A roof of the strong twin rime conjecture Men-Jw Ho # (retired), Chou-Jung Hsu, *, Wi-Jne Ho b Dertment of Industril Mngement # Nn Ki University of Technology Nn-Tou 54, Tiwn b Dertment of Medicinl Botnicl nd Helth Alictions D-Yeh University Dcun Chnghu 559, Tiwn * Corresonding uthor Tel: +886-49-563489 et 57; F: +886-49-56584 E-mil ddress: jrsheu@nkutedutw
A roof of the strong twin rime conjecture Abstrct For integers nd k, let T;k denote the number of twin rime irs, k with distnce k nd (not k T;k for ll k Logiclly, ; roose sliding model to estimte g ; nd g ; ) Let g ; T denote the verge of T k should be function of g ; T We first, T Second, derive the reltions between T ;k T from the sieve structure Third, settle the errors cused by the deendence of rimes Keywords: Sieve, Twin rime, Sliding model Introduction For integers nd k, let T;k denote the number of twin rime irs, k with distnce k nd nother twin rime irs Note tht twin rime irs, 6 my involves, Here the distnce does not men g ectly Let C 06606 be the twin rime constnt We try to rove the conjecture roosed by Hrdy nd Littlewood [] in 93 tht (equivlently) T ; k ~ C () k ; 3 ln
Generl nottions Throughout this er, the following nottions denote s follows, y, k, i, n: Integers, * : Primes n : Multile of i : i -th rime 3 k N : Set of ositive integers not greter thn k 4 T;k : Denotes s bove Note tht it is for 5 T ; y : Averge of ; g T k for ll k y but not k 6 : Denotes the correction fctors for the deendence of rimes The suffies re ttched to the fctors only to distinguish vrious cses 7 : Number of rimes not greter thn Sliding model A sliding model is shown in Figure While the row (B) is sliding to the right ste by ste nd totlly by, ll intersections of the rimes in row (B) nd row (A) re twin rime irs The intersections for ste k re the twin rime irs with distnce k For emle, while sliding to the right, the intersections for ste =, 4, 6, re: Ste, 3,5, 5,7,,3, 7,9, Ste 4, 3, 7, 7,, 3,7, 9, 3, Ste 6, 5,, 7,3,,7, 3,9, 3
Figure Sliding model to form twin rime irs, for ste = 6 (A) 3 5 7-3 - 7 9-3 - - 9 3 ste = 6 (B) 3 5 7-3 - 7 9-3 - - 9 3 Totlly intersection (,7) Lemm Tg ; ~ ~ ln () Proof: According to the rime number theorem, t T ; k ~ dudt (3) k t ln tln u We give n roimte estimtion for the integrl Let Q q denote the number of intersections of rime q in row (B), while q slides to the right totlly by For 3 q 4 3 ~ ~ ~ ln ln ln ln ln, Q For n rbitrrily smll ositive rel, if, For q r with r, r r r r Qr r r~ ~ ln r ln r ln r ln r ~ ~ ~ ln r ln r ln ln Thus k t T ; k ~ dudt ~ t ln tln u g T ;k k ; ~ ~ ln T 4
With the roimte estimtion, we hve ; ~ ; ~ ; ~ ; Tg Tg Tg Tg ~ ~ 4 ln (4) We highlight the key concets used in the following discussions Key concet It imlies tht T; k with smll distnce is similr to tht with lrge one, lthough the rimes re infinitely srse s Logiclly, ; T k must be function of g ; T;k from the sieve structure to ehibit tht ; T In Section 6, we give derivtion of T k involves g ; T 3 Deendence of rimes For the sieves in this er, the difference between the intervl, nd, is ignored For Ertosthenes sieve, ~ ln (5) According to Mertens third theorem, e ~ 0, where is Euler s constnt The fctor is to correct the reltive errors cused by the deendence of rimes for Ertosthenes sieve Tht is, the ctul (corrected) frction of the crdinlity of the shifted set over is According to Lemm, we dd fctor ~0 ; Tg for () to hve 5
Lemm Let nd denote s bove Then Tg ; 3 (6) Proof: By (5), we hve Tg ; 3 Key concet Ignoring the errors cused by the deendence of rimes, for the number of rimes, roduct eression is derived from the sieve structure Since the roduct is logicl nd meningful, it should be n essentil comonent of in (5) The deendence of rimes my distort the meningful roduct derived from sieve, but it never violtes the nture of the sieve structure which sreds uniformly nd infinitely ll over the entire integer sequence For our trget conjecture, we derive some meningful roduct eressions from sieve, nd then dd fctor to correct the errors distorted by the deendence of rimes 4 First derivtion of T ; k from sieve Let the nottions for Lemm 3 denote s follows: Φ k: Ertosthenes sieve for k 0 N Φ k;k : Sieve to eliminte the elements, first the even numbers, second n k with 6
