Two Approache to Provng Goldbach Conecture By Bernard Farley Adved By Charle Parry May 3 rd 5
A Bref Introducton to Goldbach Conecture In 74 Goldbach made h mot famou contrbuton n mathematc wth the conecture that all even number can be expreed a the um of two prme (currently, h conecture tated a all even number greater than can be expreed a the um of two prme nce no longer condered a prme a t wa n Goldbach tme []) A of yet, no proof of Goldbach Conecture ha been found Th conecture ha been hown to be correct for a large amount of number ung numercal calculaton Some example are 3 7, 8 7, 3 97, and o on The tudy of Goldbach conecture ha led to very great achevement nce 9 For example, I M Vnogradov proved n937 that every uffcently large odd number can be expreed a the um of three prme Alo, the nvetgaton of Goldbach conecture ha been a catalyt for the creaton and development of everal number-theoretc method whch are ueful n number theory and other feld of mathematc In 9, the mathematcan Hlbert gave a famou peech at the nd Internatonal Congre of Mathematc held n Par where he propoed 3 problem for mathematcan n the th century Goldbach conecture wa part of one of thoe problem Then n 9, Hardy ad that Goldbach conecture not only one of the mot famou and dffcult problem n number theory, but alo n the whole of mathematc The frt great achevement toward Goldbach conecture were obtaned n the 9 The frt of whch wa n 93 when, ung the crcle method, Brth
mathematcan Hardy and Lttlewood proved that every uffcently large odd nteger the um of three odd prme and almot all even nteger are um of two prme, provded that the grand Remann hypothe aumed to be true In 99, Norwegan mathematcan Brun etablhed, ung h eve method, that every large even number the um of two number each havng at mot nne prme factor Then n 93, ung Brun method along wth h own dea of the denty of an nteger equence, Ruan mathematcan Schnrelman proved that every uffcently large nteger the um of at mot c prme for a fxed number c Then n 937, the Ruan mathematcan Vnogradov wa able to remove, ung the crcle method and h method on the etmaton of exponental um wth prme varable, the dependence on the grand Remann hypothe and therefore provde uncondtonal proof of Hardy and Lttlewood fndng And fnally, after mprovement on Brun method and h reult, Chnee mathematcan Chen Jng Run wa able to prove that every large even nteger the um of a prme and a product of at mot two prme n 966 [] Unfortunately, my mathematcal bacground not uffcent to undertand the advanced method of thee mathematcan However, th dd not end my curoty for the problem Snce learnng about t n hgh chool I have fddled wth t much n my pare tme In fact, t wa n hgh chool that I realzed an equvalent tatement to Goldbach conecture whch eemed to mae t more approachable It wa not untl the latter part of my college year that I had the mathematc bacground to tart attacng th equvalent tatement The purpoe of my reearch wa to approach Goldbach Conecture ung th equvalent tatement n hope of dcoverng nght nto an alternatve way of provng Goldbach Conecture For my reearch Mathematca wa the 3
