ ♦❢ ✜♥❞✐♥❣ ♣r✐♠❡s ✉♣ t♦✱ s❛②✱ 2 32 ❛s •✐♠✐t❡❞ ❜② t❤❡ •❛r❣❡ ♣r✐♠❡ ❜♦✉♥❞ ✉s❡❞ ✐♥ t❤❡ ◆❋❙ s✐❡✈✐♥❣ st❡♣✱ t❤❡ ✈❛•✉❡ ♦❢ B 2 ✇✐•• ✉s✉❛••② ❜❡ ♦❢ t❤❡ ♦r❞❡r ♦❢ ❛ ❢❡✇ t❤♦✉s❛♥❞✱ ❛♥❞ ❛ ❝❤♦✐❝❡ d = 210 ✇♦r❦s ✇❡•• ✐♥ t❤✐s ❝❛s❡✳ ❲✐t❤ B 2 = 5000, i 1 = 24 ❛♥❞ |S| = 24, s♦ t❤❡ t✇♦ •✐sts ♦❢ f i ❛♥❞ g j ❛r❡ ❛❜♦✉t ❡q✉❛••② •❛r❣❡✱ ❛ss✉♠✐♥❣ s♠❛•• i 0 ✳ ❋♦r s♠❛••❡r B 2 ✱ ❛ s♠❛••❡r d ✐s ♣r❡❢❡r❛❜•❡✱ ❢♦r ❡①❛♠♣•❡ d = 90 ❢♦r B 1 = 100, B 2 = 1000. ❲❡ ❤❛✈❡ ❝❤♦s❡♥ i 1 ❛s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ❜❛s❡❞ ♦♥ B 2 ✱ ❜✉t ✇❡ ♠❛② r❡❞✉❝❡ i 1 ②❡t ✐❢ [i 1 d -d/2, i 1 d + d/2] ❞♦❡s ♥♦t ✐♥❝•✉❞❡ ❛♥② ♣r✐♠❡s ✉♣ t♦ B 2 ✱ ❛♥❞ s♦ ♦❜t❛✐♥ t❤❡ ✜♥❛• ✈❛•✉❡ ♦❢ i 1 . ❍❛✈✐♥❣ ❝❤♦s❡♥ d, S, ❛♥❞ i 1 , ✇❡ ❝❛♥ ❝❤♦♦s❡ T. ❲❡ s❛② ❛ ♣r✐♠❡ q ∈ ]B 1 , B 2 ] ✐s ❝♦✈❡r❡❞ ❜② ❛♥ (i, j)✲♣❛✐r ✐❢ q | id ± j; ❛ss✉♠✐♥❣ t❤❛t ♦♥•② t❤❡ •❛r❣❡st ♣r✐♠❡ ❢❛❝t♦r ♦❢ ❛♥② id ± j ✈❛•✉❡ •✐❡s ✐♥ ]B 1 , B 2 ] , ❡❛❝❤ ♣❛✐r ♠❛② ❝♦✈❡r ✉♣ t♦ t✇♦ ♣r✐♠❡s✳ ❋♦r ❡❛❝❤ ♣r✐♠❡ q ∈ ]B 1 , B 2 ] ✇❡ ♠❛r❦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡♥tr② a[q] ✐♥ ❛♥ ❛rr❛② t♦ s✐❣♥✐❢② ❛ ♣r✐♠❡ t❤❛t ②❡t ♥❡❡❞s t♦ ❜❡ ❝♦✈❡r❡❞✳ ▲❡t r ❜❡ t❤❡ s♠❛••❡st ♣r✐♠❡ ♥♦t ❞✐✈✐❞✐♥❣ d✳ ❚❤❡♥ q | id ± j ❛♥❞ q = id ± j ✐♠♣•✐❡s q = (id ± j)/s ✇✐t❤ s ≥ r s✐♥❝❡ id ± j ⊥ d, t❤✉s q ≤ (id ± j)/r. ❍❡♥❝❡ ❝♦♠♣♦s✐t❡ ✈❛•✉❡s ♦❢ id±j ✇✐t❤ i ≤ i 1 ❝❛♥ ❝♦✈❡r ♦♥•② ♣r✐♠❡s ✉♣ t♦ ⌊(i 1 d+d/2)/r⌋✱ ❛♥❞ ❡❛❝❤ ♣r✐♠❡ q > ⌊(i 1 d + d/2)/r⌋ ❝❛♥ ❜❡ ❝♦✈❡r❡❞ ♦♥•② ❜② q = id ± j. ■♥ ❛ ✜rst ♣❛ss✱ ✇❡ ❡①❛♠✐♥❡ ❡❛❝❤ ♣r✐♠❡ q, ⌊(i 1 d + d/2)/r⌋ < q ≤ B 2 , ❤✐❣❤❡st t♦ •♦✇❡st ❛♥❞ t❤❡ (i, j)✲♣❛✐r ❝♦✈❡r✐♥❣ t❤✐s ♣r✐♠❡✳ ❚❤✐s ♣❛✐r ✐s t❤❡ ♦♥•② ✇❛② t♦ ❝♦✈❡r q ❛♥❞ ♠✉st ❡✈❡♥t✉❛••② ❜❡ ✐♥❝•✉❞❡❞ ✐♥ T. ■❢ t❤✐s (i, j)✲♣❛✐r ❛•s♦ ❝♦✈❡rs ❛ s♠❛••❡r ♣r✐♠❡ q ′ ❛s ❛ ❝♦♠♣♦s✐t❡ ✈❛•✉❡✱ t❤❡♥ a[q ′ ] ✐s ✉♥✲♠❛r❦❡❞✳ ■♥ ❛ s❡❝♦♥❞ ♣❛ss✱ ✇❡ •♦♦❦ ❢♦r ❛❞❞✐t✐♦♥❛• (i, j)✲♣❛✐rs t❤❛t ❝♦✈❡r t✇♦ ♣r✐♠❡s✱ ❜♦t❤ ✈✐❛ ❝♦♠♣♦s✐t❡ ✈❛•✉❡s✳ ❲❡ ❡①❛♠✐♥❡ ❡❛❝❤ (i, j)✲♣❛✐r ✇✐t❤ i 0 ≤ i ≤ i 1 ❤✐❣❤❡st t♦ •♦✇❡st✱ ❛♥❞ j ∈ S. ■❢ t❤❡r❡ ❛r❡ t✇♦ ♣r✐♠❡s q ′ ❛♥❞ q ′′ ♠❛r❦❡❞ ✐♥ t❤❡ ❛rr❛② t❤❛t ❛r❡ ❝♦✈❡r❡❞ ❜② t❤❡ (i, j)✲♣❛✐r ✉♥❞❡r ❡①❛♠✐♥❛t✐♦♥✱ t❤❡♥ a[q ′ ] ❛♥❞ a[q ′′ ] ❛r❡ ✉♥✲♠❛r❦❡❞✱ ❛♥❞ a[id -j] ✐s ♠❛r❦❡❞ ✐♥st❡❛❞✳ ■♥ t❤❡ t❤✐r❞ ♣❛ss✱ ✇❡ ❝♦✈❡r t❤❡ r❡♠❛✐♥✐♥❣ ♣r✐♠❡s q ≤ ⌊(i 1 d + d/2)/r⌋ ✉s✐♥❣ (i, j)✲♣❛✐rs ✇✐t❤ •❛r❣❡ i, ✐❢ ♣♦ss✐❜•❡✱ ❤♦♣✐♥❣ t❤❛t ✇❡ ♠❛② ✐♥❝r❡❛s❡ t❤❡ ✜♥❛• i 0 ✈❛•✉❡✳ ❆s ✐♥ t❤❡ s❡❝♦♥❞ ♣❛ss✱ ✇❡ ❡①❛♠✐♥❡ ❡❛❝❤ (i, j)✲♣❛✐r ✐♥ ♦r❞❡r ♦❢ ❞❡❝r❡❛s✐♥❣ i ❛♥❞✱ ✐❢ t❤❡r❡ ✐s ❛ ♣r✐♠❡ q ′ ✇✐t❤ a[q ′ ] ♠❛r❦❡❞✱ q ′ | id ± j ❜✉t q ′ = id ± j✱ ✇❡ ✉♥✲♠❛r❦ a[q ′ ] ❛♥❞ ♠❛r❦ a[id -j] ✐♥st❡❛❞✳ ❚❤✐s ✇❛②✱ ❛•• ♣r✐♠❡s ✐♥ ]B 1 , B 2 ] ❛r❡ ❝♦✈❡r❡❞✱ ❛♥❞ ❡❛❝❤ ✇✐t❤ ❛♥ (i, j)✲♣❛✐r ✇✐t❤ t❤❡ •❛r❣❡st ♣♦ss✐❜•❡ i ≤ i 1 ✳ ❲❡ ♥♦✇ ❝❤♦♦s❡ t❤❡ ✜♥❛• i 0 ✈❛•✉❡ ❜② •♦♦❦✐♥❣ ❢♦r t❤❡ s♠❛••❡st ✭♥♦t ♥❡❝❡ss❛r✐•② ♣r✐♠❡✮ q s✉❝❤ t❤❛t ❛ a[q] ✐s ♠❛r❦❡❞✱ ❛♥❞ s❡tt✐♥❣ i 0 = ⌊q/d⌉. ❚❤❡ s❡t T ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✐♥❝•✉❞✐♥❣ ❡❛❝❤ (i, j)✲♣❛✐r ✇❤❡r❡ ❛♥ ❛rr❛② ❡•❡♠❡♥t a[id -j] ♦r a[id + j] ✐s ♠❛r❦❡❞✳ ❚❤❡ ♣❛✐rs ✐♥ T ❛r❡ st♦r❡❞ ✐♥ ♦r❞❡r ♦❢ ✐♥❝r❡❛s✐♥❣ i s♦ t❤❛t t❤❡ f i ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ s❡q✉❡♥t✐❛••② ❢♦r P✕✶ ❛♥❞ P✰✶✳ ✼✳✷ ■♥✐t✐❛•✐s❛t✐♦♥ ■♥ t❤❡ ✐♥✐t✐❛•✐s❛t✐♦♥ ♣❤❛s❡ ♦❢ st❛❣❡ ✷ ❢♦r P✕✶ ❛♥❞ P✰✶ ✭❛♥❞ s✐♠✐•❛r•② ❢♦r ❊❈▼✮✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ✈❛•✉❡s g j = V j (X 1 ) ✇✐t❤ 1 ≤ j < d/2, j ⊥ d ❛♥❞ s❡t ✉♣ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ f i = V id (X 1 ) ❢♦r i 0 ≤ i ≤ i 1 . ❚♦ ❞♦ s♦✱ ✇❡ ♥❡❡❞ ▲✉❝❛s ❝❤❛✐♥s t❤❛t ❣❡♥❡r❛t❡ ❛•• r❡q✉✐r❡❞ ✈❛•✉❡s ♦❢ id ❛♥❞ j✳ ❲❡ tr② t♦ ✜♥❞ ❛ s❤♦rt ▲✉❝❛s ❝❤❛✐♥ t❤❛t ♣r♦❞✉❝❡s ❛•• r❡q✉✐r❡❞ ✈❛•✉❡s t♦ s❛✈❡ ❣r♦✉♣ ♦♣❡r❛t✐♦♥s ✇❤✐❝❤ ❛r❡ ❝♦st•② ❡s♣❡❝✐❛••② ❢♦r ❊❈▼✳ || ♥➦ ✼✵✹✶ ✸✵ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ ▲✉❝❛s ❝❤❛✐♥s ❢♦r ✈❛•✉❡s ✐♥ ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ❛r❡ ♣❛rt✐❝✉•❛r•② s✐♠♣•❡✱ s✐♥❝❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ s✉❝❝❡ss✐✈❡ t❡r♠s ✐s ❝♦♥st❛♥t✳ ❲❡ ♠❡r❡•② ♥❡❡❞ t♦ st❛rt t❤❡ ❝❤❛✐♥ ✇✐t❤ t❡r♠s t❤❛t ❣❡♥❡r❛t❡ t❤❡ ❝♦♠♠♦♥ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ ✜rst t✇♦ t❡r♠s ♦❢ t❤❡ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✳ ❚❤❡ ✈❛•✉❡s ♦❢ j ✇✐t❤ j ⊥ d ❛♥❞ 6 | d ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t✇♦ ❛r✐t❤♠❡t✐❝ ♣r♦✲ ❣r❡ss✐♦♥s 1+6m ❛♥❞ 5+6m, ✈✐❛ t❤❡ ▲✉❝❛s ❝❤❛✐♥ 1,
❋♦r d = 210, t❤❡ r❡q✉✐r❡❞ 24 ✈❛•✉❡s ♦❢ j ❝❛♥ t❤❡r❡❢♦r❡ ❜❡ ❣❡♥❡r❛t❡❞ ✇✐t❤ ❛ ▲✉❝❛s ❝❤❛✐♥ ♦❢ •❡♥❣t❤ 37. ❚♦ ❛❞❞ t❤❡ ✈❛•✉❡s ♦❢ id ✇✐t❤ i 0 ≤ i ≤ i 1 , ✇❡ ♥❡❡❞ t♦ ❛❞❞ d, i 0 d, ❛♥❞ (i 0 + 1)d t♦ t❤❡ ❝❤❛✐♥✳ ■❢ 2 || d✱ ✇❡ ❤❛✈❡ d/2 -2 ⊥ d ❛♥❞ d/2 + 2 ⊥ d ❛♥❞ ✇❡ ❝❛♥ ❛❞❞ d t♦ t❤❡ ▲✉❝❛s ❝❤❛✐♥ ❜② ✐♥❝•✉❞✐♥❣ 4 = 2+2 ❛♥❞ d = d/2+2+d/2-2. ■❢ 4 | d, ✇❡ ❤❛✈❡ d/2 -1 ⊥ d ❛♥❞ d/2 + 1 ⊥ d ❛♥❞ ✇❡ ❝❛♥ ❛❞❞ d s✐♠♣•② ✈✐❛ d = d/2 + 1 + d/2 + 1 ❛s 2 ✐s ❛•r❡❛❞② ✐♥ t❤❡ ❝❤❛✐♥✳ ❙✐♥❝❡ i 0 ✐s ✉s✉❛••② s♠❛••✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ ❜♦t❤ i 0 d ❛♥❞ (i 0 + 1)d ❢r♦♠ d ✇✐t❤ ♦♥❡ ❜✐♥❛r② ❝❤❛✐♥✳ ❯s✐♥❣ t❤✐s ▲✉❝❛s ❝❤❛✐♥✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ ❛♥❞ st♦r❡ ❛•• t❤❡ g j = V j (X 1 ) r❡s✐❞✉❡s ❛s ✇❡•• ❛s V d (X 1 ), f i0d (X 1 ), ❛♥❞ f (i0+1)d (X 1 ). ■♥ t❤❡ ❝❛s❡ ♦❢ P✕✶ ❛♥❞ P✰✶✱ s✐♥❝❡ t❤❡ (i, j)✲♣❛✐rs ❛r❡ st♦r❡❞ ✐♥ ♦r❞❡r ♦❢ ✐♥❝r❡❛s✐♥❣ i✱ ❛•• t❤❡ f i ✈❛•✉❡s ♥❡❡❞ ♥♦t ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ❛❞✈❛♥❝❡✱ ❜✉t ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ s✉❝❝❡ss✐✈❡•② ❛s t❤❡ (i, j)✲♣❛✐rs ❛r❡ ♣r♦❝❡ss❡❞✳ ✼✳✷✳✶ ■♥✐t✐❛•✐s❛t✐♦♥ ❢♦r ❊❈▼ ❋♦r ❊❈▼ st❛❣❡ ✷ ✇❡ ✉s❡ ❝✉r✈❡s ✐♥ ▼♦♥t❣♦♠❡r② ❢♦r♠ ✇✐t❤ ♣r♦•❡❝t✐✈❡ ❝♦♦r❞✐♥❛t❡s✱ •✉st ❛s ✐♥ st❛❣❡ ✶✱ t♦ ❛✈♦✐❞ ❝♦st•② ♠♦❞✉•❛r ✐♥✈❡rs❡s✳ ❚❤❡ ✐♥✐t✐❛•✐s❛t✐♦♥ ✉s❡s t❤❡ s❛♠❡ ▲✉❝❛s ❝❤❛✐♥ ❛s ✐♥ ✼✳✷ ❢♦r t❤❡ r❡q✉✐r❡❞ ✈❛•✉❡s ♦❢ id ❛♥❞ j✱ s♦ t❤❛t id • P 1 ❛♥❞ j • P 1 ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❡✣❝✐❡♥t•②✳ ❍♦✇❡✈❡r✱ t✇♦ ♣♦✐♥ts (X 1 :: Z 1 ) ❛♥❞ (X 2 :: Z 2 ) ✐♥ ♣r♦•❡❝t✐✈❡ ❝♦♦r❞✐♥❛t❡s ❜❡✐♥❣ ✐❞❡♥t✐❝❛• ❞♦❡s ♥♦t ✐♠♣•② X 1 = X 2 ✱ ❜✉t X 1 /Z 1 = X 2 /Z 2 , ✇❤❡r❡ Z 1 ❛♥❞ Z 2 ❛r❡ ❣❡♥❡r❛••② ♥♦t ❡q✉❛•✱ s♦ t❤❡ X✲❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡s❡ ♣♦✐♥ts ❝❛♥♥♦t ❜❡ ✉s❡❞ ❞✐r❡❝t•② t♦ ❜✉✐•❞ t❤❡ •✐sts f i ❛♥❞ g j ✳ ❚❤❡r❡ ❛r❡ s❡✈❡r❛• ✇❛②s t♦ ♦✈❡r❝♦♠❡ t❤✐s ♦❜str✉❝t✐♦♥✳ ❙❡✈❡r❛• ❛✉t❤♦rs ✭❢♦r ❡①✲ ❛♠♣•❡ ❬✼✱ ✼✳✹✳✷( ♦r ❬✷✹(✮ ♣r♦♣♦s❡ st♦r✐♥❣ ❜♦t❤ X ❛♥❞ Z ❝♦♦r❞✐♥❛t❡ ✐♥ t❤❡ f i ❛♥❞ g j •✐sts✱ t❤❡♥ ❛❝❝✉♠✉•❛t✐♥❣ t❤❡ ♣r♦❞✉❝t A = (i,j)∈T ((f i ) X (g j ) Z -(g j ) X (f i ) Z ) . ❆♥ ❛❞✈❛♥t❛❣❡ ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ✐s t❤❛t t❤❡ f i ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ s❡q✉❡♥t✐❛••② ✇❤✐•❡ t❤❡ ♣r♦❞✉❝t ✐s ❜❡✐♥❣ ❛❝❝✉♠✉•❛t❡❞ ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ g j t♦ ♣r❡❝♦♠♣✉t❡ ❛♥❞ st♦r❡ ❝❛♥ ❜❡ ❝♦♥tr♦••❡❞ ❜② ❝❤♦✐❝❡ ♦❢ d, ✇❤✐❝❤ ❛••♦✇s ❊❈▼ st❛❣❡ ✷ t♦ r✉♥ ✉♥❞❡r ❡①✲ tr❡♠❡•② t✐❣❤t ♠❡♠♦r② ❝♦♥❞✐t✐♦♥s s✉❝❤ ❛s ✐♥ ❛♥ ❋P•❆ ✐♠♣•❡♠❡♥t❛t✐♦♥✳ ❚❤❡ ♦❜✈✐♦✉s ❞✐s❛❞✈❛♥t❛❣❡ ✐s t❤❛t ❡❛❝❤ (i, j)✲♣❛✐r ♥♦✇ ✉s❡s 3 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s ✐♥st❡❛❞ ♦❢ 1 ❛s ✐♥ ✭✶✵✮✳ ❆♥♦t❤❡r ❛♣♣r♦❛❝❤ ❛♥❞ ♠✉❝❤ ♣r❡❢❡r❛❜•❡ ✐♥ ❛♥ ✐♠♣•❡♠❡♥t❛t✐♦♥ ✇❤❡r❡ s✉✣❝✐❡♥t ♠❡♠♦r② ✐s ❛✈❛✐•❛❜•❡ ✐s ❝❛♥♦♥✐❝❛•✐③✐♥❣ t❤❡ ♣r❡❝♦♠♣✉t❡❞ ♣♦✐♥ts