A PROOF OF THE TWIN PRIME CONJECTURE AND OTHER POSSIBLE APPLICATIONS

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1 A PROOF OF THE TWI PRIME COJECTURE AD OTHER POSSIBLE APPLICATIOS by PAUL S. BRUCKMA 38 Frot Street, #3 aaimo, BC V9R B8 (Caada) pbruckma@hotmail.com

2 ABSTRACT : A elemetary proof of the Twi Prime Cojecture (TPC) is give. A defiite itegral expressio is obtaied for () ; applyig the Riema defiitio of the itegral, it is evetually show that this itegral is ubouded, provig TPC. We also idicate that there exist ifiitely may prime pairs (p, q) with p < q such that : q p = k, k =,, ad q - p = ; also, we idicate that there exist ifiitely may prime triplets (p, p+u, p+u+v), provided u ad v are chose so as to avoid trivial cases. KEY WORDS OR TERMS asymptotic behavior ; coutig fuctio ; prime ; Twi Prime Cojecture ; twi prime pairs ; m-tuplets of primes with specified differeces ; Sophie Germai prime pairs.

3 3 A PROOF OF THE TWI PRIME COJECTURE AD OTHER POSSIBLE APPLICATIOS by Paul S. Bruckma 3 First Street Soitula, BC V 3E (Caada) pbruckma@hotmail.com. ITRODUCTIO : The primary purpose of this paper is to prove the Twi Prime Cojecture (TPC), which states that the umber of twi prime pairs (prime pairs (p, q) such that q = p+) is ifiite. I subsequet sectios, we also prove a umber of similar ad more geeral cojectures. We let () deote the characteristic fuctio of the odd primes, where is ay positive iteger. This otatio excludes the prime, which makes the esuig aalysis a bit more coveiet. We also itroduce the polyomial fuctios f ad g :, defied as follows for = 3, 4, : f (z) = ( ) z () 3 g : (z) = ( ) ( )z. () Clearly, f ad g : are polyomials, hece etire fuctios. By the Cauchy itegral formula, it follows that for all : 3 () = i f (z) dz z. (3) Here, is ay simple closed cotour i the complex plae, take i the positive (i.e. couter-clockwise) directio ad cotaiig the origi withi its iterior.

4 We also let x be ay real value i the uit iterval I [,]. We may the express g (x) i terms of f as follows : 4 f (z) f (z) g : (x) = ( ) x dz 3 i = dz 3 z i z 3 3 x ( ), or : z g : (x) = i f (z)f (x / z) dz 3 z. (4) We may suppose that is the circle {z = r exp(i), < }, where < r. Also, assume that x = r, so that z x = r I. The g: (r ) = Sice g : (r ) must be real-valued, we the see that f (re i i r )f e (re i ) d. r g : (r ) = i cos f (re ) d. (5) ow f i (re ) = f (re i ) f (re -i ) = m i(m) r (m) ()e, hece : m3 3 f i (re ) m = r (m) ()cos{(m ) } m3 3. (6) ote that f i (re ) f (r) () f = {*()}, where *() = () -. i Sice f (re ) = (m)r m cosm + i (m)r m si m, a alterative mode of expressio m3 that avoids double sums is the followig : m3 i m f (re ) = (m)r cosm (m)r m3 m3 We make the defiitio : Also, usig (6) : F () = f i (e ) = cosp 3p 3 F () = (p q) 3p 3q m p si m si p (7) (8) cos (9) It is foud by itegratig the expressio i (9) that F ( ) d = (*()).

5 5 Sice F ( - ) = F (), we see that / We ow make the followig defiitio : F ( ) d = *() () It the follows that : F ) + F (/ - ) = J () () / 4. MORE DEFIITE ITEGRAL EXPRESSIOS : Cosider the followig special values : We may set r = i (5), ad obtai : J ( ) d = *() () *() = f () = ( ) ; (3) 3 () = g : () =. ( ) ( ) (4) 3 () = cos F ( ) d. (5) Our goal is to show that the itegral i (5) is a ubouded fuctio of. Just as was doe for the itegral i (), we may decompose the itegral i (5) ito a pair of itegrals, usig the relatios : cos ( - ) = cos, F ( - ) = F (). The () = cos F ( ) d. We repeat the process, usig the relatios : cos ( - ) = cos, F ( - ) = F () ; this yields : () = / cos F ( ) d (6) ow make the followig defiitio : F ) - F (/ - ) = G () (7) Usig the relatio cos (/ - ) = -cos, ad the defiitio i (7), we obtai :