prmary tool to collect data The dfferent functon ued to generate the data can be found n the Appendx An Equvalent Statement of Goldbach Conecture An equvalent tatement to Goldbach Conecture that for every nteger n there ext an nteger uch that n and n are prme number Theorem : For n, n p q where p and q are prme number f and only f there ext an nteger uch that n and n are prme number Proof: Suppoe n p q where p and q are prme number Oberve that n (q n) n q p and n (q n) q Thu there ext an nteger q n uch that n and n are prme number Now uppoe there ext an nteger uch that n and n are prme number Then ( n ) ( n ) n Chooe an nteger n Let π ( n ) where ( x) π equal the number of prme le than or equal to x The Chnee Remander Theorem guarantee that there ext an nteger uch that n and n are not dvble by the prme, 3,, p Theorem : For all n, there ext an nteger uch that n and n are not dvble by the prme, 3,, p Proof: Chooe a uch that a / ±n(mod p ) for By the Chnee Remander Theorem there ext a oluton modulo α (where α 3 p the th prmoral, and α for convenence) to the followng ytem of congruence equaton: 4
a (mod ) a (mod 3) a (mod p ) So then, a / ± n(mod p ) Thu n ± / (mod p ) for Therefore n and n are not dvble by the prme, 3,, p If n ± n, then n ± prme nce n ± n and n ± not dvble by all of the prme number le than or equal to n (th the eve of Eratothene) However, a oluton moduloα, o can be uch that n ± gnfcantly greater than n It value dependent on how a choen for However, f n the rght range, partcularly f n, then n and n are prme Moreover, f for every n there ext uch a then Goldbach conecture true Let loo at an example Let 5 n, then π ( 5) 4 ± n (mod ) ± n,(mod 3) ± n,3(mod 5) ± n 3,4(mod 7) Oberve that One poble way of choong a to have a, a, a, a Th gve 3 4 5 whch greater than5 5 However, f we ntead let a, a, a 4, a we get 9 < 5 and thu n 5 9 6 and 3 4 n 5 9 43 are both prme 5
The Sequence Approach A pecal equence wa created to generate a, gven an nteger eed, uch that n and n are not dvble by the prme, 3,, p It defned a follow: () Gven a potve nteger n, let π ( n ) and let be ome nteger Then defned a follow:,, If / ±n(mod p ), If p ) and / ±n(mod p ), If p ) and p ) Theorem 3: Gven n, let π ( n ), and let be ome nteger Then n ± not dvble by the prme, 3,, p Proof: If / ±n(mod p ), then / ± n(mod p ) If p ) and / ±n(mod p ), then / ± n(mod p ) Oberve that p relatvely prme to α 3 p Suppoe p ) and p ) Th only poble f > Thu < p So then / α (mod p ) and / (mod p ), but nce p ) and p ), / ± n(mod p ) Therefore, n all cae / ±n(mod p ) Furthermore, for r, r c c α c α are,, or ) But, α, where c, c,, c are contant (whch p dvde α, Thu 6