s♦ t❤❛t ❛•• ♣♦✐♥ts ❤❛✈❡ t❤❡ s❛♠❡ Z✲❝♦♦r❞✐♥❛t❡✳ ❚♦ ♣r♦❞✉❝❡ t❤❡ ❞❡s✐r❡❞ •✐sts f i ❛♥❞ g j ✱ ✇❡ t❤❡r❡❢♦r❡ ❝♦♠♣✉t❡ ❛•• t❤❡ r❡q✉✐r❡❞ ♣♦✐♥ts Q i = id • P 1 ❛♥❞ R j = j • P 1 . ■❢ ❛•• Z✲❝♦♦r❞✐♥❛t❡s ♦❢ Q i ❛♥❞ R j ❛r❡ ♠❛❞❡ ✐❞❡♥t✐❝❛•✱ Q i = R j ♦♥ E p ✐♠♣•✐❡s (Q i ) X ≡ (R j ) X (mod p), ❛s ❞❡s✐r❡❞✱ ❛♥❞ ✇❡ ♠❛② s❡t f i = (Q i ) X ❛♥❞ g j = (R j ) X . ❲❡ s✉❣❣❡st t✇♦ ♠❡t❤♦❞s ❢♦r t❤✐s✳ ❖♥❡ ✐s t♦ s❡t ❛•• Z✲❝♦♦r❞✐♥❛t❡s t♦ 1 (mod N ) ✈✐❛ (X :: Z) = (XZ -1 :: 1). ❚♦ ❞♦ s♦✱ ✇❡ ♥❡❡❞ t❤❡ ✐♥✈❡rs❡ ♠♦❞✉•♦ N ♦❢ ❡❛❝❤ Z✲❝♦♦r❞✐♥❛t❡ ♦❢ ♦✉r ♣r❡❝♦♠♣✉t❡❞ ♣♦✐♥ts✳ ❆ tr✐❝❦ ❞✉❡ t♦ ▼♦♥t❣♦♠❡r②✱ ❞❡s❝r✐❜❡❞ ❢♦r ❡①❛♠♣•❡ ✐♥ ❬✻✱ ✶✵✳✸✳✹(✱ r❡♣•❛❝❡s n ♠♦❞✉•❛r ✐♥✈❡rs❡s ♦❢ r❡s✐❞✉❡s r k ♠♦❞✉•♦ N ✱ 1 ≤ k ≤ n✱ ❜② 1 ♠♦❞✉•❛r ✐♥✈❡rs❡ ❛♥❞ 3n -3 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛✲ t✐♦♥s✳ ❚❤✐s ✇❛② ✇❡ ❝❛♥ ❝❛♥♦♥✐❝❛•✐③❡ ❛ ❜❛t❝❤ ♦❢ n ♣♦✐♥ts ✇✐t❤ 4n -3 ♠♦❞✉•❛r ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✸✶ ♠✉•t✐♣•✐❝❛t✐♦♥s ❛♥❞ 1 ♠♦❞✉•❛r ✐♥✈❡rs❡✳ ◆♦t ❛•• ♣♦✐♥ts ♥❡❡❞❡❞ ❢♦r t❤❡ f i ❛♥❞ g j •✐sts ♥❡❡❞ t♦ ❜❡ ♣r♦❝❡ss❡❞ ✐♥ ❛ s✐♥❣•❡ ❜❛t❝❤❀ ✐❢ ♠❡♠♦r② ✐s ✐♥s✉✣❝✐❡♥t✱ t❤❡ ♣♦✐♥ts ♥❡❡❞❡❞ ❢♦r f i ❝❛♥ ❜❡ ♣r♦❝❡ss❡❞ ✐♥ s❡✈❡r❛• ❜❛t❝❤❡s ✇❤✐•❡ ♣r♦❞✉❝t ✭✶✵✮ ✐s ❜❡✐♥❣ ❛❝❝✉♠✉•❛t❡❞✳ ❆ ❢❛st❡r ♠❡t❤♦❞ ✇❛s s✉❣❣❡st❡❞ ❜② P✳ )✐♠♠❡r♠❛♥♥✳ •✐✈❡♥ n ≥ 2 ♣♦✐♥ts P 1 , . . . , P n ✱ P i = (X i :: Z i )✱ ✇❡ s❡t ❛•• Z✲❝♦♦r❞✐♥❛t❡s t♦ 1≤i≤n Z i ❜② ♠✉•t✐♣•②✲ ✐♥❣ ❡❛❝❤ X k ❜② T k = 1≤i≤n,i =k Z i . ❚❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ❡✣❝✐❡♥t•② ❜② ❝♦♠♣✉t✐♥❣ t✇♦ •✐sts s k = 1≤i≤k Z i ❛♥❞ t k = k<i≤n Z i ❢♦r 1 ≤ k < n✱ ❡❛❝❤ ❛t t❤❡ ❝♦st ♦❢ n -2 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s✳ ◆♦✇ ✇❡ ❝❛♥ s❡t T 1 = t 1 , T n = s n-1 , ❛♥❞ T i = s i-1 t i ❢♦r 1 < i < n✱ ❛•s♦ ❛t t❤❡ ❝♦st ♦❢ n -2 ♠✉•t✐♣•✐❝❛t✐♦♥s✳ ▼✉•t✐♣•②✐♥❣ X i ❜② T i ❝♦sts ❛♥♦t❤❡r n ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s ❢♦r ❛ t♦t❛• ♦❢ ♦♥•② 4n -6 ♠♦❞✲ ✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s✱ ✇✐t❤♦✉t ❛♥② ♠♦❞✉•❛r ✐♥✈❡rs✐♦♥✳ ❆•❣♦r✐t❤♠ ✹ ✐♠♣•❡♠❡♥ts t❤✐s ✐❞❡❛✳ ❙✐♥❝❡ t❤❡ ❝♦♠♠♦♥ Z✲❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❝❛♥♦♥✐❝❛•✐③❡❞ ♣♦✐♥ts ✐s t❤❡ ♣r♦❞✉❝t ♦❢ ❛•• ♣♦✐♥ts idP 1 ❛♥❞ jP 1 , t❤❡ ❝♦♠♣•❡t❡ s❡t ♦❢ ♣♦✐♥ts ♥❡❡❞❡❞ ❢♦r t❤❡ f i ❛♥❞ g j •✐sts ♠✉st ❜❡ ♣r♦❝❡ss❡❞ ✐♥ ❛ s✐♥❣•❡ ❜❛t❝❤✳ ■♥t❡r❡st✐♥❣•②✱ ✐❢ t❤❡ ❝✉r✈❡ ♣❛r❛♠❡t❡rs ❛r❡ ❝❤♦s❡♥ s✉❝❤ t❤❛t t❤❡ ❝✉r✈❡ ✐♥✐t✐❛•✲ ✐s❛t✐♦♥ ❝❛♥ ❜❡ ❞♦♥❡ ✇✐t❤ ♠♦❞✉•❛r ❞✐✈✐s✐♦♥ ❜② s♠❛•• ❝♦♥st❛♥ts r❛t❤❡r t❤❛♥ ✇✐t❤ ❛ ♠♦❞✉•❛r ✐♥✈❡rs❡✱ t❤❡♥ ❊❈▼ ✐♠♣•❡♠❡♥t❡❞ t❤✐s ✇❛② ❞♦❡s ♥♦t ✉s❡ ❛♥② ♠♦❞✉•❛r ✐♥✈❡rs❡s ❛t ❛••✱ ✇✐t❤♦✉t s❛❝r✐✜❝✐♥❣ t❤❡ ♦♣t✐♠❛• ❝♦st ♦❢ 1 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥ ♣❡r (i, j)✲♣❛✐r ✐♥ st❛❣❡ ✷✳ ■♥♣✉t✿ n ≥ 2✱ ❛♥ ✐♥t❡❣❡r N ✱ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r Z 1 , . . . , Z n , r❡s✐❞✉❡s ♠♦❞✉•♦ N ❉❛t❛✿ s, ❛ r❡s✐❞✉❡ ♠♦❞✉•♦ N ❖✉t♣✉t✿ T 1 , . . . , T n , r❡s✐❞✉❡s ♠♦❞✉•♦ N ✇✐t❤ T i ≡ 1≤i≤n,i =k Z i (mod N ) T n-1 := Z n ❀ ❢♦r k := n -1 ❞♦✇♥t♦ 2 ❞♦ T k-1 := T k • Z k mod N ❀ s := Z 1 ❀ T 2 := T 2 • s mod N ❀ ❢♦r k := 3 t♦ n ❞♦ s := s • Z k-1 mod N ❀ T k := T k • s mod N ❀ ❆•❣♦r✐t❤♠ ✹✿ ❇❛t❝❤ ❝r♦ss ♠✉•t✐♣•✐❝❛t✐♦♥ ❛•❣♦r✐t❤♠✳ ✼✳✸ ❊①❡❝✉t✐♥❣ ♣•❛♥s ❚❤❡ st❛❣❡ ✷ ♣•❛♥ st♦r❡s t❤❡ (i, j)✲♣❛✐rs ✇❤✐❝❤ ❝♦✈❡r ❛•• ♣r✐♠❡s ✐♥ ]B1, B2]✳ ❚❤❡ f i ❛♥❞ g j •✐sts ❛r❡ ❝♦♠♣✉t❡❞ ❛s ❞❡s❝r✐❜❡❞ ✐♥ ✼✳✷✳ ❙t❛❣❡ ✷ t❤❡♥ r❡❛❞s t❤❡ st♦r❡❞ (i, j)✲♣❛✐rs✱ ❛♥❞ ♠✉•t✐♣•✐❡s f i -g j t♦ ❛♥ ❛❝❝✉♠✉•❛t♦r✿ A = (i,j)∈T (f i -g j ) mod N. ✭✶✶✮ ❙✐♥❝❡ t❤❡ ♣❛✐rs ❛r❡ st♦r❡❞ ✐♥ ♦r❞❡r ♦❢ ✐♥❝r❡❛s✐♥❣ i✱ t❤❡ ❢✉•• •✐st f i ♥❡❡❞ ♥♦t ❜❡ ♣r❡❝♦♠♣✉t❡❞ ❢♦r P✕✶ ❛♥❞ P✰✶✱ ❜✉t ❡❛❝❤ f i ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ s❡q✉❡♥t✐❛••② ❜② V (i+1)d (X 1 ) = V id (X 1 )V d (X 1 ) -V (i-1)d (X 1 ). ❆t t❤❡ ❡♥❞ ♦❢ st❛❣❡ ✷✱ ✇❡ t❛❦❡ r = gcd(A, N ), ❤♦♣✐♥❣ t❤❛t 1 ≤ r ≤ N ❛♥❞ s♦ t❤❛t r ✐s ❛ ♣r♦♣❡r ❢❛❝t♦r ♦❢ N. || ♥➦ ✼✵✹✶ ✸✷ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ ✼✳✸✳✶ ❇❛❝❦tr❛❝❦✐♥❣ ❲❡ ✇♦✉•❞ •✐❦❡ t♦ ❛✈♦✐❞ ✜♥❞✐♥❣ ❛•• ♣r✐♠❡ ❢❛❝t♦rs ♦❢ t❤❡ ✐♥♣✉t ♥✉♠❜❡r N s✐✲ ♠✉•t❛♥❡♦✉s•②✱ ✐✳❡✳✱ ✜♥❞✐♥❣ N ❛s ❛ tr✐✈✐❛• ❢❛❝t♦r✳ ❆s ✐♥ st❛❣❡ ✶✱ ❛ ❜❛❝❦tr❛❝❦✐♥❣ ♠❡❝❤❛♥✐s♠ ✐s ✉s❡❞ t♦ r❡❝♦✈❡r ❢r♦♠ s✉❝❤ ❛ s✐t✉❛t✐♦♥✳ ❙✐♥❝❡ r = gcd(A, N ) ❛♥❞ A ✐s ❛ r❡❞✉❝❡❞ r❡s✐❞✉❡ ♠♦❞✉•♦ N ✱ ✇❡ ✜♥❞ r = N ❛s ❛ ❢❛❝t♦r ✐❢ ❛♥❞ ♦♥•② ✐❢ A = 0. ❲❡ s❡t ✉♣ ❛ ✏❜❛❝❦✉♣✑ r❡s✐❞✉❡ A ′ = 1 ❛t t❤❡ st❛rt ♦❢ ❡✈❛•✉❛t✐♦♥ ♦❢ ✭✶✶✮✳ ❆t ♣❡r✐♦❞✐❝ ✐♥t❡r✈❛•s ❞✉r✐♥❣ t❤❡ ❡✈❛•✉❛t✐♦♥ ♦❢ ✭✶✶✮✱ ❢♦r ❡①❛♠♣•❡ ❡❛❝❤ t✐♠❡ t❤❛t i ✐s ✐♥❝r❡❛s❡❞✱ ✇❡ t❡st ✐❢ A = 0, ✇❤✐❝❤ ✐s ❢❛st s✐♥❝❡ t❤❡ r❡s✐❞✉❡ ❞♦❡s ♥♦t ♥❡❡❞ t♦ ❜❡ ❝♦♥✈❡rt❡❞ ♦✉t ♦❢ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ ✐❢ |❊❉❈ ✭s❡❡ ❙❡❝t✐♦♥ ✸✳✷✮ ✐s ✉s❡❞ ❢♦r t❤❡ ❛r✐t❤♠❡t✐❝✳ ■❢ A = 0✱ ✇❡ t❛❦❡ r = gcd(A ′ , N ) ❛♥❞ ❡♥❞ st❛❣❡ ✷✳ ❖t❤❡r✇✐s❡✱ ✇❡ s❡t A ′ = A. ❚❤✐s ✇❛②✱ ❛ ♣r♦♣❡r ❢❛❝t♦r ♦❢ N ❝❛♥ ❜❡ ❞✐s❝♦✈❡r❡❞ s♦ •♦♥❣ ❛s ❛•• ♣r✐♠❡ ❢❛❝t♦rs ♦❢ N ❛r❡ ♥♦t ❢♦✉♥❞ ❜❡t✇❡❡♥ t✇♦ t❡sts ❢♦r A = 0. ✼✳✹ P✰✶ ❛♥❞ ❊❈▼ st❛❣❡ ✷ ♣❡r❢♦r♠❛♥❝❡ ❚❛❜•❡s ✻ ❛♥❞ ✼ ❝♦♠♣❛r❡s t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ P✰✶ ❛♥❞ t❤❡ ❊❈▼ st❛❣❡ ✷ ✐♠♣•❡♠❡♥t❛t✐♦♥ ❢♦r ❞✐✛❡r❡♥t B 2 ✈❛•✉❡s ❛♥❞ ♠♦❞✉•✉s s✐③❡s ♦♥ ❆▼❉ P❤❡♥♦♠ ❛♥❞ ■♥t❡• ❈♦r❡ ✷ ❈P❯s✳ ■♥ ❡❛❝❤ ❝❛s❡✱ t❤❡ t✐♠✐♥❣ r✉♥ ✉s❡❞ B 1 = 10 ❛♥❞ d = 210, ❛♥❞ t❤❡ t✐♠❡ ❢♦r ❛ r✉♥ ✇✐t❤ B 1 = 10 ❛♥❞ ✇✐t❤♦✉t ❛♥② st❛❣❡ ✷ ✇❛s s✉❜tr❛❝t❡❞✳ ❈♦r❡ ✷ P❤❡♥♦♠ B 2 1 ✇♦r❞ 2 ✇♦r❞s -2 ❜✐ts 1 ✇♦r❞ 2 ✇♦
❚❛❜•❡ ✻✿ ❚✐♠❡ ✐♥ ♠✐❝r♦s❡❝♦♥❞s ❢♦r P✰✶ st❛❣❡ ✷ ✇✐t❤ ❞✐✛❡r❡♥t B 2 ✈❛•✉❡s ♦♥ 2.146 •❍③ ■♥t❡• ❈♦r❡ ✷ ❛♥❞ 2 •❍③ ❆▼❉ P❤❡♥♦♠ ❈P❯s ✼✳✺ ❖✈❡r❛•• ♣❡r❢♦r♠❛♥❝❡ ♦❢ P✕✶✱ P✰✶ ❛♥❞ ❊❈▼ ❚❛❜•❡s ✽ ❛♥❞ ✾ s❤♦✇s t❤❡ ❡①♣❡❝t❡❞ t✐♠❡ t♦ ✜♥❞ ♣r✐♠❡s ❝•♦s❡ t♦ 2 25 , 2 26 , . . . , 2 32 ❢♦r ♠♦❞✉•✐ ♦❢ 1 ✇♦r❞ ❛♥❞ ♦❢ 2 ✇♦r❞s✱ ❛♥❞ t❤❡ B 1 ❛♥❞ B 2 ♣❛r❛♠❡t❡rs ❝❤♦s❡♥ t♦ ♠✐♥✐♠✐③❡ t❤❡ ❡①♣❡❝t❡❞ t✐♠❡✳ ❚❤❡ ♣r♦❜❛❜✐•✐t② ❡st✐♠❛t❡ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❡••✐♣t✐❝ ❝✉r✈❡ ✇✐t❤ r❛t✐♦♥❛• 12 t♦rs✐♦♥ ❛♥❞ ♣❛r❛♠❡t❡r k = 2 ✐♥ ✻✳✷✳ ❚❤❛t t❤❡ B 1 ❛♥❞ B 2 ♣❛r❛♠❡t❡rs ❛r❡ ♥♦t ♠♦♥♦t♦♥♦✉s•② ✐♥❝r❡❛s✐♥❣ ✇✐t❤ ❢❛❝t♦r s✐③❡ ✐s ❞✉❡ t♦ t❤❡ ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✸✸ ❈♦r❡ ✷ P❤❡♥♦♠ B 2 1 ✇♦r❞ 2 ✇♦r❞s -2 ❜✐ts 1 ✇♦r❞ 2 ✇♦
❚❛❜•❡ ✼✿ ❚✐♠❡ ✐♥ ♠✐❝r♦s❡❝♦♥❞s ❢♦r ❊❈▼ st❛❣❡ ✷ ✇✐t❤ ❞✐✛❡r❡♥t B 2 ✈❛•✉❡s ♦♥ 2.146 •❍③ ■♥t❡• ❈♦r❡ ✷ ❛♥❞ 2 •❍③ ❆▼❉ P❤❡♥♦♠ ❈P❯s ❢❛❝t t❤❛t t❤❡ ❡①♣❡❝t❡❞ t✐♠❡ t♦ ✜♥❞ ❛ ♣r✐♠❡ ❢❛❝t♦r ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ B 1 ❛♥❞ B 2 ✐s ✈❡r② ✢❛t ❛r♦✉♥❞ t❤❡ ♠✐♥✐♠✉♠✱ s♦ t❤❛t ❡✈❡♥ s♠❛•• ♣❡rt✉r❜❛t✐♦♥s ♦❢ t❤❡ t✐♠✐♥❣s ♥♦t✐❝❡❛❜•② ❛✛❡❝t t❤❡ ♣❛r❛♠❡t❡rs ❝❤♦s❡♥ ❛s ♦♣t✐♠❛•✳ n B 1 B 2 Pr♦❜✳ 1 ✇♦r❞ 2 ✇♦
❚❛❜•❡ ✽✿ ❊①♣❡❝t❡❞ t✐♠❡ ✐♥ ♠✐❝r♦s❡❝♦♥❞s ❛♥❞ ♣r♦❜❛❜✐•✐t② t♦ ✜♥❞ ♣r✐♠❡ ❢❛❝t♦rs ❝•♦s❡ t♦ 2 n ♦❢ ❝♦♠♣♦s✐t❡s ✇✐t❤ 1 ♦r 2 ✇♦r❞s ✇✐t❤ P✕✶ ♦♥ 2 •❍③ ❆▼❉ P❤❡♥♦♠ ❈P❯s✳ ❚❤❡ B 1 ❛♥❞ B 2 ♣❛r❛♠❡t❡rs ❛r❡ ❝❤♦s❡♥ t♦ ♠✐♥✐♠✐③❡ t❤❡ t✐♠❡✴♣r♦❜❛❜✐•✐t② r❛t✐♦✳ ✽ ❈♦♠♣❛r✐s♦♥ t♦ ❤❛r❞✇❛r❡ ✐♠♣•❡♠❡♥t❛t✐♦♥s ♦❢ ❊❈▼ ❙❡✈❡r❛• ❤❛r❞✇❛r❡ ✐♠♣•❡♠❡♥t❛t✐♦♥s ♦❢ ❊❈▼ ❢♦r ✉s❡ ❛s ❛ ❝♦❢❛❝t♦r✐③❛t✐♦♥ ❞❡✈✐❝❡ ✐♥ ◆❋❙ ❤❛✈❡ ❜❡❡♥ ❞❡s❝r✐❜❡❞ r❡❝❡♥t•②✱ ❜❛s❡❞ ♦♥ t❤❡ ♣r♦♣♦s❡❞ ❞❡s✐❣♥ ✏❙❍❆|❑✑ ❜② ❋r❛♥❦❡ ❡t ❛•✳ ❬✶✶( ❙❍❆|❑ ✐s ❛ ❤❛r❞✇❛r❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ •◆❋❙ ❢♦r ❢❛❝t♦r✐♥❣ 1024✲❜✐t ✐♥t❡❣❡rs ✇❤✐❝❤ ✉s❡s ❊❈▼ ❢♦r ❝♦❢❛❝t♦r✐③❛t✐♦♥ ❛❢t❡r s✐❡✈✐♥❣✳ ❚❤❡ ✐❞❡❛ ♦❢ ✐♠♣•❡♠❡♥t✐♥❣ •◆❋❙ ✐♥ ❤❛r❞✇❛r❡ ✐s ✐♥s♣✐r❡❞ ❜② t❤❡ ♦❜s❡r✈❛t✐♦♥ ♦❢ ❇❡r♥st❡✐♥ ❬✶( || ♥➦ ✼✵✹✶ ✸✹ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ n B 1 B 2 Pr♦❜✳ 1 ✇♦r❞ 2 ✇♦