6 6 () = / 4 cos G ( ) d (8) It is i this latter form that we will obtai the order of magitude of (). Clearly, the values of cos over the iterval [, /4] are o-egative ad decreasig. It would therefore be helpful if we could determie the behavior of the fuctio G ( ) over this same iterval. 3. ESTIMATES OF ITEGRALS : The followig estimate for the itegral i () is appropriate for the Riema defiitio of the itegral : * (;) = F (k / ) k (9) Usig the same reasoig o the itegral i (), we may also defie # ( ; ) = k J (k / 4) () We fid that if is replaced by i (9), we obtai the followig relatio, after simplificatio : * (; ) = # ( ; ) F ( / 4) () Thus, the defiitios i (9) ad () are essetially equivalet for large. O the other had, based o the same reasoig applied to the itegrals i (6) ad (8), make the followig defiitios : * (; ) = cos(k / )F (k / ) k ; () # (; ) = cos(k / )G (k / 4) k ; (3) ote the followig : *( ; ) ~ # * ( ; ) ~ *() ; (; ) ~ # ( ; ) ~ ( ), as (4)

7 7 Clearly, it follows from (8) that F () > for all. This, of course, implies that J () > for all. We the see that the summads i (9) ad () are positive ; therefore the estimates i (9) ad () must also be positive. I poit of fact, we are oly moderately iterested i such estimates ; these are provided oly as umerical checks i the subsequet computatios. Our real iterest lies i the estimates give by () ad (3). If i () is replaced by, we obtai, after simplificatio : * # ( ; ) = ( ; ) (5) We the see that the two estimates i () ad (3) are essetially equivalet. Heceforth, we therefore use oly the estimate i (3), ad omit the superscript otatio #. For computatioal purposes, we have used =, = 5 ; after some rather extesive computatios, we fid that (; * 5) = Also, we fid that (; 5) = * The true values for = are as follows : () = 67, ad ( ) = 35. The computatios imply that the estimate i (3) is apparetly domiated by the value of G (). This observatio, of course, eeds to be cofirmed. The goal of the developmet i the followig sectio is to reach this coclusio, which will imply TPC. 4. ESTIMATE OF G ( AD PROOF OF TPC : Our startig poit is the followig geeralizatio of the Prime umber Theorem (PT) : a p ~ p a, a =,,, (6) (a ) a x This may be obtaied by partial itegratio of the auxiliary fuctio dx ; the derivatio x of (6) is left as a exercise for the reader. It might be oted that this appears i SIREV, i slightly differet form []. ote that for a =, (6) yields PT. The asymptotic behavior is, of course, exhibited as. We may also remark that the sum i (6) is ot substatially chaged, i a asymptotic sese, if we exclude the prime p = ad restrict p to be odd.

8 8 I this sectio, we use the relatio i (6) to derive certai relatios that evaluate the quatities F () ad G () for all (positive) values of. For our purpose, we restrict for ow to the iterval [, /4]. Make the followig defiitios : As we recall, by (8) : C () cos p, S () si p (7) 3p 3p Usig (7) : C () = C () ~ k ( ) k ( ) 3p k k (k)! (k F () = {C ()} + {S ()} (8) k k ) k k (p) k k = ( ) p. ow usig (6), we obtai : (k)! (k)!, or : k 3p C () ~ si() (9) Also, S ()= 3p k ( ) k k (p) (k )! k k k ( ) p ~ k (k )! 3p k or : ( ) k k (k )! (k k ), S () ~ cos() (3) It the follows from (8) that F () ~ 4 si ( / ) (3) We ow make the relatively iocuous assumptio that is a multiple of 4, or = 4M, say. With this assumptio, si{(/ - )/} = si{m - /} = (-) M- si(/) ; the F (/ - ) ~ 4 si ( / ) { / } (3) From the defiitio give i (7), it follows that if, / :