/ ± n(mod p ) Then / ±n(mod p ) for and therefore r u n ± not dvble by the prme, 3,, p For an example, agan let loo at 5 n Then π ( 5) 4 ± n (mod ) ± n,(mod 3) ± n,3(mod 5) ± n 3,4(mod 7) Oberve that Let ue the eed Then Oberve that / ± n(mod ) So then Oberve that n(mod 3) and / ± n(mod 3) So then 3 Oberve that 3 n(mod 5) and 3 3 6 n(mod 5) So then 3 6 5 Oberve that 5 / ± n(mod 7) So then 3 5 5 5 67 and 5 5 37 67 and 37 are not dvble by, 3, 5, or 7 4 3 Alo note that 37 and 67 are prme number For a gven n, (where π ( n ) ) behave omewhat erratcally for varyng value of The followng are value for, gven n and : 7
Table n {-,-99} {-99,-99} {-98,-63} {-97,-63} {-96,-63} {-95,-63} {-94,-63} {-93,-63} {-9,3} {-9,3} {-9,3} {-89,3} {-88,3} {-87,3} {-86,-8} {-85,-8} {-84,-8} {-83,-8} {-8,-8} {-8,-8} {-8,4} {-79,4} {-78,4} {-77,4} {-76,4} {-75,4} {-74,4} {-73,4} {-7,4} {-7,4} {-7,4} {-69,4} {-68,-63} {-67,-63} {-66,-63} {-65,-63} {-64,-63} {-63,-63} {-6,-57} {-6,-57} {-6,-57} {-59,-57} {-58,-57} {-57,-57} {-56,89} {-55,89} {-54,89} {-53,89} {-5,89} {-5,89} {-5,-39} {-49,-39} {-48,-39} {-47,-39} {-46,-39} {-45,-39} {-44,-39} {-43,-39} {-4,-39} {-4,-39} {-4,-39} {-39,-39}, {, } {-38,-3} {-37,-3} {-36,-3} {-35,-3} {-34,-3} {-33,-3} {-3,-7} {-3,-7} {-3,-7} {-9,-7} {-8,-7} {-7,-7} {-6,89} {-5,89} {-4,89} {-3,89} {-,89} {-,89} {-,3} {-9,3} {-8,3} {-7,3} {-6,3} {-5,3} {-4,3} {-3,3} {-,3} {-,3} {-,3} {-9,3} {-8,-3} {-7,-3} {-6,-3} {-5,-3} {-4,-3} {-3,-3} {-,3} {-,3} {,3} {,3} {,3} {3,3} {4,39} {5,39} {6,39} {7,39} {8,39} {9,39} {,3} {,3} {,3} {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {9,3} {,3} {,3} {,7} {3,7} {4,7} {5,7} {6,7} {7,7} {8,63} {9,63} {3,63} {3,63} {3,63} {33,63} {34,39} {35,39} {36,39} {37,39} {38,39} {39,39} {4,8} {4,8} {4,8} {43,8} {44,8} {45,8} {46,8} {47,8} {48,8} {49,8} {5,8} {5,8} {5,57} {53,57} {54,57} {55,57} {56,57} {57,57} {58,63} {59,63} {6,63} {6,63} {6,63} {63,63} {64,379} {65,379} {66,379} {67,379} {68,379} {69,379} {7,8} {7,8} {7,8} {73,8} {74,8} {75,8} {76,8} {77,8} {78,8} {79,8} {8,8} {8,8} {8,97} {83,97} {84,97} {85,97} {86,97} {87,97} {88,3} {89,3} {9,3} {9,3} {9,3} {93,3} {94,99} {95,99} {96,99} {97,99} {98,99} The value that are bolded are one where uch that n ± are prme number A good porton of the value n the table ft uch a condton Th obervaton combned wth other obervaton of data lead to the followng conecture: Conecture : For all nteger n there ext an nteger uch that n where a defned n () Thu mplyng that n ± are prme number It follow from th that Goldbach conecture true nce ( n ) ( n ) n Alo, note from the data n Table that there a great amount of repetton n the value for for n (but a we wll ee oon, there repetton n value of for all n ) If we loo at value of n modulo 6 we get ome explanaton for the repetton We are 8