❚❛❜•❡ ✾✿ ❊①♣❡❝t❡❞ t✐♠❡ ✐♥ ♠✐❝r♦s❡❝♦♥❞s ❛♥❞ ♣r♦❜❛❜✐•✐t② ♣❡r ❝✉r✈❡ t♦ ✜♥❞ ♣r✐♠❡ ❢❛❝t♦rs ❝•♦s❡ t♦ 2 n ♦❢ ❝♦♠♣♦s✐t❡s ✇✐t❤ 1 ♦r 2 ✇♦r❞s ✇✐t❤ ❊❈▼ ♦♥ 2 •❍③ ❆▼❉ P❤❡♥♦♠ ❈P❯s✳ ❚❤❡ B 1 ❛♥❞ B 2 ♣❛r❛♠❡t❡rs ❛r❡ ❝❤♦s❡♥ ❡♠♣✐r✐❝❛••② t♦ ♠✐♥✐♠✐③❡ t❤❡ ❡①♣❡❝t❡❞ t✐♠❡✳ t❤❛t ❞❡❞✐❝❛t❡❞ ❤❛r❞✇❛r❡ ❝♦✉•❞ ❛❝❤✐❡✈❡ ❛ s✐❣♥✐✜❝❛♥t•② •♦✇❡r ❝♦st ✐♥ t❡r♠s ♦❢ ❆r❡❛✲ ❚✐♠❡ ♣r♦❞✉❝t t❤❛♥ ❛ s♦❢t✇❛r❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ t❤❛t ✉s❡s s✐❡✈✐♥❣ ♦♥ ❛ r❡❣✉•❛r P❈✳ ❍❡ ♣r♦♣♦s❡s✱ ❛♠♦♥❣ ♦t❤❡r ❛•❣♦r✐t❤♠s✱ t♦ ✉s❡ ❊❈▼ ❢♦r t❤❡ s♠♦♦t❤♥❡ss t❡st✳ P❡•③• ❡t ❛•✳ ❬✷✹( ♣r❡s❡♥t ❛ s❝❛•❛❜•❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ st❛❣❡ ✶ ❛♥❞ st❛❣❡ ✷ ❢♦r ✐♥♣✉t ♥✉♠❜❡rs ♦❢ ✉♣ t♦ ✷✵✵ ❜✐ts✱ ❜❛s❡❞ ♦♥ ❳✐•✐♥① ❱✐rt❡①✷✵✵✵❊✲✻ ❋P•❆s ✇✐t❤ ❛♥ ❡①t❡r♥❛• ♠✐❝r♦❝♦♥tr♦••❡r✳ ❚❤❡✐r ❞❡s✐❣♥ ❤❛s ♦♥❡ ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥ ✉♥✐t ♣❡r ❊❈▼ ✉♥✐t✱ ❛♥❞ ❡❛❝❤ ❊❈▼ ✉♥✐t ♣❡r❢♦r♠s ❜♦t❤ st❛❣❡ ✶ ❛♥❞ st❛❣❡ ✷✳ ❚❤❡② ♣r♦♣♦s❡ ✉s✐♥❣ t❤❡ ❜♦✉♥❞s B 1 = 910 ❛♥❞ B 2 = 57000 ❢♦r ✜♥❞✐♥❣ ♣r✐♠❡s ♦❢ ✉♣ t♦ ❛❜♦✉t 40 ❜✐ts✳ ❚❤❡② ✉s❡ ❝✉r✈❡s ✐♥ ▼♦♥t❣♦♠❡r② ❢♦r♠ ✭✸✮ ❛♥❞ ❛ ❜✐♥❛r② ▲✉❝❛s ❝❤❛✐♥ ❢♦r st❛❣❡ ✶ t❤❛t ✉s❡s 13740 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s ✭✐♥❝•✉❞✐♥❣ sq✉❛r✐♥❣s✮✱ ❛♥❞ ❡st✐♠❛t❡ t❤❛t ❛♥ ♦♣t✐♠✐③❡❞ ▲✉❝❛s ❝❤❛✐♥ ❝♦✉•❞ ❞♦ ✐t ✐♥ ≈ 12000 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s✳ ❚❤❡② ✉s❡ ❛♥ ❡♥❤❛♥❝❡❞ st❛♥❞❛r❞ st❛❣❡ ✷ t❤❛t ✉s❡s 3 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s ♣❡r (i, j)✲♣❛✐r✱ s❡❡ ✼✳✷✳ ❲✐t❤ ❛ ✈❛•✉❡ d = 210, t❤❡② ❡st✐♠❛t❡ 303 ♣♦✐♥t ❛❞❞✐t✐♦♥s ❛♥❞ 14 ♣♦✐♥t ❞♦✉❜•✐♥❣s ✐♥ t❤❡ ✐♥✐t✐❛•✐s❛t✐♦♥ ♦❢ st❛❣❡ ✷✱ ❛♥❞ 13038 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s ❢♦r ❝♦••❡❝t✐♥❣ t❤❡ ♣r♦❞✉❝t ✭✶✵✮ ✇✐t❤ 4346 ❞✐✛❡r❡♥t (i, j)✲♣❛✐rs ❢♦r ❛ t♦t❛• ♦❢ 14926 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s ✐♥ st❛❣❡ 2✳ ❍♦✇❡✈❡r✱ t♦ ♠✐♥✐♠✐③❡ t❤❡ ❆❚ ♣r♦❞✉❝t✱ t❤❡② ♣r♦♣♦s❡ ✉s✐♥❣ d = 30 ✇✐t❤ ❛ t♦t❛• ♦❢ 24926 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s ✐♥ st❛❣❡ ✷✳ ■♥ ♦✉r ✐♠♣•❡♠❡♥t❛t✐♦♥✱ st❛❣❡ ✶ ✇✐t❤ B 1 = 910 ❛♥❞ P|❆❈✲❣❡♥❡r❛t❡❞ ❝❤❛✐♥s ✭✉s✐♥❣ ❝♦st 6 ❢♦r ♣♦✐♥t ❛❞❞✐t✐♦♥✱ 5 ❢♦r ❞♦✉❜•✐♥❣✱ 0.5 ❢♦r ❡❛❝❤ ❜②t❡ ❝♦❞❡ ❛♥❞ 0.5 ❢♦r ❡❛❝❤ ❜②t❡ ❝♦❞❡ ❝❤❛♥❣❡ ❛s ♣❛r❛♠❡t❡rs ❢♦r r❛t✐♥❣ ❝❛♥❞✐❞❛t❡ ❝❤❛✐♥s✮ ✉s❡s 11403 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s✱ 83% ♦❢

  ♠❜❡r ♦❢ ♣r✐♠❡s p ✐♥ 2 30 , 2 30 + 10 8 ✇❤❡r❡ t❤❡ •❛r❣❡st ♣r✐♠❡ ❢❛❝✲ t♦r ♦❢ p -1✱ r❡s♣❡❝t✐✈❡•② t❤❡ ♥✉♠❜❡r ♦❢ P♦••❛r❞✲|❤♦ ✐t❡r❛t✐♦♥s t♦ ✜♥❞ p✱ ✐s ✐♥ [100n, 100n + 99], n ∈ N. ❚❤❡ •❡❢t ❣r❛♣❤ s❤♦✇s 0 ≤ n ≤ 1000✱ t❤❡ r✐❣❤t ❣r❛♣❤ s❤♦✇s ❛ ③♦♦♠ ♦♥ 0 ≤ n ≤ 100✳ || ♥➦ ✼✵✹✶ ✻ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ ✷ ❚r✐❛• ❉✐✈✐s✐♦♥ ❇❡❢♦r❡ ❢❛❝t♦r✐♥❣ ♦❢ t❤❡ ♥♦♥✲s✐❡✈❡❞ ❝♦❢❛❝t♦r ♦❢ t❤❡ ♣♦•②♥♦♠✐❛• ✈❛•✉❡s ✐♥t♦ •❛r❣❡ ♣r✐♠❡s ❝❛♥ ❝♦♠♠❡♥❝❡✱ t❤❡ ❝♦❢❛❝t♦r ♥❡❡❞s t♦ ❜❡ ❞❡t❡r♠✐♥❡❞ ❜② ❞✐✈✐❞✐♥❣ ♦✉t ❛•• t❤❡ ❢❛❝t♦r ❜❛s❡ ♣r✐♠❡s✳ ❋♦r ♠❡❞✐✉♠ s✐③❡ ❢❛❝t♦r ❜❛s❡ ♣r✐♠❡s✱ s❛② •❛r❣❡r t❤❛♥ ❛ ❢❡✇ ❤✉♥❞r❡❞ ♦r ❛ ❢❡✇ t❤♦✉s❛♥❞✱ ❛ s✐❡✈✐♥❣ t❡❝❤♥✐q✉❡ ✭✑r❡✲s✐❡✈✐♥❣✑✮ ❝❛♥ ❜❡ ✉s❡❞ ❛❣❛✐♥ t❤❛t st♦r❡s t❤❡ ♣r✐♠❡s ✇❤❡♥ r❡✲s✐❡✈✐♥❣ ❤✐ts ❛ •♦❝❛t✐♦♥ ♣r❡✈✐♦✉s•② ♠❛r❦❡❞ ❛s ✑•✐❦❡•② s♠♦♦t❤✳✑ ❋♦r •❛r❣❡ ❢❛❝t♦r ❜❛s❡ ♣r✐♠❡s✱ s❛② •❛r❣❡r t❤❛♥ ❛ ❢❡✇ t❡♥ t❤♦✉s❛♥❞✱ t❤❡ ♥✉♠❜❡r ♦❢ ❤✐ts ✐♥ t❤❡ s✐❡✈❡ ❛r❡❛ ✐s s♠❛•• ❡♥♦✉❣❤ t❤❛t t❤❡ ♣r✐♠❡s ❝❛♥ ❜❡ st♦r❡❞ ❞✉r✐♥❣ t❤❡ ✐♥✐t✐❛• s✐❡✈✐♥❣ ♣r♦❝❡ss ✐ts❡•❢✳ ❋♦r t❤❡ s♠❛••❡st ♣r✐♠❡s✱ ❤♦✇❡✈❡r✱ r❡✲s✐❡✈✐♥❣ ✐s ✐♥❡✣❝✐❡♥t✱ ❛♥❞ ❛ tr✐❛• ❞✐✈✐s✐♦♥ t❡❝❤♥✐q✉❡ s❤♦✉•❞ ❜❡ ✉s❡❞✳ ❚❤✐s ❙❡❝t✐♦♥ ❡①❛♠✐♥❡s ❛ ❢❛st tr✐❛• ❞✐✈✐s✐♦♥ r♦✉t✐♥❡✱ ❜❛s❡❞ ♦♥ ✐❞❡❛s ❜② ▼♦♥t❣♦♠❡r② ❛♥❞ •r❛♥•✉♥❞❬✶✸( ❬✷✶(✱ t❤❛t ♣r❡❝♦♠♣✉t❡s s❡✈❡r❛• ✈❛•✉❡s ♣❡r ❝❛♥❞✐❞❛t❡ ♣r✐♠❡ ❞✐✈✐s♦r t♦ s♣❡❡❞ ✉♣ t❤❡ ♣r♦❝❡ss✳