9 9 4 si ( / ) G () ~, ( / ) (33) It is ot difficult to show that G () = 4 4, () 4,3 (), where 4,j () = {p : p j (mod 4)}, with j = or 3. We kow that the sets 4, () ad 4,3 () are equiumerous, meaig that 4, () ~ 4,3 () as. This implies that each such quatity is ~ * () as. Therefore, we see that G () ~ {*()}. The first positive value of that occurs as the argumet of G () i formula (3) is = /4. We let = k/4, for k, the values that appear as positive argumets of G i (3). Makig this substitutio i (33), this yields : G (k/4) ~ 64 si (k / 8) k (k) ( k), or equivaletly, as : si(k / 8 G (k/4) ~ k / 8 k, (k =,,, ) (34) ( k) We ote that over the iterval [, ], the quatity k is a decreasig fuctio of k ( k) with rage [, ]. As we already kow, F () = * () ~, by PT. I the form give by (34), it is clear that if we let k, G () ~ ~ {*()} = F (), oce agai. ow the asymptotic formula give i (34) is valid as, but says othig about asymptotic behavior as : However, if we are to apply the formula i (3), we must cosider to be icreasig idefiitely, as well as. We may get aroud this difficulty by supposig that ad icrease idefiitely i a depedet maer ; we fid it coveiet to suppose that =. Heceforth, asymptotic behavior will implicitly be described as. I our computatios, we assumed that =, = 5. As a further refiemet i our computatios, we may replace the quatity asymptotically. We the make the defiitio : by * (), which is certaily valid

10 si{k / 4 G # (k / ) = * () k / 4 k, k =,,, / (35) ( k) We may also defie G # () = * (), the limitig case i (35) as k. We may also exted the domai of k to k = +/, +/,,, ad make the additioal defiitio for k = : G # ( / ) = * (). As we have see (from (5), (3) ad our other assumptios): / () ~ cos(k / )G (k / ). We may also assert that G (k / ) ~ G # (k / ). k It the follows that : / () # ~ cos(k / )G (k / ) k (36) The advatage of usig G # (k / ) i (36), as opposed to G (k / ), is that G # (k / ) is o-egative over k[, /) ; clearly, G # (k / ) = iff k = 4, 8,, 4[M/]. The same may ot be said of G (k / ), as verified by computatio. Although icidetal here, we may show from the defiitios that the G ' s ad # G s share some iterestig properties. Sice a descriptio of these properties would be distractig to the arrative, we idicate these i the Appedix. Returig to the expressio i (36), its other compoet, amely cos(k/), is of course positive ad decreasig over k[, /) (ad vaishes for k = /). We may therefore assert the followig cosequece of (36) : () > G # () (37) Sice G () ~, it follows that as : () > (38)

11 Sice is a ubouded fuctio of, this implies TPC! O the other had, the followig kow result is attributable to Bru : () = O Together with (38), this further shows that 5. THE STROG TWI PRIME COJECTURE : (39) () (4) The famous Hardy-Littlewood Cojecture give i [] (also kow as the strog Twi Prime cojecture, which we deote as STPC) is a extesio of TPC, ad provides a estimate for the value of (), which we ow kow is a ubouded fuctio of : () ~ C ~ C dx (4) x Here, C (the so-called Twi Primes Costat) is give by C =, where the p3 (p ) product is over all odd primes. Approximately, C o attempt will be made i this paper to prove STPC. 6. A MORE GEERAL COJECTURE : It would appear that the method idicated i this paper may be used similarly (with certai modificatios), to prove a variety of more geeral umber-theoretic cojectures. However, o rigorous proofs are give i this sectio or subsequet sectios ; we merely idicate a skeleto of a proof of how a rigorous developmet for the more geeral cases might proceed. The most obvious geeralizatio that comes to mid is the cojecture that k () is ubouded ; here, k () is the coutig fuctio of the prime pairs (p, p+k) with p. Our startig poit for this is the more geeral fuctio : 3

12 g k: (z) = ( ) ( k)z, = 3, 4, ; k=,, (4) 3 Icidetally, this explais the use of the otatio g : (x) used i the case of the twi primes. The first o-zero expoet appearig i the expasio of (4) eed ot ecessarily be equal to 3 ; for example, if k = 3, such expoet is 5, sice the first such prime pair of the type (p, p+6) is (5, ) ad ot (3, 9). If the coutig fuctio of the prime pairs (p, p+k) is deoted as k (), ote that k () = g k: (). A developmet similar to the precedig oe yields the followig : k () i cosk f (e ) d (43) By the same process as used previously for the twi primes, we fid that k () / i cosk f (e ) d (44) As before, we may deduce that k () is a ubouded fuctio of. 7. PRIME TRIPLETS : Aother applicatio that comes to mid ivolves the fuctio,4 (), which couts the umber of prime triplets (p, p+, p+6) with p. I order to prove that,4 () is ubouded, we would begi with the polyomial fuctio : g,4 : (z) = ( ) ( ) ( 6) z, = 3, 4, (45) 3 I a more geeral case, we would defie g u,v: (x) =( ) ( u) ( u v)x. ote that, i order to avoid trivial cases (e.g., u = v = ), we would eed to impose additioal coditios o u ad v, i order to esure that oe of the triplet compoets are divisible by 3 (except possibly 3 itself) ; we fid that we must exclude the cases where 3