loong at n modulo 6 becaue α 6 (we alo could loo at n modulo α 3 3, α 4, etc) However, before we do that, we mut how that for n and two nteger and,, f where u v, then Suppoe Oberve that, a wa hown n the proof for Theorem 3, u v u v / ± n(mod p ) for v Thu by the defnton of u v, u u u v So then v v It follow that Now let n (mod 6) Then n (mod ) and n (mod3) We wll conder the eed,,, 3, 4, and 5 where (mod 6) n order to dcover the behavor of all eed modulo 6 Then (mod ) and n / ± (mod ) Thu Oberve that / ± n(mod3) n Then Now conder the eed, but Then Conder the eed Then (mod ) and / ± n(mod ) Then 3 Oberve that 3 n(mod 3) and 3 3 / ± n(mod 3) Thu 5 Now conder the eed 3 3 3 So then 3 Conder the eed 4 4 4 (mod ) and 4 4 / ± n(mod ) Thu 4 5 So then Fnally, conder the eed 5 4 Thu 5 5 Therefore for n (mod 6) and (mod6), 5 and 3 4 5 Th approach can be ued on the other value of n modulo 6 to get mlar reult The reult can be ummarzed a follow: 9
() For (mod 6), If (mod 6) n If,5(mod 6) and 3 4 5 3 4 5 6 n If,4(mod 6) 3 n If 3(mod 6) n and 3 4 Th reult can be teted on an example Let n 6 So then n (mod 6) Table n 6, {, } {,7} {,7} {,35} {3,35} {4,35} {5,35} {6,7} {7,7} {8,7} {9,7} {,7} {,7} {,83} So the pattern n () eem to hold Note that the relaton n () do not ndcate when new value for wll occur, but rather whch one are defntely the ame A more pecfc et of relaton can mot lely be found by loong at n modulo hgher prmoral Another nteretng apect of the ere that for n, t doe not converge Theorem 4: For n and an nteger, doe not converge Proof: Suppoe converged Then for ome l It wa l l l hown n the proof for Theorem 3 / ±n(mod p ) for l l r r rt Then n l pm pm p m for ome t where p t m, pm, pm are dtnct t prme wth, r, r,, r beng ther repectve non-negatve exponent and wth t
r r rt m > l But then n p p p (mod p ) Thu l m m mt m l n(mod pm ) Then > m l m l Th a contradcton Therefore doe not converge Now we wll nvetgate, for a gven n, whch value of gve uch that n The followng table a table of value of uch that n for n 5 and for 3 98
Table 3 5 n, }, {, 48 3 {-9858,7} {-9857,7} {-9856,7} {-9855,7} {-9854,7} {-9853,7} {-985,83} {-985,83} {-985,83} {-9849,83} {-9848,83} {-9847,83} {-988,7} {-987,7} {-986,7} {-985,7} {-984,7} {-983,7} {-75,-89} {-749,-89} {-748,-89} {-747,-89} {-746,-89} {-745,-89} {-744,-89} {-743,-89} {-74,-89} {-74,-89} {-74,-89} {-739,-89} {-7,-89} {-79,-89} {-78,-89} {-77,-89} {-76,-89} {-75,-89} {-74,-89} {-73,-89} {-7,-89} {-7,-89} {-7,-89} {-79,-89} {-78,-83} {-77,-83} {-76,-83} {-75,-83} {-74,-83} {-73,-83} {-54,-89} {-539,-89} {-538,-89} {-537,-89} {-536,-89} {-535,-89} {-534,-89} {-533,-89} {-53,-89} {-53,-89} {-53,-89} {-59,-89} {-5,-89} {-59,-89} {-58,-89} {-57,-89} {-56,-89} {-55,-89} {-54,-89} {-53,-89} {-5,-89} {-5,-89} {-5,-89} {-499