  s✐❞❡r ❛ ♣❛rt✐❝✉•❛r ✐♥t❡❣❡r N = ℓ j=0 n j β j ✱ ❛♥❞ ❛ ♣❛rt✐❝✉•❛r ♣r✐♠❡ p ∈ P ✳ ❚❤❡ ❛•❣♦r✐t❤♠ ✜rst ❞♦❡s ❛ s❡♠✐✲r❡❞✉❝t✐♦♥ ♠♦❞✉•♦ p t♦ ♦❜t❛✐♥ ❛ s✐♥❣•❡✲ ✇♦r❞ ✐♥t❡❣❡r ❝♦♥❣r✉❡♥t t♦ N (mod p)✱ t❤❡♥ t❡sts t❤✐s s✐♥❣•❡✲✇♦r❞ ✐♥t❡❣❡r ❢♦r ❞✐✈✐s✐❜✐•✐t② ❜② p✳ ❚♦ ❞♦ s♦✱ ✇❡ ❝♦♠♣✉t❡ r = n 0 + ℓ j=1 n j w j ≤ (β -1)(ℓ(p -1) + 1)✳ ❚♦ s✐♠♣•✐❢② t❤❡ ♥❡①t st❡♣s✱ ✇❡ r❡q✉✐r❡ p < β ℓ ✳ ❊✈❡♥ ❢♦r β = 2 32 ✱ ℓ = 4✱ t❤✐s ❣✐✈❡s p < 32768 ✇❤✐❝❤ ✐s ❡❛s✐•② s✉✣❝✐❡♥t ❢♦r tr✐❛• ❞✐✈✐s✐♦♥ ✐♥ ◆❋❙✳❲✐t❤ t❤✐s ❜♦✉♥❞ ♦♥ p✱ ✇❡ ❤❛✈❡ r < (β -1)(

  ❢r♦♠ ❝❛❝❤❡ ♠❛② ✐♥tr♦❞✉❝❡ ❛❞❞✐t✐♦♥❛• •❛t❡♥❝②✳ ❚♦ ♠❡❛s✉r❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ tr✐❛• ❞✐✈✐s✐♦♥ ❝♦❞❡✱ ✇❡ ❞✐✈✐❞❡ 10 7 ❝♦♥s❡❝✉t✐✈❡ ✐♥t❡❣❡rs ♦❢ 1, . . . , 5 ✇♦r❞s ❜② t❤❡ ✜rst n = 256, 512, 1024, ❛♥❞ 2048 ♦❞❞ ♣r✐♠❡s ♦♥ ❛ 2 •❍③ ❆▼❉ P❤❡♥♦♠ ❈P❯✱ s❡❡ ❋✐❣✉r❡ ✷✳ ❚❤❡ ❤✐❣❤❡r t✐♠✐♥❣s ♣❡r tr✐❛• ❞✐✈✐s✐♦♥ ❢♦r n = 256 ❛r❡ ❞✉❡ t♦ t❤❡ ❛❞❞✐t✐♦♥❛• ❝♦st ♦❢ ❞✐✈✐❞✐♥❣ ♦✉t ❢♦✉♥❞ ❞✐✈✐s♦rs✱ ✇❤✐❝❤ ❤❛s ❛ ❣r❡❛t❡r r❡•❛t✐✈❡ ❝♦♥tr✐❜✉t✐♦♥ ❢♦r s♠❛••❡r ♣r✐♠❡s ✇❤✐❝❤ ❞✐✈✐❞❡ ♠♦r❡ ❢r❡q✉❡♥t•②✳ ❚❤❡ t✐♠✐♥❣ ❢♦r ℓ = 4, n = 2048 ✐s ❝•♦s❡ t♦ t❤❡ ♣r❡❞✐❝t❡❞ 23 ❝•♦❝❦ ❝②❝•❡s✳ ❚❤❡ s✉❞❞❡♥ ✐♥❝r❡❛s❡ ❢♦r n = 2048 ✐♥ t❤❡ ❝❛s❡ ♦❢ N ✇✐t❤ ♦♥❡ ✇♦r❞ ✐s ❞✉❡ t♦ ❝❛❝❤✐♥❣✿ ✇✐t❤ 7 st♦r❡❞ ✈❛•✉❡s ✭p✱ p ✐♥✈ ✱ p •✐♠ ✱ w 1,...,4 ✮ ♦❢ 8 ❜②t❡s ❡❛❝❤✱ n = 2048 ❤❛s ❛ t❛❜•❡ ♦❢ ♣r❡❝♦♠♣✉t❡❞ ✈❛•✉❡s ♦❢ s✐③❡ 112❦❇✱ ✇❤✐❝❤ ❡①❝❡❡❞s t❤❡ •❡✈❡•✲✶ ❞❛t❛ ❝❛❝❤❡ s✐③❡ ♦❢ 64❦❇ ♦❢ t❤❡ P❤❡♥♦♠✳ ❋♦r •❛r❣❡ s❡ts ♦❢ ❝❛♥❞✐❞❛t❡ ♣r✐♠❡s✱ t❤❡ s❡q✉❡♥t✐❛• ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣r❡❝♦♠♣✉t❡❞ ❞❛t❛ ❝❛✉s❡ ❢r❡q✉❡♥t ♠✐ss❡s ✐♥ t❤❡ •❡✈❡•✲✶ ❝❛❝❤❡✱ ❛♥❞ t❤❡ tr✐❛• ❞✐✈✐s✐♦♥s ❢♦r N ♦❢ ♦♥•② ♦♥❡ ✇♦r❞ ❛r❡ ❢❛st ❡♥♦✉❣❤ t❤❛t tr❛♥s❢❡r r❛t❡ ❢r♦♠ t❤❡ •❡✈❡•✲✷ ❝❛❝❤❡ •✐♠✐ts t❤❡ ❡①❡❝✉t✐♦♥✳ ❚❤✐s ❝♦✉•❞ ❜❡ ❛✈♦✐❞❡❞ ❜② ❝♦♠♣✉t✐♥❣ ❢❡✇❡r w i ❝♦♥st❛♥ts ✭✐✳❡✳✱ ❝❤♦♦s✐♥❣ ❛ s♠❛••❡r ℓ✮ ✐❢ t❤❡ N ❛r❡ ❦♥♦✇♥ t♦ ❜❡ s♠❛••✱ ♦r st♦r✐♥❣ t❤❡ w i ✐♥ s❡♣❛r❛t❡ ❛rr❛②s r❛t❤❡r t❤❛♥ ✐♥t❡r•❡❛✈❡❞✱ s♦ t❤❛t t❤❡

  t❤❡ ✜❣✉r❡ ❢♦r t❤❡ ❜✐♥❛r② ▲✉❝❛s ❝❤❛✐♥✳ ✭❯s✐♥❣ ❝❤❛✐♥s ❢♦r ❝♦♠♣♦s✐t❡ ✈❛•✉❡s ✇❤❡r❡ t❤❡ r❡s✉•t✐♥❣ ❝❤❛✐♥ ✐s s❤♦rt❡r t❤❛♥ t❤❡ ❝♦♥✲ ❝❛t❡♥❛t❡❞ ❝❤❛✐♥s ❢♦r t❤❡ ❢❛❝t♦rs ✐s ♥♦t ❝✉rr❡♥t•② ✉s❡❞ ❛♥❞ ❝♦✉•❞ ♣r♦❜❛❜•② r❡❞✉❝❡ t❤✐s ✜❣✉r❡ ❜② ❛ ❢❡✇ ♠♦r❡ ♣❡r❝❡♥t✳✮ ❖✉r st❛❣❡ ✷ ✇✐t❤ B 2 = 57000 ❛♥❞ d = 210 ✉s❡s 290 ♣♦✐♥t ❛❞❞✐t✐♦♥s✱ 13 ♣♦✐♥t ❞♦✉❜•✐♥❣s✱ 1078 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s ❢♦r ♣♦✐♥t ❝❛♥♦♥✐❝❛•✐③❛t✐♦♥ ❛♥❞ 4101 ♣❛✐rs ✇❤✐❝❤ ❝♦st 1 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥ ❡❛❝❤✱ ❢♦r ❛ t♦t❛• ♦❢ 6945 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s✳ ❚❤❡ ❝♦st ♦❢ ❝♦♠♣✉t✐♥❣ ❛♥❞ ❝❛♥♦♥✐✲ ❝❛•✐③✐♥❣ t❤❡ ♣♦✐♥ts idP 1 ❤❛s ❛ r❡•❛t✐✈❡•② •❛r❣❡ s❤❛r❡ ✐♥ t❤✐s ✜❣✉r❡✱ s✉❣❣❡st✐♥❣ t❤❛t ❛ ✈❛•✉❡ ❢♦r d s✉❝❤ t❤❛t B 2 /(dφ(d)) ✐s ❝•♦s❡r t♦ 1 ♠✐❣❤t r❡❞✉❝❡ t❤❡ t♦t❛• ♠✉•t✐✲ ♣•✐❝❛t✐♦♥ ❝♦✉♥t✳ ■♥ ❛ ❤❛r❞✇❛r❡ ✐♠♣•❡♠❡♥t❛t✐♦♥✱ t❤❡ ❡①tr❛ ♠❡♠♦r② r❡q✉✐r❡♠❡♥ts ♠❛② ♠❛❦❡ •❛r❣❡r d ✈❛•✉❡s ✐♥❡✣❝✐❡♥t ✐♥ t❡r♠s ♦❢ t❤❡ ❆❚ ♣r♦❞✉❝t✱ ❜✉t t❤✐s ✐s ♥♦t ❛♥ ✐ss✉❡ ✐♥ ❛ s♦❢t✇❛r❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦♥ ❛ ♥♦r♠❛• P❈✳ ■♥ ♦✉r ✐♠♣•❡♠❡♥t❛t✐♦♥✱ d = 630 ♣r♦✈✐❞❡s t❤❡ ♠✐♥✐♠✉♠ t♦t❛• ♥✉♠❜❡r ♦❢ 5937 ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s ✐♥ ■◆|■❆ ❆ ❙♦❢t✇❛r❡ ■♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✸✺ st❛❣❡ ✷✱ ♦♥•② 40% ♦❢ t❤❡ ♥✉♠❜❡r r❡♣♦rt❡❞ ❜② P❡•③• ❡t ❛•✳ ❢♦r d = 210✱ ❛♥❞ ♦♥•② 