13 u v (mod 3). Assumig that such coditios are i place, we make the followig 3 defiitio : u,v () =( ) ( u) ( u v) = g u,v: () (46) 3 We proceed by a process similar to the developmet idicated i Sectio. The u,v () = f (z) f (z) ( ) ( u) dz u v i = z () ( u) uv i z z 3 3 dz, or by referece to the defiitio i (44) : u,v () = i f (z)g z u: uv ( / z) dz (47) After further simplificatio, we obtai : u, v () = i(uv) i e f (e )g (e i u: Re It may be show that the expressio i (5) has the order ) d. (48) 3, which i tur shows that u, v () is ubouded. Clearly, this process ca be exteded to deal with ay m-tuples of primes that are to be spaced apart o-trivially at pre-determied itervals. 8. SOPHIE GERMAI PRIME PAIRS : We may also deal i like maer with the Sophie Germai primes, aother class of prime pairs that has ot bee discussed i the precedig treatmet. The Sophie Germai primes, amed i hoor of the early 9 th cetury Frech female mathematicia, are those primes p such that p+ is also prime. The iitial fuctios that we use to deal with these are g SG: (z) = ( ) ( ) z, ad SG () = ( ) ( ) = g SG: (). The proof that 3 there exist ifiitely may Sophie Germai pairs (p, p+) follows by a process that is aaous to that previously used. We omit the derivatio, ad simply idicate the result : 3 SG () i i i Re e f (e )f (e ) d (49) The itegrad reduces to the followig : p cos(p ) si si(p ) cos ;

14 4 i this expressio, all sums are over odd primes p. Agai usig the estimates previously derived i (9) ad (3) for C () ad S (), we obtai the followig estimates : cos( p ) = cos C () - si S () ~ cos si si ( cos), or cos( p ) ~ cos( ) si (5) Also, si( p ) = cos S () + si C () ~ si si cos ( cos), or The, after simplificatio : si( p ) ~ si( ) si (5) cos p cos(p ) si si(p ) ~ si si( / ) cos(( / ) ). (5) We see that this has the same order of magitude as F () ; by the same reasoig as applied to the case of the twi primes, it follows that the itegral i (49) is ubouded. I this case, as well as i certai more geeral situatios ivolvig m-tuples of primes, the applicable expressios appear to led themselves to comparable aalysis. o doubt, may other iterestig examples will occur to the reader. Some of these, at least, appear to be susceptible to the method idicated i this paper (with suitable modificatios). I all such cases, the applicable coutig fuctio is show to be ubouded from the uboudedess of the correspodig itegral. Geerally, such m-tuples of primes will be of order m, which is a ubouded fuctio of.

15 5 APPEDIX The followig properties are easily proved by the appropriate defiitios, over the domai k =,,, : G # (k / ) = - ( / k / ) ; G (k / ) = - ( / k / ) ; (i) G # Equivaletly, if k is replaced by k, the egative of the origial term is obtaied ; special cosideratios must be made at the ed poits k = ad k =. The relatios i (i) easily imply the followig : k G # G (k / ) ; G (k / ) ; (ii) G # k ( / 4) ; ( / 4) ; (iii) Also, the followig easily follows from the defiitio of G i (7) : / G G Aother iterestig result is the followig : ( ) d =, (iv) / k # G (k / ) ~.5 (v) Proof of (v) : From the defiitio of # G give i (35), the special value at k =, ad PT, we see that the sum i (v) (call it S ) is ~ 6 { / si (k / 4) 6 k k k / si (k / 4) }. k As, the secod sum i the precedig expressio teds to vaish. If J represets the sum k / 4 (k ) show that J, we the see that S ~ as. The S ~ 8 6 J 6 J 4. Also, it is easily 5, after simplificatio, which proves (v).

16 6 I view of the last result, the formula give i (36) ad STPC (assumig the latter to be true), we may regard the average cosie appearig i (36) to equal C /(.5) = (.8)C, which is approximately equal to.583. By this computatio, such average appearig i (36) is roughly 58. REFERECE []. G. H. Hardy & J. E. Littlewood. Some problems of Partitio umerorum III : O the expressio of a umber as a sum of primes. Acta Math. (93) 44, -7. Reprited i Collected Works of G. H. Hardy, Vol. I : 56-63, Claredo Press, Oxford, 966 []. SIAM Review (SIREV) Problem Sectio. Problem 8-5. Proposed by Paul S. Bruckma. MATHEMATICS SUBJECT CLASSIFICATIOS : P3, A4

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