,-89} {-498,-83} {-497,-83} {-496,-83} {-495,-83} {-494,-83} {-493,-83} {-348,7} {-347,7} {-346,7} {-345,7} {-344,7} {-343,7} {-34,-7} {-34,-7} {-34,-7} {-339,-7} {-338,-7} {-337,-7} {-38,7} {-37,7} {-36,7} {-35,7} {-34,7} {-33,7} {-,3} {-,3} {-,3} {-9,3} {-8,3} {-7,3} {-9,3} {-9,3} {-9,3} {-89,3} {-88,3} {-87,3} {-38,7} {-37,7} {-36,7} {-35,7} {-34,7} {-33,7} {-8,7} {-7,7} {-6,7} {-5,7} {-4,7} {-3,7} {-44,-89} {-439,-89} {-438,-89} {-437,-89} {-436,-89} {-435,-89} {-434,-89} {-433,-89} {-43,-89} {-43,-89} {-43,-89} {-49,-89} {-4,-89} {-49,-89} {-48,-89} {-47,-89} {-46,-89} {-45,-89} {-44,-89} {-43,-89} {-4,-89} {-4,-89} {-4,-89} {-399,-89} {-398,-83} {-397,-83} {-396,-83} {-395,-83} {-394,-83} {-393,-83} {-338,-3} {-337,-3} {-336,-3} {-335,-3} {-334,-3} {-333,-3} {-7,-57} {-7,-57} {-7,-57} {-69,-57} {-68,-57} {-67,-57} {-48,-3} {-47,-3} {-46,-3} {-45,-3} {-44,-3} {-43,-3} {-3,-89} {-9,-89} {-8,-89} {-7,-89} {-6,-89} {-5,-89} {-4,-89} {-3,-89} {-,-89} {-,-89} {-,-89} {-9,-89} {-8,-3} {-7,-3} {-6,-3} {-5,-3} {-4,-3} {-3,-3} {-,-7} {-,-7} {-,-7} {-9,-7} {-8,-7} {-7,-7} {-6,-7} {-5,-7} {-4,-7} {-3,-7} {-,-7} {-,-7} {-,-89} {-99,-89} {-98,-89} {-97,-89} {-96,-89} {-95,-89} {-94,-89} {-93,-89} {-9,-89} {-9,-89} {-9,-89} {-89,-89} {-88,-83} {-87,-83} {-86,-83} {-85,-83} {-84,-83} {-83,-83} {-8,-47} {-8,-47} {-8,-47} {-79,-47} {-78,-47} {-77,-47} {-76,-7} {-75,-7} {-74,-7} {-73,-7} {-7,-7} {-7,-7} {-5,-47} {-5,-47} {-5,-47} {-49,-47} {-48,-47} {-47,-47} {-8,-3} {-7,-3} {-6,-3} {-5,-3} {-4,-3} {-3,-3} {-,-87} {-,-87} {-,-87} {-9,-87} {-8,-87} {-7,-87} {-,-99} {-9,-99} {-8,-99} {-7,-99} {-6,-99} {-5,-99} {-4,-99} {-3,-99} {-,-99} {-,-99} {-,-99} {-99,-99} {-98,47} {-97,47} {-96,47} {-95,47} {-94,47} {-93,47} {-9,-87} {-9,-87} {-9,-87} {-89,-87} {-88,-87} {-87,-87} {-68,47} {-67,47} {-66,47} {-65,47} {-64,47} {-63,47} {-6,-57} {-6,-57} {-6,-57} {-59,-57} {-58,-57} {-57,-57} {-56,-} {-55,-} {-54,-} {-53,-} {-5,-} {-5,-} {-38,7} {-37,7} {-36,7} {-35,7} {-34,7} {-33,7} {-3,-7} {-3,-7} {-3,-7} {-9,-7} {-8,-7} {-7,-7} {-6,-} {-5,-} {-4,-} {-3,-} {-,-} {-,-} {-,} {-9,} {-8,} {-7,} {-6,} {-5,} {-4,} {-3,} {-,} {-,} {-,} {-9,} {-8,7} {-7,7} {-6,7} {-5,7} {-4,7} {-3,7} {-,3} {-,3} {,3} {,3} {,3} {3,3} {,} {,} {,} {3,} {4,} {5,} {6,} {7,} {8,} {9,} {,} {,} {,7} {3,7} {4,7} {5,7} {6,7} {7,7} {5,57} {53,57} {54,57} {55,57} {56,57} {57,57} {8,87} {83,87} {84,87} {85,87} {86,87} {87,87} {88,3} {89,3} {9,3} {9,3} {9,3} {93,3} {94,99} {95,99} {96,99} {97,99} {98,99} {99,99} {,47} {3,47} {4,47} {5,47} {6,47} {7,47} {8,3} {9,3} {,3} {,3} {,3} {3,3} {4,47} {43,47} {44,47} {45,47} {46,47} {47,47} {54,89} {55,89} {56,89} {57,89} {58,89} {59,89} {6,7} {6,7} {6,7} {63,7} {64,7} {65,7} {66,7} {67,7} {68,7} {69,7} {7,7} {7,7} {7,7} {73,7} {74,7} {75,7} {76,7} {77,7} {78,83} {79,83} {8,83} {8,83} {8,83} {83,83} {84,89} {85,89} {86,89} {87,89} {88,89} {89,89} {,7} {3,7} {4,7} {5,7} {6,7} {7,7} {8,3} {9,3} {,3} {,3} {,3} {3,3}