24% ♦❢ t❤❡✐r ♥✉♠❜❡r ❢♦r d = 30. ❚❤❡s❡ ✜❣✉r❡s s✉❣❣❡st t❤❛t ❛ s♦❢t✇❛r❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ♦♥ ❛ ♥♦r♠❛• P❈ ❡♥•♦②s ❛♥ ❛❞✈❛♥t❛❣❡ ♦✈❡r ❛♥ ✐♠♣•❡♠❡♥t❛t✐♦♥ ✐♥ ❡♠❜❡❞❞❡❞ ❤❛r❞✇❛r❡ ❜② ❤❛✈✲ ✐♥❣ s✉✣❝✐❡♥t ♠❡♠♦r② ❛✈❛✐•❛❜•❡ t❤❛t ❝❤♦✐❝❡ ♦❢ ❛•❣♦r✐t❤♠s ❛♥❞ ♦❢ ♣❛r❛♠❡t❡rs ❛r❡ ♥♦t ❝♦♥str❛✐♥❡❞ ❜② ♠❡♠♦r②✱ ✇❤✐❝❤ s✐❣♥✐✜❝❛♥t•② r❡❞✉❝❡s t❤❡ ♥✉♠❜❡r ♦❢ ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥s ✐♥ st❛❣❡ ✷✳ ❚❤✐s ♣r♦❜•❡♠ ♠✐❣❤t ❜❡ r❡❞✉❝❡❞ ❜② s❡♣❛r❛t✐♥❣ t❤❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ st❛❣❡ ✶ ❛♥❞ st❛❣❡ ✷ ✐♥ ❤❛r❞✇❛r❡✱ s♦ t❤❛t ❡❛❝❤ st❛❣❡ ✶ ✉♥✐ts ♥❡❡❞s ♦♥•② ✈❡r② •✐tt•❡ ♠❡♠♦r② ❛♥❞ ❢♦r✇❛r❞s ✐ts ♦✉t♣✉t t♦ ❛ st❛❣❡ ✷ ✉♥✐t ✇❤✐❝❤ ❤❛s ❡♥♦✉❣❤ ♠❡♠♦r② t♦ ❝♦♠♣✉t❡ st❛❣❡ ✷ ✇✐t❤ ❛ s♠❛•• ♠✉•t✐♣•✐❝❛t✐♦♥ ❝♦✉♥t✱ ✇❤✐•❡ t❤❡ st❛❣❡ ✶ ✉♥✐t ♣r♦❝❡ss❡s t❤❡ ♥❡①t ✐♥♣✉t ♥✉♠❜❡r✳ •❛• ❡t ❛•✳ ❬✶✷( ✐♠♣r♦✈❡ ♦♥ t❤❡ ❞❡s✐❣♥ ❜② P❡•③• ❡t ❛•✳ ♠❛✐♥•② ❜② ✉s❡ ♦❢ ❛ ♠♦r❡ ❡✣❝✐❡♥t ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥✱ ❜② ❛✈♦✐❞✐♥❣ •✐♠✐t❛t✐♦♥s ❞✉❡ t♦ t❤❡ ♦♥✲❝❤✐♣ ❜•♦❝❦ |❆▼ ✇❤✐❝❤ ❛••♦✇s t❤❡♠ t♦ ✜t ♠♦r❡ ❊❈▼ ✉♥✐ts ♣❡r ❋P•❆✱ ❛♥❞ r❡♠♦✈✐♥❣ t❤❡ ♥❡❡❞ ❢♦r ❛♥ ❡①t❡r♥❛• ♠✐❝r♦❝♦♥tr♦••❡r✳ ❚❤❡ ❛•❣♦r✐t❤♠ ♦❢ ❊❈▼ st❛❣❡ ✶ ❛♥❞ st❛❣❡ ✷ ✐s ❡ss❡♥t✐❛••② t❤❡ s❛♠❡ ❛s t❤❛t ♦❢ P❡•③• ❡t ❛•✳ ❚❤❡② r❡♣♦rt ❛♥ ♦♣t✐♠❛• ♣❡r❢♦r♠❛♥❝❡✴❝♦st r❛t✐♦ ♦❢ 311 ❊❈▼ r✉♥s ♣❡r s❡❝♦♥❞ ♣❡r ✩100 ❢♦r ❛♥ ✐♥♣✉t ♥✉♠❜❡r ♦❢ ✉♣ t♦ 198 ❜✐ts ✇✐t❤ B 1 = 910, B 2 = 57000, d = 210, ✉s✐♥❣ ❛♥ ✐♥❡①♣❡♥s✐✈❡ ❙♣❛rt❛♥ ✸❊ ❳❈✸❙✶✻✵✵❊✲✺ ❋P•❆ ❢♦r t❤❡✐r ✐♠♣•❡♠❡♥t❛t✐♦♥✳ ❚❤❡② ❛•s♦ ❝♦♠♣❛r❡ t❤❡✐r ✐♠♣•❡♠❡♥t❛t✐♦♥ t♦ ❛♥ ❊❈▼ ✐♠♣•❡♠❡♥t❛t✐♦♥ ✐♥ s♦❢t✇❛r❡✱ •▼P✲❊❈▼ ❬✶✵(✱ r✉♥♥✐♥❣ ♦♥ ❛ P❡♥t✐✉♠ ✹✱ ❛♥❞ ❝♦♥❝•✉❞❡ t❤❛t t❤❡✐r ❞❡s✐❣♥ ♦♥ ❛ •♦✇✲❝♦st ❙♣❛rt❛♥ ✸ ❋P•❆ ♦✛❡rs ❛❜♦✉t 10 t✐♠❡s ❜❡tt❡r ♣❡r❢♦r♠❛♥❝❡✴❝♦st r❛t✐♦ t❤❛♥ •▼P✲❊❈▼ ♦♥ ❛ P❡♥t✐✉♠ ✹✳ ❍♦✇❡✈❡r✱ •▼P✲❊❈▼ ✐s ❛ ♣♦♦r ❝❛♥❞✐❞❛t❡ ❢♦r ❛ss❡ss✐♥❣ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❊❈▼ ✐♥ s♦❢t✇❛r❡ ❢♦r ✈❡r② s♠❛•• ♥✉♠❜❡rs ✇✐t❤ •♦✇ B 1 ❛♥❞ B 2 ✈❛•✉❡s✳ •▼P✲❊❈▼ ✐s ♦♣t✐♠✐③❡❞ ❢♦r s❡❛r❝❤✐♥❣ •❛r❣❡ ♣r✐♠❡ ❢❛❝t♦rs ✭❛s •❛r❣❡ ❛s r❡❛s♦♥❛❜•② ♣♦ss✐❜•❡ ✇✐t❤ ❊❈▼✮ ♦❢ ♥✉♠❜❡rs ♦❢ ❛t •❡❛st ❛ ❤✉♥❞r❡❞ ❞✐❣✐ts s✐③❡ ❜② ✉s❡ ♦❢ ❛s②♠♣t♦t✐❝❛••② ❢❛st ❛•❣♦r✐t❤♠s ✐♥ ♣❛rt✐❝✉•❛r ✐♥ st❛❣❡ ✷✱ s❡❡ ❬✷✾(✳ ❋♦r ✈❡r② s♠❛•• ✐♥♣✉t✱ t❤❡ ❢✉♥❝t✐♦♥ ❝❛•• ❛♥❞ •♦♦♣ ♦✈❡r❤❡❛❞ ✐♥ ♠♦❞✉•❛r ❛r✐t❤♠❡t✐❝ ❛♥❞ t❤❡ ❝♦st ♦❢ ❣❡♥❡r❛t✐♥❣ ▲✉❝❛s ❝❤❛✐♥s ♦♥✲t❤❡✲✢② ✐♥ st❛❣❡ ✶ ❞♦♠✐♥❛t❡s t❤❡ ❡①❡❝✉t✐♦♥ t✐♠❡❀ •✐❦❡✇✐s❡ ✐♥ st❛❣❡ ✷✱ t❤❡ ✐♥✐t✐❛•✐s❛t✐♦♥ ♦❢ t❤❡ ♣♦•②♥♦♠✐❛• ♠✉•t✐✲ ♣♦✐♥t ❡✈❛•✉❛t✐♦♥ ❛♥❞ ❛❣❛✐♥ ❢✉♥❝t✐♦♥ ❝❛•• ❛♥❞ •♦♦♣ ♦✈❡r❤❡❛❞ ✇✐•• ❞♦♠✐♥❛t❡✱ ✇❤✐•❡ t❤❡ B 2 ✈❛•✉❡ ✐s ❢❛r t♦♦ s♠❛•• t♦ •❡t t❤❡ ❛s②♠♣t♦t✐❝❛••② ❢❛st st❛❣❡ ✷ ✭✇✐t❤ t✐♠❡ ✐♥ Õ( √ B 2 )✮ ♠❛❦❡ ✉♣ ❢♦r t❤❡ ♦✈❡r❤❡❛❞✳ ❉❡ ▼❡✉•❡♥❛❡r ❡t✳ ❛•✳ ❬✽( ❢✉rt❤❡r ✐♠♣r♦✈❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡✴❝♦st✲r❛t✐♦ ❜② ✉s✐♥❣ ❛ ❤✐❣❤✲♣❡r❢♦r♠❛♥❝❡ ❳✐•✐♥① ❱✐rt❡①✹❙❳ ❋P•❆ ✇✐t❤ ❡♠❜❡❞❞❡❞ ♠✉•t✐♣•✐❡rs ✐♥st❡❛❞ ♦❢ ✐♠♣•❡♠❡♥t✐♥❣ t❤❡ ♠♦❞✉•❛r ♠✉•t✐♣•✐❝❛t✐♦♥ ✇✐t❤ ❣❡♥❡r❛•✲♣✉r♣♦s❡ •♦❣✐❝✳ ❚❤❡② ✐♠♣•❡♠❡♥t ♦♥•② st❛❣❡ ✶ ♦❢ ❊❈▼ ❛♥❞ ♦♥•② ❢♦r ✐♥♣✉t ♦❢ ✉♣ t♦ 135 ❜✐ts✳ ❖♥❡ ❊❈▼ ✉♥✐t ✉t✐•✐③❡s ❛•• ♠✉•t✐♣•✐❡rs ♦❢ t❤❡ s❡•❡❝t❡❞ ❋P•❆✱ s♦ ♦♥❡ ❊❈▼ ✉♥✐t ✜ts ♣❡r ❞❡✈✐❝❡✳ ❇② s❝❛•✐♥❣ t❤❡ t❤r♦✉❣❤♣✉t ♦❢ t❤❡ ❞❡s✐❣♥ ♦❢ •❛• ❡t ❛•✳ t♦ 135✲❜✐t ✐♥♣✉t✱ t❤❡② ❝♦♥❝•✉❞❡ t❤❛t t❤❡✐r ❞❡s✐❣♥ ♦✛❡rs ❛ 15.6 t✐♠❡s ❜❡tt❡r ♣❡r❢♦r♠❛♥❝❡✴❝♦st r❛t✐♦✳ ■♥ ♣❛rt✐❝✉•❛r✱ ❛ss✉♠✐♥❣ ❛ ❝♦st ♦❢ ✩116 ♣❡r ❞❡✈✐❝❡✱ t❤❡② st❛t❡ ❛ t❤r♦✉❣❤♣✉t ♦❢ 13793 ❊❈▼ st❛❣❡ ✶ ✇✐t❤ B 1 = 910 ♣❡r s❡❝♦♥❞ ♣❡r ✩100. ❲❡ ❝♦♠♣❛r❡ t❤❡ ❝♦st ♦❢ ✜♥❞✐♥❣ 40✲❜✐t ❢❛❝t♦rs ✉s✐♥❣ ♦✉r s♦❢t✇❛r❡ ✐♠♣•❡♠❡♥✲ t❛t✐♦♥ ♦❢ ❊❈▼ ✇✐t❤ t❤❛t ❣✐✈❡♥ ❜② ❞❡ ▼❡✉•❡♥❛❡r ❡t ❛•✳ ❖✉r ✐♠♣•❡♠❡♥t❛t✐♦♥ ✐s ❝✉rr❡♥t•② •✐♠✐t❡❞ t♦ ♠♦❞✉•✐ ♦❢ s✐③❡ 2 ✇♦r❞s ✇✐t❤ t❤❡ t✇♦ ♠♦st s✐❣♥✐✜❝❛♥t ❜✐ts ③❡r♦✱ ♦r 126 ❜✐ts ♦♥ ❛ 64✲❜✐t s②st❡♠✱ ✇❤❡r❡❛s t❤❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❞❡ ▼❡✉•❡✲ ♥❛❡r ❡t ❛•✳ ❛••♦✇s 135✲❜✐t ♠♦❞✉•✐✳ ❊①t❡♥❞✐♥❣ ♦✉r ✐♠♣•❡♠❡♥t❛t✐♦♥ t♦ ♥✉♠❜❡rs ♦❢ 3 ✇♦r❞s ✐s ✐♥ ♣r♦❣r❡ss✱ ❜✉t ♥♦t ❢✉♥❝t✐♦♥❛• ❛t t❤✐s t✐♠❡✳ ❲❡ ❡①♣❡❝t t❤❛t ❊❈▼ ✇✐t❤ 3✲✇♦r❞ ♠♦❞✉•✐ ✇✐•• t❛❦❡ ❛❜♦✉t t✇✐❝❡ ❛s •♦♥❣ ❛s ❢♦r 2✲✇♦r❞ ♠♦❞✉•✐✳ ❋♦r || ♥➦ ✼✵✹✶ ✸✻ ❆•❡①❛♥❞❡r ❑r✉♣♣❛ ❉❡✈✐❝❡ ❳❈✹❱❙❳✷✺✲✶✵ P❤❡♥♦♠ ✾✸✺✵❡ P❤❡♥♦♠ ■■ ❳✹ ✾✺✺ ❈•♦❝❦ r❛t❡ 0.22

