UBC Undergraduate Research

Prime generating Lucas sequences Liu, Paul; Estrin, Ron Apr 30, 2011

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P❘■▼❊ ●❊◆❊❘❆❚■◆● ▲❯❈❆❙ ❙❊◗❯❊◆❈❊❙  P❆❯▲ ▲■❯ ✫ ❘❖◆ ❊❙❚❘■◆  ❙❝✐❡♥❝❡ ❖♥❡ Pr♦❣r❛♠  ❚❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❇r✐t✐s❤ ❈♦❧✉♠❜✐❛  ❱❛♥❝♦✉✈❡r✱ ❈❛♥❛❞❛  ❆♣r✐❧ ✷✵✶✶  ✶  P❘■▼❊ ●❊◆❊❘❆❚■◆● ▲❯❈❆❙ ❙❊◗❯❊◆❈❊❙  ✷  ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♣r✐♠❡ ♥✉♠❜❡rs ✐♥ ▲✉❝❛s s❡q✉❡♥❝❡s ✇❛s ✐♥✈❡st✐❣❛t❡❞ ❜② ✐♥❞❡♣❡♥❞❡♥t❧② ❝❤❛♥❣✲ ✐♥❣ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡s ❛♥❞ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❝♦♥st❛♥ts ✐♥ t❤❡ r❡❝✉rs✐✈❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ s❡q✉❡♥❝❡✳ ❚❤❡ ♣r✐♠❡ ❞✐str✐❜✉t✐♦♥ ✇❛s ♦❜t❛✐♥❡❞ ❜② ❝♦✉♥t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ♣r✐♠❡s ✐♥ t❤❡ ✜rst ✶✵✵✵ t❡r♠s ♦❢ ✈❛r✐♦✉s ▲✉❝❛s s❡✲ q✉❡♥❝❡s✳ ■t ✇❛s ❢♦✉♥❞ t❤❛t ❜♦t❤ s♠❛❧❧❡r s❡❡❞s ❛♥❞ s♠❛❧❧❡r ♠✉❧t✐♣❧✐❡rs ♣r♦❞✉❝❡❞ ♠♦r❡ ♣r✐♠❡s ♦♥ ❛✈❡r❛❣❡ t❤❛♥ ✐❢ t❤❡ s❡❡❞s ❛♥❞ ♠✉❧t✐♣❧✐❡rs ✇❡r❡ ❧❛r❣❡✳ ■t ✇❛s ❛❧s♦ ❞❡t❡r♠✐♥❡❞ t❤❛t ❝❤❛♥❣✐♥❣ t❤❡ ✐♥✐t✐❛❧ s❡❡❞s ♣r♦❞✉❝❡❞ ♠♦r❡ ♣r✐♠❡s ❛♥❞ ♠♦r❡ ✈❛r✐❛t✐♦♥ ✐♥ ♣r✐♠❡ ❝♦✉♥ts t❤❛♥ ❝❤❛♥❣✐♥❣ t❤❡ ♠✉❧t✐♣❧✐❡rs✳ ❆❜str❛❝t✳  ■♥tr♦❞✉❝t✐♦♥✳ ❚❤❡ s❡❛r❝❤ ❢♦r ♣r✐♠❡ ♥✉♠❜❡rs ✐s ❛♥ ❛❝t✐✈❡ ❛s♣❡❝t ♦❢ ♠❛t❤❡♠❛t✐❝s t❤❛t ❛♣♣❡❛rs ✐♥ s❡✈❡r❛❧  ✜❡❧❞s✱ ♠♦st ❝♦♠♠♦♥ ♦❢ ✇❤✐❝❤ ✐s ✐♥ ♥✉♠❜❡r t❤❡♦r②✳  Pr✐♠❡ ♥✉♠❜❡rs ❢♦✉♥❞ ✐♥ ▲✉❝❛s s❡q✉❡♥❝❡s ❤❛✈❡ ❜❡❡♥  ♦❢ ✐♥t❡r❡st ❢♦r s♦♠❡ t✐♠❡✱ ❛♥❞ ❤❛✈❡ ❧❡❞ t♦ s❡✈❡r❛❧ ❞✐s❝♦✈❡r✐❡s ❛♥❞ t❤❡ ❝r❡❛t✐♦♥ ♦❢ ♣r✐♠❡ t❡sts s✉❝❤ ❛s t❤❡ ▲✉❝❛s✲▲❡❤♠❡r Pr✐♠❡ ❚❡st✳ ❚❤❡s❡ r❛t❤❡r s♣❡❝✐❛❧ ♥✉♠❜❡rs ❤❛✈❡ ❢♦✉♥❞ ✈❛r✐♦✉s ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❡✈❡r②❞❛② ❧✐❢❡✱ s✉❝❤ ❛s ❛❧❣♦r✐t❤♠s ❛♥❞ ♠❡t❤♦❞s ❢♦r ❞❛t❛ ❡♥❝r②♣t✐♦♥✳ ▲❡t ✉s ❞❡✜♥❡ ✭✐♥❢♦r♠❛❧❧②✮ ▲✉❝❛s s❡q✉❡♥❝❡s ❛s t❡r♠s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ s❡q✉❡♥❝❡✿  M (a, b, P, Q) : an = P · an−1 − Q · an−2 ,  a1 = a,  a2 = b  ❆♠♦♥❣ t❤❡ ♠♦st ✇❡❧❧✲❦♥♦✇♥ ▲✉❝❛s s❡q✉❡♥❝❡s ❛r❡ t❤❡ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡ ✭an  1)✱  ❛♥❞ ✐ts ❝♦♠♣❧❡♠❡♥t ▲✉❝❛s s❡q✉❡♥❝❡ ✭✇❤❡r❡  √  √  P+ D 2  ❛♥❞  = an−1 + an−2 , a1 = 0, a2 =  a1 = 2, a2 = 1✮✳  ▲✉❝❛s s❡q✉❡♥❝❡s ❝❛♥ ❛❧s♦ ❜❡ ❞❡✜♥❡❞ ❜❛s❡❞ ♦♥ t❤❡✐r ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧  α=  ✭✶✮  β=  P− D , 2  X 2 − P X + Q✱  ✇❤♦s❡ r♦♦ts✿  D = P 2 − 4Q  α2 −β 2 n n α−β ✭✇❤❡r❡ a1 = 0, a2 = 1✮✱ ❛♥❞ Vn = α + β ✭✇❤❡r❡ a1 = 2, a2 = 1✮ ✇❤✐❝❤ ❛r❡ ❝♦♠♣❧❡♠❡♥t ▲✉❝❛s s❡q✉❡♥❝❡s✳ ❚❤❡ ▲✉❝❛s s❡q✉❡♥❝❡ ❢♦r ✈❛❧✉❡s ♦❢ P = 3, Q = 2✱ ❝r❡❛t❡s t❤❡ s❡q✉❡♥❝❡ Un = 2n − 1✱ ✇❤❡r❡ ♣r✐♠❡ Un ❛r❡ ❦♥♦✇♥ ❛s ▼❡rs❡♥♥❡ ♣r✐♠❡s✱ ❛♥❞ U43112609 ❬✷❪✱ ✐s t❤❡ ❝✉rr❡♥t❧② ❧❛r❣❡st  ❝r❡❛t❡ t❤❡ s❡q✉❡♥❝❡s  Un =  ❦♥♦✇♥ ♣r✐♠❡✳ ❚❤✉s ✐t ✐s ❛♣♣❛r❡♥t t❤❛t ✐♥t❡r❡st✐♥❣ ❡✈❡♥ts ❝❛♥ ❜❡ s❡❡♥ ✇✐t❤ ♣r✐♠❡s ✐♥ ▲✉❝❛s s❡q✉❡♥❝❡s✳  ❚❤✐s ♣❛♣❡r ✉♥❞❡rt❛❦❡s tr❡♥❞s ✐♥ ♣r✐♠❡ ❞❡♥s✐t② ✐♥ ❣❡♥❡r❛❧ ▲✉❝❛s s❡q✉❡♥❝❡s ✐♥ t✇♦ ❛s♣❡❝ts ✲ ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢  a1 , a 2 ✱  ❛♥❞ ✈❛r✐♦✉s ✈❛❧✉❡s ♦❢  P, Q✳  ❋r♦♠ t❤✐s✱ s❡✈❡r❛❧ ✐♠♣♦rt❛♥t ♦❜s❡r✈❛t✐♦♥s ❛r❡ ♠❛❞❡ ❜❡❢♦r❡ t❤❡ s❡❛r❝❤  ❜❡❣✐♥s s♦ ❛s t♦ r❡str✐❝t t❤❡ s❡❛r❝❤ s♣❛❝❡✳  ❈♦♥s✐❞❡r ✈❛r✐♦✉s ✈❛❧✉❡s ♦❢  a1 = a, a2 = b ✭P = 1, Q = −1✮✱  s✉❝❤ t❤❛t t❤❡ ▲✉❝❛s s❡q✉❡♥❝❡ ✐s ♥♦t t❤❡  ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡✳ ❚❤❡♥ t❤❡ s❡q✉❡♥❝❡ ✇♦✉❧❞ ❜❡ ✇r✐tt❡♥ ❛s✿  M (a, b, 1, −1) : a, b, a + b, a + 2b, 2a + 3b, 3a + 5b, ...  ✭✷✮  ❈♦♠♣❛r❡ t❤✐s t♦ t❤❡ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡✿  Fn = 1, 1, 2, 3, 5, 8... ❲❡ ❝❛♥ s❡❡ t❤❛t s❡q✉❡♥❝❡ ✭✷✮ ✐s s✐♠♣❧② ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡s✱ s✉❝❤ t❤❛t✿  M (a, b, 1, −1) : an = a · Fn−2 + b · Fn−1 ❚❤✐s s❡q✉❡♥❝❡ ✇✐❧❧ ❜❡ ❞❡✜♥❡❞ ❛s t❤❡ ❣❡♥❡r❛❧ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡✱ ❞❡✜♥❡  Ln  ❛s s❡q✉❡♥❝❡ ♦❜t❛✐♥❡❞ ❢r♦♠  Gn (2, 1)✳  Gn (a, b)  ✭✸✮ ❢♦r  a1 = a  ❛♥❞  a 2 = b✳  ▲❡t ✉s ❛❧s♦  ❍❡♥❝❡ ✇❡ ♠❛❦❡ ♦✉r ✜rst ✐♠♣♦rt❛♥t ♦❜s❡r✈❛t✐♦♥✳ ■❢  a  ❛♥❞  b  ❛r❡ ♥♦t ❝♦♣r✐♠❡ ✭t❤❡② s❤❛r❡ ❛ ❝♦♠♠♦♥ ❞✐✈✐s♦r✮ t❤❡♥ t❤❡r❡ ✐s ♥♦ ❤♦♣❡ ✐♥ ♦❜t❛✐♥✐♥❣ ❛♥② ♣r✐♠❡s ✭❡①❝❡♣t ❢♦r t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡s ✐❢ t❤❡② ❤❛♣♣❡♥ t♦ ❜❡ ♣r✐♠❡✮✳ ❈♦♥s✐❞❡r ❛ ❝♦♠♠♦♥ ❞✐✈✐s♦r✱  a = dr  ❛♥❞  b = ds  ❢♦r s♦♠❡  d✱  ❜❡t✇❡❡♥  a  ❛♥❞  b✳  ❚❤❡♥ s✐♥❝❡  r, s  M (a, b, 1, −1) : an = a · Fn−2 + b · Fn−1 = d(r · Fn−2 + s · Fn−1 ). ❚❤❡ ❡①❛❝t s❛♠❡ ♦❜s❡r✈❛t✐♦♥ ❝❛♥ ❜❡ ♠❛❞❡ ❢♦r ✈❛r✐♦✉s ✈❛❧✉❡s ♦❢  P, Q✳  ■❢ ✇❡ ♠✉❧t✐♣❧② t❤❡ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡  ❜② ♥♦♥✲❝♦♣r✐♠❡ ✈❛❧✉❡s✱ t❤❡♥ ✐t ✐s ❡q✉❛❧ t♦ ♠✉❧t✐♣❧②✐♥❣ t❤❡ ❡♥t✐r❡ s❡q✉❡♥❝❡ ❜② s♦♠❡ ♥✉♠❜❡r✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t ❛♥② r❡s✉❧t❛♥t ✈❛❧✉❡ ❤❛s ❛ ❞✐✈✐s♦r t❤❛t ✐s ♥♦t ✐ts❡❧❢✳  P❘■▼❊ ●❊◆❊❘❆❚■◆● ▲❯❈❆❙ ❙❊◗❯❊◆❈❊❙  ✸  ■t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t ♣r✐♠❡ ♥✉♠❜❡rs ❛r❡ q✉✐t❡ r❛♥❞♦♠ ✐♥ t❤❡✐r ❛♣♣❡❛r❛♥❝❡✳ ▲❡t ✉s ❞❡✜♥❡ ✏Pr✐♠❡ ❉❡♥s✐t② ❉✐str✐❜✉t✐♦♥✑ ❛s t❤❡ ❢r❡q✉❡♥❝② ♦❢ t❤❡ ♣r✐♠❡ ❝♦✉♥t ✐♥ t❤❡ ✜rst ✶✵✵✵ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ ❞✐✈✐❞❡❞ ❜② t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❞❛t❛ ♣♦✐♥ts ✇❡ ❤❛✈❡✳ Pr✐♠❡ ♥✉♠❜❡rs ❣r♦✇ ❛t t❤❡ r❛t❡ ♦❢ ❛♣♣r♦①✐♠❛t❡❧②  nth  n log n  ✭✇❤❡r❡  n  ✐s t❤❡  ♣r✐♠❡✮✱ ❛♥❞ s♦ ❛s ♦✉r s❡❡❞s ❛♥❞ ♠✉❧t✐♣❧✐❡rs ❜❡❝♦♠❡ ❜✐❣✱ ✇❡ ❡①♣❡❝t t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ♣r✐♠❡s t❤❛t ✇❡  ♦❜t❛✐♥ ✇✐t❤✐♥ ❛ ✶✵✵✵ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ ✇♦✉❧❞ ❧✐❦❡❧② ❞❡❝r❡❛s❡✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ Pr✐♠❡ ◆✉♠❜❡r ❚❤❡♦r❡♠ ✭P◆❚✮ st❛t❡s t❤❛t t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❤✐tt✐♥❣ ❛ ♣r✐♠❡ ♥❡❛r s♦♠❡ ♥✉♠❜❡r  N✱  ✐s ❛♣♣r♦①✐♠❛t❡❧② 1/log N ✱ ❛♥❞ s♦  t❤❡ ♣r✐♠❡ ❞❡♥s✐t② ✇♦✉❧❞ ❧✐❦❡❧② ❞❡❝r❡❛s❡ ❛s ♦✉r ♠✉❧t✐♣❧✐❡rs ❛♥❞ ✐♥✐t✐❛❧ ✈❛❧✉❡s ❜❡❝♦♠❡ ❧❛r❣❡✳  ▼❡t❤♦❞s✳ ❚❤❡ ❞❛t❛ ✇❛s ❝♦❧❧❡❝t❡❞ t❤r♦✉❣❤ ❝♦♠♣✉t❛t✐♦♥❛❧ s❡❛r❝❤❡s ♣r♦❞✉❝❡❞ ✐♥ t❤❡ ♣r♦❣r❛♠♠✐♥❣ ❧❛♥❣✉❛❣❡  ▼❛t❤❡♠❛t✐❝❛✳ ❆s ♣r✐♠❛❧✐t② t❡st✐♥❣ t♦♦❦ ✉♣ t❤❡ ❜✉❧❦ ♦❢ t❤✐s ♣r♦❥❡❝t✱ ▼❛t❤❡♠❛t✐❝❛ ✇❛s ❝❤♦s❡♥ ❢♦r ✐ts ❡✛❡❝t✐✈❡ ♣r✐♠❡ t❡st✐♥❣ ❢✉♥❝t✐♦♥ ❛s ✇❡❧❧ ❛s ✐ts ❛❜✐❧✐t② t♦ ❛♥❛❧②③❡ ♠❛ss✐✈❡ ❛♠♦✉♥ts ♦❢ ❞❛t❛✳  ❚♦ ❞❡t❡r♠✐♥❡ t❤❡ ❡✛❡❝t ♦❢ ❞✐✛❡r❡♥t ✐♥✐t✐❛❧ ✈❛❧✉❡s ♦♥ t❤❡ ♣r✐♠❡ ❞❡♥s✐t② ❢♦r  (a, b, n < 1000)  ✇❡r❡ t❡st❡❞ ❢♦r ♣r✐♠❡s✳ ❚❤❡ ❞✐✛❡r❡♥t s❡❡❞s  ✐♥ t❤❡ ✜rst ✶✵✵✵ t❡r♠s✳  (a, b)  Gn (a, b)✱  ❛❧❧ ✈❛❧✉❡s  Gn (a, b)  ✇❡r❡ t❤❡♥ r❛♥❦❡❞ ❜② t❤❡ ♥✉♠❜❡r ♦❢ ♣r✐♠❡s  ❍♦✇❡✈❡r✱ ❛s s❡❛r❝❤✐♥❣ ❡①❤❛✉st✐✈❡❧② t❤r♦✉❣❤ s✉❝❤ ❛ ❧❛r❣❡ s♣❛❝❡ r❡q✉✐r❡s ♠❛ss✐✈❡  ❝♦♠♣✉t❛t✐♦♥❛❧ ♣♦✇❡r✱ ♠❛♥② ♦♣t✐♠✐③❛t✐♦♥s ✇❡r❡ ✉s❡❞ t♦ s♣❡❡❞ ✉♣ t❤❡ s❡❛r❝❤✳ ❉✉❡ t♦ t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t ♦♥❧② ❝♦♣r✐♠❡ ♣❛✐rs ♣r♦❞✉❝❡❞ s❡q✉❡♥❝❡s ✇✐t❤ ❛♥② ♣r✐♠❡s✱ ♦♥❧② ❝♦♣r✐♠❡ ♣❛✐rs ♦❢  Gn (a, b)✱  ✇❤❡r❡  a > b✱  (a, b)  ✇❡r❡ ✉s❡❞✳  ♣r♦❞✉❝❡s ❛ s❡q✉❡♥❝❡ t❤❛t ✇♦✉❧❞ ❛❧r❡❛❞② ❤❛✈❡ ❜❡❡♥ s❡❛r❝❤❡❞ ❢♦r s♦♠❡  ❛♥❞ s♦ ✇❡ s❡t ❢♦r t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t  a < b✳  ❋✉rt❤❡r♠♦r❡✱  Gn (c, d)✱ c < d✱  ❆❞❞✐t✐♦♥❛❧❧②✱ t❤❡ ✐ss✉❡ ♦❢ ❛ s❡q✉❡♥❝❡ ✉s✐♥❣ s❡❡❞s t❤❛t ❛r❡  ❝♦♥s❡❝✉t✐✈❡ t❡r♠s ✐♥ s♦♠❡ ♣r❡✈✐♦✉s❧② s❡❛r❝❤❡❞ s❡q✉❡♥❝❡ ✇❛s ❝♦rr❡❝t❡❞ ❢♦r ❜② ♣r✉♥✐♥❣ t❤❡ ❧✐st ♦❢ r❡s✉❧ts ❛❢t❡r t❤❡ s❡❛r❝❤✳  ❚❤❡s❡ s✐♠♣❧❡ r❡str✐❝t✐♦♥s ❡❛s✐❧② r❡❞✉❝❡❞ t❤❡ s❡❛r❝❤ s♣❛❝❡ ❜② ♦✈❡r ✼✵✪✳  ❉✉❡ t♦ t❤❡ r❡❝✉rs✐✈❡  ❛❞❞✐♥❣ ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ ❣❡♥❡r❛❧ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡✱ t❤❡r❡ ✐s ❛❧✇❛②s ♦♥❡ ❡✈❡♥ ♥✉♠❜❡r ❢♦r ❡✈❡r② t✇♦ ♦❞❞ ♥✉♠❜❡rs✳ ❯s✐♥❣ t❤✐s ❢❛❝t✱ t❤❡ s❡❛r❝❤ s♣❛❝❡ ✇❛s ❢✉rt❤❡r r❡❞✉❝❡❞ ❜② t❡st✐♥❣ ♦♥❧② ♦❞❞ t❡r♠s✳ ❆❞❞✐t✐♦♥❛❧❧②✱ t♦ s♣❡❡❞ ✉♣ t❤❡ ❡①♣❧✐❝✐t t❡r♠ ❝❛❧❝✉❧❛t✐♦♥s ♦❢  Gn (a, b)✱ ❛ ❧♦♦❦✉♣ t❛❜❧❡ ♦❢ ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡rs ✇❛s ✉s❡❞ ❛s ♦♣♣♦s❡❞  t♦ ❛♥② ❛❝t✉❛❧ ❝❛❧❝✉❧❛t✐♦♥✳  ❚♦ s❡❛r❝❤ t❤r♦✉❣❤ ❞✐✛❡r❡♥t ♠✉❧t✐♣❧✐❡rs ❢♦r  200; n < 1000)  ✇❡r❡ t❡st❡❞ ❢♦r ♣r✐♠❡s✳  Mn (a, b, P, −Q)✱ ❛❧❧ ✈❛❧✉❡s ♦❢ Mn (a, b, P, −Q) (0 < P, Q < (P, Q) ✇❡r❡ r❡str✐❝t❡❞ t♦ ✉♥❞❡r ✷✵✵ ❛s ♠✉❧t✐♣❧✐❡rs  ❚❤❡ ✈❛❧✉❡s ♦❢  ❣r❡❛t❡r t❤❛♥ t❤✐s ✈❛❧✉❡ r❡q✉✐r❡❞ ❛♥ ✐♥♦r❞✐♥❛t❡ ❛♠♦✉♥t ♦❢ ❝♦♠♣✉t❛t✐♦♥ t✐♠❡ ❞✉❡ t♦ t❤❡✐r ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤✳ ❆❣❛✐♥✱ t❤❡ ❞✐✛❡r❡♥t ♠✉❧t✐♣❧✐❡rs  (P, Q)  ✇❡r❡ r❛♥❦❡❞ ❜❛s❡❞ ♦♥ ♥✉♠❜❡r ♦❢ ♣r✐♠❡s ✐♥ t❤❡ ✜rst ✶✵✵✵ t❡r♠s ❛♥❞  Mn (a, b, P, −Q) ❛♥❞ Mn (a, b, −Q, P ) ❞♦ ♥♦t ♣r♦❞✉❝❡ t❤❡ s❛♠❡ s❡q✉❡♥❝❡s✱ ♦♥❡ ❝❛♥♥♦t ✐♠♣♦s❡ t❤❛t P < Q✳ ❆❞❞✐t✐♦♥❛❧❧②✱ ❛s Mn (a, b, P, −Q) ❝❛♥♥♦t ❜❡ ❞❡❝♦♠♣♦s❡❞ ❡❛s✐❧② ✐♥t♦ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡s✱ t❤❡ r❡❝✉rs✐✈❡ ❞❡✜♥✐t✐♦♥ ♦❢ Mn ✇❛s  t❤❡ r❡q✉✐r❡♠❡♥t ♦❢ ❝♦♣r✐♠❛❧✐t② ✇❛s ✉s❡❞ t♦ r❡❞✉❝❡ t❤❡ s❡❛r❝❤ s♣❛❝❡✳  ❍♦✇❡✈❡r✱ ❛s  ✉s❡❞ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ t❡r♠s ✐♥st❡❛❞ ♦❢ ✉s✐♥❣ ❛ ❧♦♦❦✉♣ t❛❜❧❡✳ ❚❤❡♥ t♦ ❣❛✐♥ s♦♠❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❤♦✇ ✐♥✐t✐❛❧ ✈❛❧✉❡s ❝❤❛♥❣❡❞ t❤❡ ❡✛❡❝t ♦❢ ❞✐✛❡r❡♥t ♠✉❧t✐♣❧✐❡rs✱  Fn (P, −Q)  ✇❛s ✐♥✈❡st✐❣❛t❡❞ ❢♦r ♦♥❡ tr✐❛❧✱ ❛♥❞  Ln (P, −Q)  ❢♦r ❛♥♦t❤❡r✳  ❘❡s✉❧ts✳ ❆t ❛♥ ✐♥✐t✐❛❧ ❣❧❛♥❝❡✱ t❤❡ ❤✐st♦❣r❛♠ ♦❢ ♣r✐♠❡ ♥✉♠❜❡rs ✇✐t❤✐♥ t❤❡ ✜rst ✶✵✵✵ t❡r♠s s❤♦✇ ❛ s❧✐❣❤t❧②  s❦❡✇❡❞ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✱ ❜✉t ❜② ✉s✐♥❣ ▼❛t❤❡♠❛t✐❝❛✬s ❋✐♥❞❉✐str✐❜✉t✐♦♥P❛r❛♠❡t❡rs ❢✉♥❝t✐♦♥✱ ✐t ✇❛s ❞❡t❡r✲ ♠✐♥❡❞ t❤❛t ✐t ✐s ♥♦t q✉✐t❡ s✉❝❤ ❛ ❞✐str✐❜✉t✐♦♥✳  P❘■▼❊ ●❊◆❊❘❆❚■◆● ▲❯❈❆❙ ❙❊◗❯❊◆❈❊❙  ✹  ❋✐❣✉r❡ ✶✿ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ♦❢ ♣r✐♠❡ ♥✉♠❜❡rs ✐♥ t❤❡ ✜rst ✶✵✵✵ t❡r♠s ♦❢ t❤❡ ❋✐❜♦♥❛❝❝✐ ❙❡q✉❡♥❝❡ ✇✐t❤ ❝❤❛♥❣✐♥❣ ✐♥✐t✐❛❧ s❡❡❞s✳ ❚❤❡ ♠❡❛♥ ✐s ✶✺✳✸ ♣r✐♠❡s ✇✐t❤ ❛ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ ✹✳✾✳ ❘❡❣❛r❞❧❡ss✱ ✐t ❝❛♥ ❜❡ s❡❡♥ t❤❛t t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♣r✐♠❡ ♥✉♠❜❡rs ✐s q✉✐t❡ r❛♥❞♦♠ ❝♦♥s✐❞❡r✐♥❣ ❤♦✇ ❝❧♦s❡❧② t❤❡✐r ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ r❡s❡♠❜❧❡s ❛ ●❛✉ss✐❛♥ ♦♥❡✳ ✇❤✐❝❤ ♣r♦❞✉❝❡❞ ✹✻ ♣r✐♠❡s✱ ❢♦❧❧♦✇❡❞ ❜②  Gn (179, 937)  ❚❤❡ ♠♦st ♣r✐♠❡ r✐❝❤ s❡q✉❡♥❝❡ ✇❛s  ✇❤✐❝❤ ♣r♦❞✉❝❡❞ ✹✸ ♣r✐♠❡s✳  Gn (32, 341)✱  ❚❤❡s❡ ❛r❡ ✐♥t❡r♠❡❞✐❛t❡  s✐③❡❞ ✈❛❧✉❡s ✇✐t❤✐♥ t❤❡ s❡❛r❝❤✱ ❜✉t ❛❢t❡r ❢✉rt❤❡r ❛♥❛❧②s✐s✱ ✐t ❛♣♣❡❛rs t❤❛t t❤❡ s♠❛❧❧❡r s❡❡❞s ♣r♦❞✉❝❡ ♦♥ ❛✈❡r❛❣❡ ♠♦r❡ ♣r✐♠❡s✳  ❋✐❣✉r❡ ✷✿ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ♦❢ ♣r✐♠❡ ♥✉♠❜❡rs ✐♥ t❤❡ ✜rst ✶✵✵✵ t❡r♠s ♦❢ t❤❡ ▲✉❝❛s ❙❡q✉❡♥❝❡ ✭▲❡❢t✮ ❛♥❞ ❋✐❜♦♥❛❝❝✐ ❙❡q✉❡♥❝❡ ✭❘✐❣❤t✮ ✇✐t❤ ❝❤❛♥❣✐♥❣ ♠✉❧t✐♣❧✐❡rs✱ ✇❤✐❝❤ ❛♣♣❡❛r q✉✐t❡ s✐♠✐❧❛r✳ ❚❤❡ ♠❡❛♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♣r✐♠❡s ❢♦✉♥❞ ✐♥ ▲✉❝❛s ❙❡q✉❡♥❝❡s ✐s ✻✳✶✻ ❛❣❛✐♥st ❋✐❜♦♥❛❝❝✐ ❙❡q✉❡♥❝❡s ✇✐t❤ ✻✳✵✼✳ ❋r♦♠ ❛❜♦✈❡✱ ✇❡ s❡❡ t❤❛t t❤❡ ▲✉❝❛s s❡q✉❡♥❝❡ s❤♦✇s ❛ s❧✐❣❤t❧② ❤✐❣❤❡r ❛✈❡r❛❣❡ ❢♦r ♣r✐♠❡s✱ ❜✉t ♦✈❡r❛❧❧ t❤❡② ❛r❡ ✈❡r② s✐♠✐❧❛r✳ ■♥t❡r❡st✐♥❣❧②✱ t❤❡ ❤✐❣❤❡st ♣r♦❞✉❝t✐♦♥ ♦❢ ♣r✐♠❡s ✐♥ t❤❡ ▲✉❝❛s s❡q✉❡♥❝❡ ❝❛♠❡ ♦✉t t♦ ❜❡ t❤❡ ♦♥❡ ✇✐t❤ ♥♦ ♠✉❧t✐♣❧✐❡rs ❛t ❛❧❧  Ln ✭✸✱✶✵✮  ❛♥❞  Ln ✭✶✶✱✻✮  Ln ✭✶✱✶✮✱  ❛♥❞  Ln ✭✺✱✹✷✮✱  ✇❤✐❝❤ ♣r♦❞✉❝❡❞ ✷✻ ♣r✐♠❡s✳  ❚❤❡② ✇❡r❡ ❢♦❧❧♦✇❡❞ ❜②  ✇✐t❤ ✷✹ ♣r✐♠❡s✳ ❚❤❡ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡ s❤♦✇❡❞ s✐♠✐❧❛r tr❡♥❞s ✇✐t❤ ❧♦✇ ♠✉❧t✐♣❧✐❡rs  ❜✉t ♣r♦❞✉❝✐♥❣ ❢❡✇❡r ♣r✐♠❡s✱ ✇✐t❤  Fn ✭✻✱✶✷✺✮  ❛♥❞  Fn ✭✶✶✱✾✵✮  ♣r♦❞✉❝✐♥❣ ♦♥❧② ✷✸ ♣r✐♠❡s✳ ❚❤✉s✱ ✐t ❝❛♥ ❜❡ s❡❡♥  t❤❛t ❝❤❛♥❣✐♥❣ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡s ♦❢ t❤❡ s❡q✉❡♥❝❡s ❛✛❡❝ts t❤❡ ♣r✐♠❡ ♣r♦❞✉❝t✐♦♥ ♠✉❝❤ ♠♦r❡✱ ❛♥❞ ♣r♦❞✉❝❡s ♠❛♥② ♠♦r❡ ♣r✐♠❡s t❤❛♥ ❝❤❛♥❣✐♥❣ t❤❡ ♠✉❧t✐♣❧✐❡rs✳  ❋✐❣✉r❡ ✸✿ ❚❤❡ ❢r❡q✉❡♥❝② ♦❢ ♣r✐♠❡s ✐♥ t❤❡ s♠❛❧❧❡st ✶✵✪ ♦❢ ♦✉r s❡❡❞ s❡❛r❝❤ s♣❛❝❡ ✭❧❡❢t✮ ❛♥❞ t❤❡ ❜✐❣❣❡st ✶✵✪ ✭r✐❣❤t✮✳ ▼❡❛♥s ♦❢ s♠❛❧❧❡st s❡❡❞s ❢♦✉♥❞ t♦ ❜❡ ✶✻✳✽ ❝♦♠♣❛r❡❞ t♦ t❤❡ ✶✵✪ ❜✐❣❣❡st s❡❡❞s ❢♦✉♥❞ t♦ ❜❡ ✶✹✳✼✳ ❙❡❡ ❛♣♣❡♥❞✐① ❢♦r ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ s✐③❡ ♦❢ s❡❡❞s ❢♦r t❤❡ ❤✐st♦❣r❛♠s✳ ❋r♦♠ t❤❡ ✶✵✻✶✼✷ ❞❛t❛ ♣♦✐♥ts ❝♦❧❧❡❝t❡❞ ❜② ✈❛r②✐♥❣ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡s ♦❢  Gn ✱  ✐t ❝❛♥ ❜❡ s❡❡♥ t❤❛t t❤❡ s♠❛❧❧❡r  s❡❡❞s ❜❡❧♦♥❣✐♥❣ ✐♥ t❤❡ ❧♦✇❡r ✶✵✪ ♣r♦❞✉❝❡ ✶ ♣r✐♠❡ ❤✐❣❤❡r ♦♥ ❛✈❡r❛❣❡ t❤❛♥ t❤❡ ❧❛r❣❡r s❡❡❞s✳ ❚♦ ❞❡t❡r♠✐♥❡ ✐❢  P❘■▼❊ ●❊◆❊❘❆❚■◆● ▲❯❈❆❙ ❙❊◗❯❊◆❈❊❙  ✺  t❤❡r❡ ❡①✐st❡❞ ❛ s✐❣♥✐✜❝❛♥t ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ ❛✈❡r❛❣❡s✱ ❛ ♥✉❧❧ ❤②♣♦t❤❡s✐s ♦❢ ♥♦ ❞✐✛❡r❡♥❝❡ ✇❛s ❛ss✉♠❡❞ ❛♥❞ ❛ ▲♦❝❛t✐♦♥ ❊q✉✐✈❛❧❡♥❝❡ ❚❡st ✇❛s ♣❡r❢♦r♠❡❞✳  ❚❤❡ r❡s✉❧t✐♥❣ ♣✲✈❛❧✉❡ ✇❛s ❛♣♣r♦①✐♠❛t❡❧②  2.8 · 10−25 ✱  ✐♥❞✐❝❛t✐♥❣ t❤❛t t❤❡r❡ ✐s ❛ ✈❡r② s✐❣♥✐✜❝❛♥t ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♠❡❛♥s ♦❢ t❤❡ s♠❛❧❧❡r s❡❡❞s ❛♥❞ t❤❡ ❧❛r❣❡r s❡❡❞s✳  ❋✐❣✉r❡ ✹✿ ❚❤❡ ❢r❡q✉❡♥❝② ♦❢ ♣r✐♠❡s ✐♥ t❤❡ s♠❛❧❧❡st ✶✵✪ ♦❢ ♦✉r ♠✉❧t✐♣❧✐❡r s❡❛r❝❤ s♣❛❝❡ ✭❧❡❢t✮ ❛♥❞ t❤❡ ❜✐❣❣❡st ✶✵✪ ✭r✐❣❤t✮✳ ❙♠❛❧❧❡r ♠✉❧t✐♣❧✐❡rs ②✐❡❧❞❡❞ ❛♥ ❛✈❡r❛❣❡ ♦❢ ✶✵ ♣r✐♠❡s✱ ✺ ♠♦r❡ ♦♥ ❛✈❡r❛❣❡ t❤❛♥ ❜✐❣ ♠✉❧t✐♣❧✐❡rs✳ ❙❡❡ ❛♣♣❡♥❞✐① ❢♦r ❤♦✇ ♠✉❧t✐♣❧✐❡rs ✇❡r❡ s❡♣❛r❛t❡❞ ❢♦r t❤❡ ❤✐st♦❣r❛♠s✳ ❋r♦♠ t❤❡ ✷✹✹✻✸ ❞❛t❛ ♣♦✐♥ts ❝♦❧❧❡❝t❡❞ ❜② ✈❛r②✐♥❣ t❤❡ ♠✉❧t✐♣❧✐❡rs ♦❢  Ln ✱  ✐t ❝❛♥ ❜❡ s❡❡♥ t❤❛t t❤❡ ♠✉❧t✐♣❧✐❡rs  ❜❡❧♦♥❣✐♥❣ ✐♥ t❤❡ ❧♦✇❡r ✶✵✪ ♣r♦❞✉❝❡ ✺ ♣r✐♠❡s ❤✐❣❤❡r ♦♥ ❛✈❡r❛❣❡ t❤❛♥ t❤❡ ❧❛r❣❡r ♠✉❧t✐♣❧✐❡rs ✭t❤❡ s❛♠❡ ❤♦❧❞s ❢♦r ✈❛r②✐♥❣ t❤❡ ♠✉❧t✐♣❧✐❡rs ♦❢ t❤❡ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡✮✳ ❙✐♠✐❧❛r t♦ t❤❡ ❛♥❛❧②s✐s ❜❡t✇❡❡♥ ❞✐✛❡r❡♥❝❡ s❡❡❞s✱ ❛ ▲♦❝❛t✐♦♥ ❊q✉✐✈❛❧❡♥❝❡ ❚❡st ✇❛s ♣❡r❢♦r♠❡❞ ❛♥❞ ❛ ♣✲✈❛❧✉❡ ♦❢  4.8 · 10−67  ✇❛s ❝❛❧❝✉❧❛t❡❞✱ ✐♥❞✐❝❛t✐♥❣ t❤❛t t❤❡  ❧♦✇❡r ✶✵✪ ❛r❡ ❛❧♠♦st ❝❡rt❛✐♥❧② ❞✐✛❡r❡♥t t❤❛♥ t❤❡ t♦♣ ✶✵✪✳  ❉✐s❝✉ss✐♦♥✳ ❋r♦♠ t❤❡ P◆❚✱ ✇❡ ❝❛♥ s❡❡ t❤❛t ❢♦r ❧❛r❣❡  t♦  ln N ✳  ❋r♦♠ t❤❡ ❡q✉❛t✐♦♥s ♦❢  t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ♦❢  αn ✳  Vn  ❛♥❞  N✱  t❤❡ ♣r♦❜❛❜✐❧✐t② ❞❡❝r❡❛s❡s ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧  Un ✱ ✇❡ ❝❛♥ ❛❧s♦ s❡❡ t❤❛t t❤❡ ❣r♦✇t❤ ♦❢ ▲✉❝❛s ♥✉♠❜❡rs ❢♦❧❧♦✇s r♦✉❣❤❧②  ❍❡♥❝❡✱  P (prime) =  1 1 = ln N n ln α  ❛♥❞ s♦ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♣r✐♠❡ ▲✉❝❛s ❞❡❝r❡❛s❡s ❜② r♦✉❣❤❧②  1 n✳  ❚❤✐s r♦✉❣❤ ❝❛❧❝✉❧❛t✐♦♥ s❤♦✇s ✉s t❤❡ ♣r✐♠❡ ❞❡♥s✐t② ♦❢ ▲✉❝❛s s❡q✉❡♥❝❡s ✐s ✐♥t✐♠❛t❡❧② ❝♦♥♥❡❝t❡❞ ✇✐t❤ t❤❡ ❣r♦✇t❤ ♦❢ ✐ts t❡r♠s✳ ❆s ❡①♣❡❝t❡❞✱ ✇❡ t❤❡♥ s❡❡ t❤❛t t❤❡ s♠❛❧❧ s❡❡❞s ❛♥❞ s♠❛❧❧ ♠✉❧t✐♣❧✐❡rs ♦❢ t❤❡ ❡①❛♠✐♥❡❞ ▲✉❝❛s s❡q✉❡♥❝❡s ♣r♦❞✉❝❡s s✐❣♥✐✜❝❛♥t❧② ♠♦r❡ ♣r✐♠❡s ♦♥ ❛✈❡r❛❣❡ t❤❛♥ t❤❡ ❧❛r❣❡ ♣r✐♠❡s ❛♥❞ ❧❛r❣❡ ♠✉❧t✐♣❧✐❡rs✳ ▼♦r❡♦✈❡r✱ ✐♥❝r❡❛s✐♥❣ t❤❡ s✐③❡ ♦❢ t❤❡ ♠✉❧t✐♣❧✐❡rs ❛❧s♦ ❤❛s ❛ ♠✉❝❤ ❣r❡❛t❡r ❡✛❡❝t t❤❛♥ ✐♥❝r❡❛s✐♥❣ t❤❡ s✐③❡ ♦❢ t❤❡ s❡❡❞s✱ ❛s ✐♥❝r❡❛s✐♥❣ ♠✉❧t✐♣❧✐❡r s✐③❡ ✐♥❝r❡❛s❡s  α✳  ❚❤✐s ❡①♣❧❛✐♥s ✇❤② t❤❡r❡ ✇❛s s✉❝❤ ❛ s✐❣♥✐✜❝❛♥t ❞✐✛❡r❡♥❝❡  ❜❡t✇❡❡♥ t❤❡ ❜♦tt♦♠ ✶✵✪ ♦❢ t❤❡ ♠✉❧t✐♣❧✐❡r ❞✐str✐❜✉t✐♦♥ ❝♦♠♣❛r❡❞ t♦ t❤❡ t♦♣ ✶✵✪✳ ■t ❛❧s♦ ❡①♣❧❛✐♥s t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ❛s ✇❡❧❧✳ ❙✐♥❝❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣r✐♠❡s ❜❡❝♦♠❡ ♠♦r❡ s♣❛rs❡ ❛s t❤❡ ♥✉♠❜❡rs ❣r♦✇ ❧❛r❣❡✱ ❢❡✇❡r ♣r✐♠❡s ✇✐❧❧ ❜❡ ♦❜t❛✐♥❡❞ ❜② t❤❡ ❤✐❣❤❡r ✈❛❧✉❡s ♠♦r❡ ♦❢t❡♥✳ ❚❤✉s✱ t❤❡ ❞✐str✐❜✉t✐♦♥ ✇✐❧❧ ❜❡ s❦❡✇❡❞ t♦ t❤❡ ❧❡❢t ❛s ✇❡ ❝❛♥ s❡❡ ♦❜✈✐♦✉s❧② s❡❡✳  ❲❡ ❝❛♥ ❛❧s♦ ♣r♦✈✐❞❡ ❛♥ ❡st✐♠❛t❡ ❢♦r t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ ♣r✐♠❡s ✭E({an ⑥✮✮ ✐♥ ❛ ❣✐✈❡♥ ✭✶✵✵✵ t❡r♠✮ ▲✉❝❛s s❡q✉❡♥❝❡ ❜② s✐♠♣❧② s✉♠♠✐♥❣ ✉♣ t❤❡ ✜rst ✶✵✵✵ ♣r♦❜❛❜✐❧✐t✐❡s✿  1000  E({Mn }) = n=1  1 ln Mn  P❘■▼❊ ●❊◆❊❘❆❚■◆● ▲❯❈❆❙ ❙❊◗❯❊◆❈❊❙  ✻  ❋✐❣✉r❡ ✺✿ ❖✉r ♣r✐♠❡ ❞❡♥s✐t② ❡st✐♠❛t❡ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♣r✐♠❡s t♦ ❜❡ ②✐❡❧❞❡❞ ❢♦r ✈❛r✐♦✉s s❡❡❞ ✈❛❧✉❡s ✭❧❡❢t✮ ❛♥❞ ♠✉❧t✐♣❧✐❡r ✈❛❧✉❡s ✭r✐❣❤t✮✳ ❚❤❡ ♠❡❛♥ ♦❢ t❤❡ ❧❡❢t ❞✐str✐❜✉t✐♦♥ ✐s ❢♦✉♥❞ t♦ ❜❡ ✾✳✸✸ ✇✐t❤ ❛ ✵✳✷✺ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥✱ ✇❤✐❧❡ t❤❡ ♠❡❛♥ ♦❢ t❤❡ r✐❣❤t ❞✐str✐❜✉t✐♦♥ ✐s ✶✳✼✵ ✇✐t❤ ❛ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ ✵✳✸✽✳ ❚❤❡② ❛r❡ ❜♦t❤ ❧❛r❣❡ ✉♥❞❡r❡st✐♠❛t❡s ♦❢ t❤❡ ❛❝t✉❛❧ ♠❡❛♥✱ ❜✉t t❤❡ ❞✐str✐❜✉t✐♦♥ ❛♣♣❡❛rs s✐♠✐❧❛r✳ ❚❤♦✉❣❤ ✉s✐♥❣ t❤❡ P◆❚ ♣r♦✈✐❞❡s ❛♥ ❡①tr❡♠❡❧② ♣♦♦r ❡st✐♠❛t❡ ♦❢ ♣r✐♠❡ ♣r♦❜❛❜✐❧✐t✐❡s ✐♥ ❣❡♥❡r❛❧ ✭❛s s❡❡♥ ❢r♦♠ t❤❡ ♠✉❧t✐♣❧✐❡rs ❡st✐♠❛t❡✮✱ ✇❡ ❝❛♥ s❡❡ ✭❢♦r ❞✐✛❡r❡♥t s❡❡❞ ✈❛❧✉❡s ✐♥ ♣❛rt✐❝✉❧❛r✮ s♦♠❡ s✐♠✐❧❛r✐t✐❡s ❜❡t✇❡❡♥ t❤❡ ❞✐str✐❜✉t✐♦♥s ♣r♦❞✉❝❡❞ ❜② t❤❡ P◆❚ ❣r❛♣❤ ❛♥❞ t❤❡ ❞✐str✐❜✉t✐♦♥ ♣r♦❞✉❝❡❞ ❜② t❤❡ ❛❝t✉❛❧ ❞❛t❛✳ ❚❤❡ ❞✐str✐❜✉t✐♦♥s ♦❜s❡r✈❡❞ ✐♥ t❤❡ ❡st✐♠❛t❡ ♦❢ ♣r✐♠❡ ♣r♦❜❛❜✐❧✐t✐❡s✱ ♣r✐♠❡s ✇✐t❤✐♥ ✈❛r✐♦✉s ♠✉❧t✐♣❧✐❡rs ♦❢ ❋✐❜♦♥❛❝❝✐ ❛♥❞ ▲✉❝❛s s❡q✉❡♥❝❡s✱ ❛♥❞ ♣r✐♠❡s ✇✐t❤✐♥ ▲✉❝❛s s❡q✉❡♥❝❡s ✇✐t❤ ❞✐✛❡r❡♥t ✐♥✐t✐❛❧ s❡❡❞s ❛❧❧ ❛♣♣❡❛r str❛♥❣❡❧② s✐♠✐❧❛r✳ ❚❤✐s ✐s s♦♠❡✇❤❛t ❡①♣❡❝t❡❞✱ ❞✉❡ t♦ t❤❡ P◆❚ ♣r❡❞✐❝t✐♥❣ t❤❛t ❜♦t❤ s♠❛❧❧❡r ✐♥✐t✐❛❧ ✈❛❧✉❡s ❛♥❞ s♠❛❧❧❡r ♠✉❧t✐♣❧✐❡rs ✇♦✉❧❞ ❣❡♥❡r❛❧❧② ②✐❡❧❞ ❛ ❤✐❣❤❡r ♥✉♠❜❡r ♦❢ ♣r✐♠❡s✱ ❛❧t❤♦✉❣❤ ✇❤② t❤❡s❡ ✈❛r✐♦✉s s✐t✉❛t✐♦♥s ❧❡❛❞ t♦ s✉❝❤ s✐♠✐❧❛r ❣r❛♣❤s ✐s ✉♥❦♥♦✇♥✳ ◆♦ ❝♦♠♠♦♥ ❞✐str✐❜✉t✐♦♥ ❛♣♣❡❛r❡❞ t♦ ♠♦❞❡❧ t❤✐s ❞❛t❛✱ ❛s ♥♦r♠❛❧✱ P♦✐ss♦♥✱ ❣❛♠♠❛ ♥♦r ❜❡t❛✲♣r✐♠❡ ❞✐str✐❜✉t✐♦♥ ❣❛✈❡ s❛t✐s❢❛❝t♦r② r❡s✉❧ts✳ ❯s✐♥❣ ✈❛r✐♦✉s st❛t✐st✐❝❛❧ ❞✐str✐❜✉t✐♦♥ t❡sts✱ t❤❡ ♣✲✈❛❧✉❡ ♦❢ t❤❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ♦❜t❛✐♥❡❞ ❢r♦♠ ▼❛t❤❡♠❛t✐❝❛✬s ❝❛❧❝✉❧❛t✐♦♥s ♣r♦❞✉❝❡❞ ❛ ✈❛❧✉❡ ♦♥ t❤❡ ♦r❞❡r ♦❢  10−14 ✱  ✐♥❞✐❝❛t✐♥❣ t❤❛t ✐t ✐s ✈❡r② ✉♥❧✐❦❡❧② t❤❛t t❤❡ ❞❛t❛ ❝❛♠❡ ❢r♦♠ t❤❡ ❞✐str✐❜✉t✐♦♥✳ ❆ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ s❡❡♠❡❞ ❧✐❦❡ t❤❡ ♠♦st ❧✐❦❡❧② ❝❛♥❞✐❞❛t❡ ❛s s❤♦✇♥ ❜② ●❛❧❧❛❣❤❡r ❬✹❪ ✇❤❡♥ t❤❡ ❜✐♥s t❡♥❞ t♦ ✐♥✜♥✐t②✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ✐t t♦♦ ❢❛✐❧❡❞ t❤❡ st❛t✐st✐❝❛❧ ♠♦❞❡❧ t❡sts✱ ♦♥ t❤❡ ♦r❞❡r ♦❢  10−14  ❛s ✇❡❧❧✱ s❤♦✇✐♥❣ t❤❛t t❤❡ ❞✐str✐❜✉t✐♦♥ ✐s ✉♥❧✐❦❡❧②✳  ❖t❤❡r ❞✐str✐❜✉t✐♦♥s s✉❝❤ ❛s t❤❡ ❜❡t❛✲♣r✐♠❡✱ ❜✐♥♦♠✐❛❧✱ ❛♥❞ ❣❛♠♠❛ ❞✐str✐❜✉t✐♦♥s ✇❡r❡ ❛tt❡♠♣t❡❞ ❛♥❞ ❛❧s♦ ❢❛✐❧❡❞ ✇✐t❤ ♣✲✈❛❧✉❡s ♦♥ t❤❡ ♦r❞❡r ♦❢  10−15  ❛♥❞  10−14 ✳  ❖❞❞❧②✱ ❜❛s❡❞ ♦♥ t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❣r❛♣❤s✱ t❤❡ ❣❛♠♠❛  ❞✐str✐❜✉t✐♦♥ ❛♣♣❡❛r❡❞ t♦ ❜❡ t❤❡ ❝❧♦s❡st✱ ❜✉t ♣r♦❞✉❝❡❞ t❤❡ ❧♦✇❡st ♣✲✈❛❧✉❡✱ ✐♥❞✐❝❛t✐♥❣ ✐t ✇❛s st✐❧❧ ❛ ♣♦♦r ✜t✳  ❆ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✐s ✉♥❧✐❦❡❧② ❢♦r t✇♦ ❧❛r❣❡ r❡❛s♦♥s✱ ❛s ✐t ❛ss✉♠❡s ❛♥ ❡✈❡♥ ❛♥❞ r❛♥❞♦♠ ❞✐str✐❜✉t✐♦♥ ♦❢ ❞❛t❛ ♦♥ ❡✐t❤❡r s✐❞❡ ♦❢ t❤❡ ♠❡❛♥✱ ❛♥❞ ✐t ✐s ❛ ❝♦♥t✐♥✉♦✉s ❞✐str✐❜✉t✐♦♥ ✇❤❡r❡ ♦✉r ❞❛t❛ ✐s ❝❧❡❛r❧② ❞✐s❝r❡t❡✳ ❆ ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ ✐s ♦❜✈✐♦✉s❧② ✇r♦♥❣ ❛s ✇❡❧❧✱ s✐♥❝❡ ✇❡ ❛r❡ ❧♦♦❦✐♥❣ ❛t t❤❡ ♥✉♠❜❡r ♦❢ ♣r✐♠❡s ✐♥ t❤❡ ✜rst ✶✵✵✵ t❡r♠s ♦❢ ❛ s❡q✉❡♥❝❡✱ ♥♦t s✉❝❝❡ss✴❢❛✐❧✉r❡ ❡①♣❡r✐♠❡♥ts ✇✐t❤ ♦♥❧② t✇♦ ♦✉t❝♦♠❡s✳  ❆❞❞✐t✐♦♥❛❧❧②✱ ❞✉❡ t♦ ❝❡rt❛✐♥ s♣❡❝✐❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ▲✉❝❛s s❡q✉❡♥❝❡s✱ t❤❡ ♣r✐♠❡s ♦❜t❛✐♥❡❞ ❛r❡ ♥♦t ❡①❛❝t❧② r❛♥❞♦♠✳  ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ❡❛s✐❧② s❡❡ ❢r♦♠ t❤❡ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡ t❤❛t t❤❡r❡ ❛r❡ t✇♦ ♦❞❞ t❡r♠s ❢♦r  ❡✈❡r② ❡✈❡♥ t❡r♠✳ ■❢ ♦✉r s❛♠♣❧❡ ✇❛s ❞r❛✇♥ ❢r♦♠ ❛ r❛♥❞♦♠ s❡t ♦❢ ✐♥t❡❣❡rs✱ t❤❡r❡ ✇♦✉❧❞ ❜❡ ❛ ✺✵✪ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞r❛✇✐♥❣ ❛♥ ❡✈❡♥ t❡r♠ ✈❡rs✉s ❛♥ ♦❞❞ t❡r♠✳  ❍♦✇❡✈❡r✱ ❜❡❝❛✉s❡ ♦❢ t❤❡ r❡❝✉rs✐✈❡ ❛❞❞✐♥❣ ✐♥✈♦❧✈❡❞ ✐♥ t❤❡  ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡✱ t❤❡r❡ ✐s ❛ t✇♦✲t❤✐r❞s ♣r♦❜❛❜✐❧✐t② ♦❢ ❞r❛✇✐♥❣ ❛♥ ♦❞❞ ♥✉♠❜❡r ❛♥❞ ♦♥❡✲t❤✐r❞ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞r❛✇✐♥❣ ❛♥ ❡✈❡♥ ♥✉♠❜❡r✳ ▼♦r❡♦✈❡r✱ t❤❡ r❡❝✉rs✐✈❡ ❛❞❞✐♥❣ ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡ ❛❧s♦ ❣✉❛r❛♥t❡❡s ❛ ✉♥✐q✉❡ ♣r✐♠❡ ❢❛❝t♦r ❛♣♣❡❛r✐♥❣ ✐♥ ❡❛❝❤ ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡r t❤❛t ❤❛s ♥♦t ❜❡❡♥ ❛ ❢❛❝t♦r ♦❢ ♣r❡✈✐♦✉s ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡rs ❬✸❪✳ ❚❤✉s✱ t❤❡ str✉❝t✉r❡ ❜❡❤✐♥❞ ❢❛❝t♦rs ❛♣♣❡❛r✐♥❣ ✐♥ r❡❣✉❧❛r ✐♥t❡❣❡rs ✭❛ ❢❛❝t♦r ♦❢ ✷ ❢♦r ❡✈❡r② s❡❝♦♥❞ t❡r♠✱ ✸ ❢♦r ❡✈❡r② t❤✐r❞ t❡r♠✱ ❡t❝✳✮ ❞✐s❛♣♣❡❛rs ❡♥t✐r❡❧② ✐♥ t❤❡ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡✳ ❙✉❝❤ ♣r♦♣❡rt✐❡s ❛❧s♦ ❛♣♣❡❛r ✐♥ ♦t❤❡r ▲✉❝❛s s❡q✉❡♥❝❡s ❛s ✇❡❧❧✱ s✉❝❤ ❛s t❤❡ ▲✉❝❛s s❡q✉❡♥❝❡ ❞❡s❝r✐❜✐♥❣ ▼❡rs❡♥♥❡ ♥✉♠❜❡rs✳ ❚❤✉s✱ ❝❤❛♥❣✐♥❣ ❜♦t❤ t❤❡ s❡❡❞s ❛♥❞ t❤❡ ♠✉❧t✐♣❧✐❡rs s✉✛❡r ❢r♦♠ t❤❡ ❛❜♦✈❡ ♣r♦♣❡rt✐❡s ❛♥❞ ②✐❡❧❞ ❞✐✣❝✉❧t✐❡s ✇❤❡♥ ♠♦❞❡❧❡❞ ❜② ❛ s✐♠♣❧❡ ❞✐str✐❜✉t✐♦♥✳ ❈♦♥❝❧✉s✐♦♥✳ ❆❢t❡r ❝♦♠♣✐❧✐♥❣ ❛❧❧ ♦❢ t❤❡ ♣r✐♠❡s ❣❡♥❡r❛t❡❞ ❜② ▲✉❝❛s s❡q✉❡♥❝❡s ❛♥❞ ♣❡r❢♦r♠✐♥❣ ✈❛r✐♦✉s st❛t✐s✲  t✐❝❛❧ t❡sts ♦♥ ✈❛r✐♦✉s s✉❜s❡❝t✐♦♥s ♦❢ ✐t✱ ❧♦✇❡r ✐♥✐t✐❛❧ s❡❡❞s ✇❡r❡ ❢♦✉♥❞ t♦ ❜❡ st❛t✐st✐❝❛❧❧② s✐❣♥✐✜❝❛♥t ❢r♦♠ ❧❛r❣❡r s❡❡❞s✱ ♣r♦❞✉❝✐♥❣ ♦♥ ❛✈❡r❛❣❡ ✶ ♠♦r❡ ♣r✐♠❡✳ ❙✐♠✐❧❛r❧②✱ s♠❛❧❧❡r ♠✉❧t✐♣❧✐❡rs ✇❡r❡ ❢♦✉♥❞ t♦ ♣r♦❞✉❝❡ ✶ ♠♦r❡ ♣r✐♠❡ ♦♥ ❛✈❡r❛❣❡✳ ❆❧t❤♦✉❣❤✱ ●❛❧❧❛❣❤❡r ❬✹❪ ♣r❡❞✐❝ts t❤❛t t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡s❡ ♣r✐♠❡s s❤♦✉❧❞ ❢♦❧❧♦✇ ❛ P♦✐ss♦♥  P❘■▼❊ ●❊◆❊❘❆❚■◆● ▲❯❈❆❙ ❙❊◗❯❊◆❈❊❙  ✼  ❞✐str✐❜✉t✐♦♥ ❛s t❤❡ ✐♥t❡r✈❛❧s t❡♥❞ t♦ ✐♥✜♥✐t②✱ ♦✉r ❞❛t❛ ❞♦❡s ♥♦t ♣❛rt✐❝✉❧❛r❧② s❤♦✇ t❤✐s✱ ❝♦♥s✐❞❡r✐♥❣ ✇❡ ✉s❡❞ ✜♥✐t❡ ❜✐♥s ❢♦r ♦♥❧② t❤❡ ✜rst ✶✵✵✵ t❡r♠s ✐♥ ❛ ▲✉❝❛s s❡q✉❡♥❝❡✳  ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♣r✐♠❡s ✇❤✐❝❤ ❛♣♣❡❛r❡❞  s✐♠✐❧❛r ✐♥ ❛❧❧ t❡sts✱ r❡♠❛✐♥s ✉♥❦♥♦✇♥✱ ❛♥❞ r❡q✉✐r❡s ❢✉rt❤❡r ✐♥✈❡st✐❣❛t✐♦♥ t♦ ❝♦♥✜r♠ t❤❛t ✐t ✇♦✉❧❞ t❡♥❞ t♦ ❛ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥✳ ❆❝❦♥♦✇❧❡❞❣♠❡♥ts✳ ❚❤♦✉❣❤ t❤❡r❡ ❛r❡ ♠❛♥② ✇❤♦ ❝♦♥tr✐❜✉t❡❞ t♦ t❤❡ ✇r✐t✐♥❣ ♦❢ t❤✐s ♣❛♣❡r✱ ✇❡ ✇♦✉❧❞ ❧✐❦❡  t♦ t❤❛♥❦ t✇♦ ♣❛rt✐❡s ✐♥ ♣❛rt✐❝✉❧❛r✳ ❲✐t❤♦✉t t❤❡✐r ❡✛♦rts ❛♥❞ ❛♥❞ t✐♠❡✱ ❝♦♠♣❧❡t✐♦♥ ♦❢ t❤✐s ♣❛♣❡r ✇♦✉❧❞ ♥♦t ❤❛✈❡ ❜❡❡♥ ♣♦ss✐❜❧❡✳ ❋♦r ❤✐s ❡①❝❡❧❧❡♥t ❛❞✈✐❝❡✱ s❤❛r♣ ✇✐t ❛♥❞ ✏r✉❣❣❡❞ ❣♦♦❞ ❧♦♦❦s✱✑ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ ♦✉r ❛❞✈✐s♦r✱ Pr♦❢❡ss♦r ❋♦❦✲❙❤✉❡♥ ▼❛tt❤❡✇ ▲❡✉♥❣✳ ❋♦r ❛♥ ❡①❝❡❧❧❡♥t ❛♥❞ t❤♦r♦✉❣❤ ❥♦❜ ♦❢ ❡❞✐t✐♥❣ t❤✐s ♣❛♣❡r✱ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ P❛✉❧ ❑❛♣♦s ❛♥❞ ❇❡♥♥❡t ▲❡✉♥❣✳ ❆❞❞✐t✐♦♥❛❧ ❣r❛t✐t✉❞❡ ❣♦❡s t♦✇❛r❞s ❛❧❧ ✇❤♦ ❤❛✈❡ ❤❡❧♣❡❞ ❜✉t ✇❤♦s❡ ♥❛♠❡s ❛r❡ ♥♦t ❡①♣❧✐❝✐t❧② ♠❡♥t✐♦♥❡❞✳  ❘❡❢❡r❡♥❝❡s ❬✶❪ ❘✐❜❡♥❜♦✐♠✱ P✳ ✭✷✵✵✵✮✳ ▼② ◆✉♠❜❡rs✱ ▼② ❋r✐❡♥❞s✳ P♦♣✉❧❛r ▲❡❝t✉r❡s ♦♥ ◆✉♠❜❡r ❚❤❡♦r②✳ ◆❡✇ ❨♦r❦✿ ❙♣r✐♥❣❡r✳ ❬✷❪ ✭✷✵✶✶✮ ●r❡❛t ■♥t❡r♥❡t ▼❡rs❡♥♥❡ Pr✐♠❡ ❙❡❛r❝❤✳ ❘❡tr✐❡✈❡❞ ❢r♦♠✿ ❤tt♣✿✴✴✇✇✇✳♠❡rs❡♥♥❡✳♦r❣✴✳ ❬✸❪ ❈❛r♠✐❝❤❛❡❧✱ ❘✳❉✳ ❖♥ t❤❡ ◆✉♠❡r✐❝❛❧ ❋❛❝t♦rs ♦❢ t❤❡ ❆r✐t❤♠❡t✐❝ ❋♦r♠s αn ± β n ✳ ❚❤❡ ❆♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝s ✶✺✱ ✸✵✲✹✽ ✭✶✾✶✸✲✶✾✶✹✮✳ ❬✹❪ ●❛❧❧❛❣❤❡r✱ P✳ ❳✳ ❖♥ t❤❡ ❉✐str✐❜✉t✐♦♥ ♦❢ Pr✐♠❡s ✐♥ ❙❤♦rt ■♥t❡r✈❛❧s✳ ▼❛t❤❡♠❛t✐❦❛ ✷✸✱ ✹✲✾ ✭✶✾✼✻✮✳ ❆♣♣❡♥❞✐①✳ ❘❛♥❦✐♥❣ s②st❡♠ ♦❢ t❤❡ ♠✉❧t✐♣❧✐❡rs ❛♥❞ s❡❡❞s✳ ❚♦ ❣r♦✉♣ t❤❡ ♠✉❧t✐♣❧✐❡rs ❛♥❞ s❡❡❞s ✐♥t♦ t❤❡ t♦♣ ✶✵✪ ❛♥❞ ❜♦tt♦♠  ✶✵✪✱ s❝♦r❡s ✇❡r❡ ❛ss✐❣♥❡❞ t♦ ❡❛❝❤ ♣❛✐r ♦❢  (P, Q)  ❛♥❞  (a, b)  ❜❛s❡❞ ♦♥ t❤❡ ❣r♦✇t❤ r❛t❡ ♦❢ t❤❡✐r s❡q✉❡♥❝❡s✳  ❋♦r ❞✐✛❡r❡♥t s❡❡❞s✱ s✐♥❝❡ t❤❡ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡ ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜②  ϕn Fn = r♦✉♥❞ √ , 5 ✇❡ ❝❛♥ ❛♣♣r♦①✐♠❛t❡  Gn  ❛s  ϕn−2 ϕn−1 Gn = a · Fn−2 + b · Fn−1 ≈ a · √ + b · √ = 5 5 √ 1+ 5 ❛♥❞ ❑ 2  =  a ϕ  + b✳  ❆s t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ♦❢  a  ❛♥❞  ✇❤❡r❡  ϕ=  b  a ϕn−1 ϕn−1 +b · √ =❑· √ ϕ 5 5  ✐s ✶✵✵✵✱ t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ♦❢ ❑ ✐s ❑max  ❚❤✉s t❤❡ ❜♦tt♦♠ ✶✵✪ ♦❢ s❡❡❞s ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s s❡❡❞s ✇❤❡r❡ ❑  < 0.1❑max  = 1000  1 ϕ  +1  ✳  ✇❤✐❧❡ t❤❡ t♦♣ ✶✵✪ ♦❢ s❡❡❞s  > 0.9❑max ✳ P+ D ✭❛s ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥✮✱ ✇❡ s❡❡ t❤❛t ❢♦r ❞✐✛❡r❡♥t ♠✉❧t✐♣❧✐❡rs✱ ❣r♦✇t❤ ✐s ❝♦♥tr♦❧❧❡❞ 2 √ P + P 2 +4Q = ❚❤✉s✱ ♦✉r ♠❛①✐♠✉♠ ❣r♦✇t❤ r❛t❡ ✇♦✉❧❞ ❜❡ ✇❤❡♥ P = Q = 200✱ αmax = 2  ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s s❡❡❞s √ ✇❤❡r❡ ❑ ❉❡✜♥✐♥❣  α=  α✳ √ 100 + 100 102 ♠♦st❧② ❜②  ✭✇❤❡r❡ t❤❡  +4Q  ❝♦♠❡s ❢r♦♠ ❝♦♠♣✉t✐♥❣ ▲✉❝❛s s❡q✉❡♥❝❡s ♦❢  ❞❡✜♥❡ t❤❡ ❜♦tt♦♠ ✶✵✪ ❛s ♠✉❧t✐♣❧✐❡rs ✇❤❡r❡  α < 0.1αmax  Ln (P, −Q)✮✳ ❋r♦♠ α > 0.9αmax ✳  ❛♥❞ t❤❡ t♦♣ ✶✵✪ ❛s  t❤✐s✱ ✇❡ ❝❛♥  

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