n an obvou upper bound for nce we want n However, the lower bound not o obvou Moreover there are curou gap of value for whch gve n whch we can ee n Table where the gap between 44 and 3 It wll be hown that thee gap can be predcted and that a lower bound for, uch that n, can be found However, frt we mut how that f p ) and p ) (or n other word ) then n (mod p ) or n α (mod p ) Suppoe p ) and p ) Then ether n(mod p ) and n(mod p ) or n(mod p ) and n(mod p ) If n(mod p ) and n(mod p ), then n n α (mod p ) and n (mod p ) If n(mod p ) and n(mod p ), then n α n(mod p ) and n α (mod p ) Therefore n α (mod p ) or n α (mod p ) Agan, let u loo at n 5 Oberve the followng table: Table 4 5 n, ( 5) 8 π n (mod p ) n α (mod p ) 3 4 4 5 5 6 4 6 7 5 6 8 5 7 3
So then for n 5, only poble for Then equal a um comprng of the term or α 3 4 5 6, and α 7 The maxmum value for nvolvng only the term or α 3, andα 4, ( ) 6 3 5 Thu n order for n 48, 5 48 or 499 The next larget poble value for 3 If 5 3 then n order for 48, 3 48 or 6 Thu there a gap [ 6,45] where for all [ 6,45], > 48 If we loo at Table 3 we ee that th true Smlarly, when conderng maxmum value for nvolvng only the term or α 3 4, and α 5, we fnd that for [ 978, 8], > 48 whch agan verfed by Table 3 The lower bound can be found by loong at the maxmum poble value for whch 3 4 5 543 So then f < 543349, then > 48 Th method can be 6 7 ued for other value of n A Countng Approach Another way to try and how that there alway ext a uch that n (and therefore n and n are prme) by ung countng method For a gven n, let π ( n ) We want uch that / ±n(mod p ) for Then for each there ether one or two bad value for modulo p More pecfcally, f n (mod p ) then there one bad value and f n / (mod p ) there are two bad value The maxmum number of oluton for that gve n ( ) n n The dea to cut out the bad value modulo p tartng wth p for and hopefully beng left wth a potve amount of good value Let loo n Then ( n ) ( ) 6 at π π and the maxmum number of oluton for 99 Oberve that (mod ) So we want to cut all where 4
99 (mod ) Th at mot 5 Th leave where (mod ) From thee we cut the bad value modulo 3 Oberve that ±,(mod 3) So then we cut from the remanng value, value of uch that (mod ) and,(mod 3) whch 99 at mot 33 3 Th leave where (mod ) and (mod3) From thee we cut the bad value modulo 5 Oberve that ± (mod 5) So then we cut from the remanng value, value of uch that (mod ), (mod3), and 99 (mod5) whch at mot 4 Th method contnued all the way up 3 5 to p p 6 3 where fnd that there are at leat 4 good value for Therefore there are at leat four way to wrte a the um of two prme (there are n fact 8 way) In general, we fnd the followng: For n, (3) Let G(n) be defned a the number of dtnct way n can be wrtten a the um of two prme (for example, G ( 5) ) (4) Let n c be defned a n c, If p n or, If p / n n (5) Let H(n) ( n ) c n (n ) α (p c n ) Then ( n) H (n) G 5