  ✶✵✿ ❈♦♠♣❛r✐s♦♥ ♦❢ ❊❈▼ ✇✐t❤ B 1 = 910, B 2 = 57000 ❢♦r 126✲❜✐t ❛♥❞ 135✲❜✐t ✐♥♣✉t ♦♥ ❛ ❱✐rt❡①✹❙❳✷✺✲✶✵ ❋P•❆ ❛♥❞ ♦♥ ❆▼❉ 64✲❜✐t ♠✐❝r♦♣r♦❝❡ss♦rs✳ t❤❡ ❝♦♠♣❛r✐s♦♥ ✇❡ ✉s❡ t✐♠✐♥❣s ❢♦r 126✲❜✐t ♠♦❞✉•✐ ✭2 ✇♦r❞s✮ ❛♥❞ ❡st✐♠❛t❡s ❢♦r 135✲❜✐t ♠♦❞✉•✐ ✭3 ✇♦r❞s✮✳ ❚❤❡ t✐♠✐♥❣s ❢♦r ♦✉r ❝♦❞❡ ❛r❡ ♦❜t❛✐♥❡❞ ✉s✐♥❣ ❛♥ ❆▼❉ P❤❡♥♦♠ ❳✹ ✾✸✺✵❡ ✇✐t❤ ❢♦✉r ❝♦r❡s ❛t 2.0 •❍③✳ ❚❤❡ ❆▼❉ 64✲❜✐t ❈P❯s ❛•• ❝❛♥ ♣❡r❢♦r♠ ❛ ❢✉•• 64 × 64✲❜✐t ♣r♦❞✉❝t ❡✈❡r② 2 ❝•♦❝❦ ❝②❝•❡s✱ ♠❛❦✐♥❣ t❤❡♠ ❛♥ ❡①❝❡••❡♥t ♣•❛t❢♦r♠ ❢♦r ♠✉•t✐✲♣r❡❝✐s✐♦♥ ♠♦❞✉•❛r ❛r✐t❤♠❡t✐❝✳ ❚❤❡ ❢❛st❡st ❆▼❉ ❈P❯ ❝✉rr❡♥t•② ❛✈❛✐•❛❜•❡ ✐s ❛ ❢♦✉r✲❝♦r❡ 3.2 •❍③ P❤❡♥♦♠ ■■ ❳✹ ✾✺✺ ❛t ❛ ❝♦st ♦❢ ❛r♦✉♥❞ ✩215 ✭r❡❣✉•❛r r❡t❛✐• ♣r✐❝❡✱ ❛❝❝♦r❞✐♥❣ t♦ ✇✇✇✳♥❡✇❡❣❣✳❝♦♠ ♦♥ ❏✉•② ✷✽t❤ ✷✵✵✾✮ ❛♥❞ ✇❡ s❝❛•❡ t❤❡ t✐♠✐♥❣s •✐♥❡❛r•② t♦ t❤❛t ❝•♦❝❦ r❛t❡✳ ❙✐♥❝❡ t❤❡ ❝♦❞❡ ✉s❡s ❛•♠♦st ♥♦ r❡s♦✉r❝❡s ♦✉ts✐❞❡ t❤❡ ❈P❯ ❝♦r❡✱ •✐♥❡❛r s❝❛•✐♥❣ ✐s r❡❛s♦♥❛❜•❡✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❝•♦❝❦ ❝②❝•❡s ✉s❡❞ ✐s ❛ss✉♠❡❞ ✐❞❡♥t✐❝❛• ❜❡t✇❡❡♥ t❤❡ P❤❡♥♦♠ ❛♥❞ P❤❡♥♦♠ ■■✳ ❙✐♠✐•❛r•②✱ r✉♥♥✐♥❣ t❤❡ ❝♦❞❡ ♦♥ n ❝♦r❡s ♦❢ ❛ ❈P❯ ✐s ❛ss✉♠❡❞ t♦ ✐♥❝r❡❛s❡ t♦t❛• t❤r♦✉❣❤♣✉t n✲❢♦•❞✳ ❚❛❜•❡ ✶✵ ❝♦♠♣❛r❡s t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ✐♥ ❤❛r❞✇❛r❡ ♦❢ ❞❡ ▼❡✉•❡♥❛❡r ❡t ❛•✳ ❛♥❞ ♦❢ ♦✉r s♦❢t✇❛r❡ ✐♠♣•❡♠❡♥t❛t✐♦♥✱ ✉s✐♥❣ t❤❡ ♣❛r❛♠❡t❡rs B 1 = 910, B 2 = 57000. ❚❤❡ s♦❢t✇❛r❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ✉s❡s d = 630 ❢♦r st❛❣❡ ✷✳ ❉❡ ▼❡✉•❡♥❛❡r ❡t ❛•✳ ❞♦ ♥♦t ✐♠♣•❡♠❡♥t st❛❣❡ ✷✱ ❜✉t ♣r❡❞✐❝t ✐ts ♣❡r❢♦r♠❛♥❝❡ ❛s ❝❛♣❛❜•❡ ♦❢ 16, 900 st❛❣❡ ✷ ♣❡r s❡❝♦♥❞ ♣❡r ❞❡✈✐❝❡✳ ❲❡ ✉s❡ t❤✐s ❡st✐♠❛t❡ ✐♥ t❤❡ ❝♦♠♣❛r✐s♦♥✳ ❚❤❡② ❛•s♦ ❣✐✈❡ t❤❡ ❝♦st ♦❢ ♦♥❡ ❳✐•✐♥① ❳❈✹❱❙❳✷✺✲✶✵ ❋P•❆ ❛s ✩116 ✇❤❡♥ ❜✉②✐♥❣ 2500 ❞❡✈✐❝❡s✳ ❚❤❡ ❝✉rr❡♥t q✉♦t❡ ❛t ✇✇✇✳♥✉❤♦r✐③♦♥s✳ ❝♦♠ ❛♥❞ ✇✇✇✳❛✈♥❡t✳❝♦♠ ❢♦r t❤✐s ❞❡✈✐❝❡ ✐s ❛❜♦✉t ✩300, ❤♦✇❡✈❡r✳ ❲❡ ❜❛s❡ t❤❡ ♣r✐❝❡ ❝♦♠♣❛r✐s♦♥ ♦♥ t❤❡ •❛tt❡r ✜❣✉r❡✳ ❖♥•② t❤❡ ❝♦st ♦❢ t❤❡ ❋P•❆ ♦r t❤❡ ❈P❯✱ r❡s♣❡❝t✐✈❡•②✱ ❛r❡ ❝♦♥s✐❞❡r❡❞✳ ❚❤❡ r❡s✉•ts s❤♦✇ t❤❛t ❛ s♦❢t✇❛r❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❝❛♥ ❝♦♠♣❡t❡ ✐♥ t❡r♠s ♦❢ ❝♦st ♣❡r ❊❈▼ tr✐❛• ✇✐t❤ t❤❡ ♣✉❜•✐s❤❡❞ ❞❡s✐❣♥s ❢♦r ❊❈▼ ✐♥ ❤❛r❞✇❛r❡✳ ❆♥ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ s♦❢t✇❛r❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ✐s ✢❡①✐❜✐•✐t②✿ ✐t ❝❛♥ r✉♥ ♦♥ ✈✐rt✉❛••② ❛♥② 64✲❜✐t P❈✱ ❛♥❞ s♦ ✉t✐•✐③❡ ♦t❤❡r✇✐s❡ ✐❞•❡ ❝♦♠♣✉t✐♥❣ r❡s♦✉r❝❡s✳ ■❢ ♥❡✇ s②st❡♠s ❛r❡ ♣✉r❝❤❛s❡❞✱ t❤❡② ✐♥✈♦•✈❡ ♦♥•② st❛♥❞❛r❞ ♣❛rts t❤❛t ❝❛♥ ❜❡ r❡❛❞✐•② ✉s❡❞ ❢♦r ❛ ✇✐❞❡ r❛♥❣❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛• t❛s❦s✳ •✐✈❡♥ ❛ ❝♦♠♣❛r❛❜•❡ ■◆|■❆

  r❞s -2 ❜✐ts

	100	11.8	35.6	9.33	24.4
	200	24.5	77.9	19.4	52.6
	300	35.3	113	27.8	76.0
	400	46.7	151	36.6	101
	500	58.7	190	46.2	127
	600	69.6	226	54.6	151
	700	82.3	266	64.5	178
	800	93.6	302	72.4	202
	900	105	342	82.5	229
	1000	117	381	92.0	255

  r❞s -2 ❜✐ts

	25 130 7000	0.359	67	176
	26 130 7000	0.297	81	213
	27 150 11000 0.290	101	264
	28 160 13000 0.256	124	324
	29 180 12000 0.220	151	395
	30 200 12000 0.188	190	496
	31 260 14000 0.182	231	604
	32 250 15000 0.147	283	744