n for ll 3, from k N Φ kφ k;k : Denote tht we first do the sieve k 0 Φ k k from the shifted set of k ; Φ 0 sieve : Crdinlity of the shifted set of the sieve Φ Φ Lemm 3 T;k k k;k 0 Φ nd net do 0 Proof: If is rime which is not n k of ny, k is not n of ny Thus k, is twin rime ir Correction fctors By Key concet, we dd the correction fctors for the deendence of rimes to the result of Lemm 3 Since the correction fctor deends on how we eliminte the elements from k the fctor should be function of nd k, for T ; T ;k,k k, 3,k k 3, 3 k in Lemm 3 Thus N,, (7),k k Key concet 3 The fctor for correcting the errors cused by the deendence of rimes must be ttched to roduct, such s occur, corresonding to sieve rocess in which the errors 3 Eqution (7) imlies tht The correction fctor is ut in front of the corresonding roduct eression 7
First, we rocess the sieve corresonding to, nd then 3 k, 3 3 Let,k be the correction fctor for 3 k, 3 4 The correction fctor for k, 3, fter the sieve of hs been 3 done, is denoted s,k 5 Thus,k,k should be the correction fctor for while,k 3 is determined, nd,k,k,k,k Fictitious itertions These re the trnsformtions used in the following derivtions (8) In the roof of Lemm 4 below, eqution (7) will be multilied by k ; 3 k ; 3, which is ckge of fictitious itertions They re not substntil itertions, nd therefore hve nothing to do with the deendence of rimes Let R ; k be generl nottion in the following derivtions, nd R; k k ; 3 k ; 3 (9) Lemm 4 T ;k,k R; k 3 (0) 8
Proof: For (7), multilied by the fictitious itertions, k,k k 3, 3 T ;,,k k k ; 3 k ; 3,k, ;, k R k k () 3 3,k R; k 3 5 Second derivtion of T; k from sieve structure For 3, n 0, nd odd numbers, if is corime of 3, n must be corime of 3 However, n nd n only hve the chnce of to be corime of 3 For the distnce k, it revels the dvntges of the number of twin corime irs with k nd the disdvntges with k Let the nottions for Lemm 3 denote s follows: ; h, h k : Twin corime ir with distnce k, where h nd h k re corimes of H, ; k : Number of ; h, h k with distnce k, nd h H, ; : Averge of H, ; k for ll k g 9
Lemm 5 Let A k, k H, ; k k, k, B k, k H, ; k k, k Then A~ B, () A~ H g, ; B ~ H g, ;, nd Proof: By the dvntges of the number of twin corime irs with k, we hve A~ B AB B B Hg, ; ~ B B B B ~ Hg ;, H g, ; A~ B~ H g, ; Let the nottions denote s follows: y; m, m k : Twin corime ir with distnce k, where m nd m k re corimes of ll y M y, ; k : Number of y; m, m k 0 with distnce k, nd m M y, ; : Averge of M y, ; k for ll k g
Let be sufficiently smll ositive rel, nd y with ln y ln Thus the errors of the deendence of rimes for the itertions of ll y cn be ignored For the itertions of ll y, M y, ; k~ M y, ; k 3 ; 3 g y (3) k y For the itertions of ll, by Key concet, we dd fctor,k to correct the errors Recll tht R; k k ; 3 M, ; k,k M, ; k R; k Since g (4) 3 ; mm, k is equivlent to the twin rime ir, (4) is rewritten s T ;k,k T ; R ; k g (5) 3 Both of the dvntges nd disdvntges of, ; H k re reltive to g, ; H Actully, the formuls shown in Lemm 5 re indeendent for the sequentil itertions of 3, 5, 7,,, nd,k is not ccumulted u to constnt while getting lrge On the contrry, decresing s reltive devition, k, ~0 Since the indeendence of Lemm 5 is not elicit, we give Section 6 to rove it By Key concet 3,,k is the correction fctor of 3, which is the only roduct remined to shoulder the errors of deendence of rime
6 Correction fctor of 3 Let y with ln y ln We trnsform the roduct eression of sieves into summtion in order tht the frctions deleted by the itertions cn be hndled sertely 3 3 5 5 7 7 * * * * y * * y* * * (6) 3* y 3* 3 * y* * 3* Lemm 6 Let,kdenote s bove Then k Proof: Let D, k,, ~0 * be the ctul frction deleted by the itertion of *, nd, k, * be the correction fctor for the itertion D, k, *, k, * * 3*,k = D, k, D, k, 3 * * 3* y y* For the itertion of *, the number of the elements deleted from is 3* N, k, 3* * * * *, nd
, k, * (7) 3* since ~0, with (7), we hve D k * * y* y *,, ~0,, * ~0, even without (7) y Actully, D k Thus *,k ~ D, k, ~ ~ C 3 Since * 3* y 3y ~ C, we hve k 3 By (8), 3 3 3 3 3, ~0 The roduct removes 3 from 3 ~ y y Thus the totl removes of for ll y become meningless, ie, D, k, * ~0 y * 3
7 Conclusions Recll tht R; k By (5) nd Lemm, we hve k ; 3, nd k T; k~ T g ; 3 k ; 3, ~0 ~ C ~C (8) k ; 3 k ; 3 ln By Lemm 4 nd dding the correction fctor,k T ;k,k R ; k 3, we hve,k,k 3 3 Comring (5) with (9),,k (9) 3,k Tg ; (0),k e Recll tht ~ nd ~0 By Lemm, Tg ; 3,k,k, k ~, k ~ e By Lemm 4, we hve e T; k~ 3 k ; 3 () 4
Etended conclusion Let y with ln y ln Let m be corime of ll y Let m i be the i -th m The seril corime distnce of m nd m d is defined s d T d denote the number of twin rime irs m, m d, in which m nd Let ; m d re rimes, with distnce m m nd d Since k ; y ~, T; d roch the sme for ll d m m References [] Hrdy, G H & J E Littlewood (93) Some roblems of rtitio numerorum III : On the eression of number s sum of rimes Act Mth 44, -70 5