6 Table 4 { } ) ( ), (, n G n H n, 3 n {,,} {3,,} {4,,} {5,,} {6,,} {7,,} {8,,} {9,,} {,,} {,,3} {,3,3} {3,,3} {4,,} {5,3,3} {6,,} {7,,4} {8,,4} {9,,} {,,3} {,3,4} {,,3} {3,,4} {4,3,5} {5,,4} {6,,3} {7,,5} {8,,3} {9,,4} {3,4,6} {3,,3} {3,,5} {33,3,6} {34,,} {35,,5} {36,4,6} {37,,5} {38,,5} {39,3,7} {4,,4} {4,,5} {4,5,8} {43,,5} {44,,4} {45,7,9} {46,,4} {47,,5} {48,5,7} {49,3,3} {5,3,6} {5,6,8} {5,,5} {53,,6} {54,5,8} {55,4,6} {56,3,7} {57,6,} {58,3,6} {59,3,6} {6,,} {6,3,4} {6,,5} {63,7,} {64,,3} {65,,7} {66,7,9} {67,3,6} {68,,5} {69,6,8} {7,4,7} {7,3,8} {7,6,} {73,3,6} {74,,5} {75,,} {76,3,4} {77,3,8} {78,7,} {79,,5} {8,3,8} {8,7,} {8,,5} {83,,6} {84,9,3} {85,3,9} {86,,6} {87,6,} {88,3,7} {89,,7} {9,9,4} {9,4,6} {9,,8} {93,6,3} {94,,5} {95,3,8} {96,7,} {97,3,7} {98,,9} {99,8,3} {,4,8} {,3,9} {,8,4} {3,4,7} {4,3,7} {5,5,9} {6,4,6} {7,,8} {8,8,3} {9,3,7} {,5,9} {,9,} {,4,7} {3,,7} {4,8,} {5,6,9} {6,,7} {7,9,5} {8,3,9} {9,5,9} {,3,8} {,5,8} {,,9} {3,9,6} {4,3,6} {5,5,9} {6,3,6} {7,4,9} {8,3,8} {9,9,4} {3,7,} {3,4,9} {3,,6} {33,6,8} {34,4,9} {35,5,9} {36,5,7} {37,4,} {38,,6} {39,5,7} {4,9,4} {4,,6} {4,4,8} {43,6,} {44,,7} {45,6,} {46,3,8} {47,,9} {48,4,8} {49,4,} {5,4,} {5,5,9} {5,3,} {53,,5} {54,5,8} {55,6,} {56,,7} {57,4,9} {58,3,} {59,,5} {6,6,} {6,5,} {6,,} {63,5,7} {64,4,} {65,9,4} {66,5,6} {67,4,} {68,4,9} {69,5,9} {7,7,3} {7,,7} {7,5,} {73,4,9} {74,,6} {75,,3} {76,5,} {77,,} {78,4,9} {79,4,} {8,8,} {8,4,8} {8,5,4} {83,,8} {84,3,8} {85,4,4} {86,,8} {87,5,} {88,3,} {89,3,} {9,6,3} {9,4,} {9,,9} {93,5,} {94,4,9} {95,9,7} {96,5,} {97,4,} {98,3,} {99,5,7} {,6,4} {,,7} {,5,} {3,5,3} {4,3,} {5,8,3} {6,5,} {7,3,} {8,6,} {9,6,} {,3,3} {,7,} {,4,} {3,3,} {4,5,9} {5,7,4} {6,4,9} {7,6,3} {8,5,} {9,3,} {,,4} {,7,3} {,4,} {3,6,} {4,6,3} {5,,7} {6,5,} {7,4,} {8,5,4} {9,5,9} {3,9,6} {3,,8} {3,5,} {33,4,3} {34,5,4} {35,,5} {36,4,3} {37,4,3} {38,9,4} {39,5,} {4,,9} {4,6,} {4,6,4} {43,5,3} {44,5,9} {45,,9} {46,6,} {47,7,3} {48,4,3} {49,5,3} {5,,3} {5,5,5} {5,,7} {53,8,5} {54,5,4} {55,5,3} {56,6,} {57,5,4} {58,7,3} {59,9,} {6,,7} {6,7,4} {6,6,} {63,6,5} {64,8,5} {65,8,4} {66,8,7} {67,5,} {68,6,3} {69,6,4} {7,,3} {7,7,} {7,6,3} {73,,3} {74,6,} {75,,9} {76,7,3} {77,7,} {78,6,} {79,6,3} {8,,8} {8,7,4} {8,5,4} {83,7,3} {84,6,3} {85,6,3} {86,8,} {87,7,6} {88,6,6} {89,7,} {9,8,9} {9,7,5} {9,7,} {93,7,3} {94,9,9} {95,,6} {96,6,5} {97,,7} {98,7,} {99,9,5} {3,5,3} Table 5 { } ) ( ), (, n G n H n, n n ncrement of {,,3} {,3,39} {,4,443} {3,3,39} {4,,3} {5,43,46} {6,,5} {7,,37} {8,5,57} {9,3,9} {,6,4} {,46,455} {,4,59} {3,4,} {4,465,497} {5,58,83} {6,5,3} {7,43,467} {8,3,} {9,,39} {,458,477} {,5,5} {,5,74} {3,498,54} {4,6,34} {5,5,33} {6,459,483} {7,33,55} {8,9,7} {9,54,554} {3,,8} {3,8,3} {3,447,464} {33,9,9} {34,4,49} {35,459,48} {36,6,73} {37,8,47} {38,437,46} {39,,3} {4,3,6} {4,438,453} {4,,3} {43,57,76} {44,458,457} {45,5,7} {46,,33} {47,443,456} {48,4,3} {49,3,35} {5,533,547} {5,4,34} {5,3,49} {53,485,49} {54,38,54} {55,6,33} {56,497,5} {57,57,79} {58,5,55} {59,449,48} {6,,9} {6,6,3} {6,456,49} {63,6,37} {64,77,97} {65,457,475} {66,4,48} {67,45,66} {68,457,45} {69,7,3} {7,,4} {7,58,65} {7,3,4} {73,35,55} {74,454,476} {75,3,39} {76,,6} {77,455,478} {78,95,38} {79,4,66} {8,458,47} {8,37,54} {8,,44} {83,483,5} {84,,8} {85,79,89} {86,458,467} {87,,49} {88,35,56} {89,5,57} {9,5,39} {9,4,48} {9,6,635} {93,3,3} {94,3,37} {95,469,473} {96,4,3} {97,,37} {98,47,487} {99,7,3} {,48,7}
The value n bold n Table 4 are the value of n where H ( n) Notce the lat of thee value n 38 Alo, H (n) a farly cloe lower bound of G (n) that grow a G (n) grow From th the followng conecture made: Conecture : For n > 38, H ( n) > And thu nce G( n) H ( n), th mple Goldbach conecture true for n > 38 Concluon The purpoe of th reearch proect wa to approach Goldbach Conecture by ung the equvalent tatement: For every nteger n, there ext an nteger uch that n and n are prme number Two approache were by ung th tatement whch reulted n two conecture The frt approach wa creatng the equence to generate a, namely for a gven nteger n Th equence wa nteretng n that t had a few rregularte, partcularly that value for were often repeated for ucceedng value of, the dtnct value of could vary greatly n ze, and alo that there were curouly large gap between group of value of whch gave n (the bound whch determned whether n ± are prme number The repeated value and gap between group of value were explaned And fnally, t wa conectured that for all nteger n there ext an nteger uch that n where that n ± a defned n () Thu mplyng are prme number It follow from th that Goldbach conecture true nce ( n ) ( n ) n 7
The econd approach wa countng or cuttng out the bad nteger value from to n and hopefully beng left wth a potve number of good value thereby provng Goldbach conecture for that partcular n Th reulted n a lower bound for G (n), namely H(n) Th lower bound turned out to actually be generally very cloe to the value of G (n), but more mportantly H(n) eemed to be greater than for n > 38 It wa then conectured that for n > 38, H ( n) > And thu nce G( n) H ( n), th mple Goldbach conecture true for n > 38 The conecture made n th paper are eentally two equvalent tatement to Goldbach conecture Hopefully thee conecture reveal new approache to olvng the problem and to mae olvng th old problem a much eaer ta 8
Bblography [] Mahoney, Mchael S Goldbach, Chrtan Dctonary of Scentfc Bography Ed Charle Coulton Gllpe New Yor: Scrbner 97-98 [] Yuan, Wang Goldbach Conecture World Scentfc Publhng, 984 Appendx Functon for calculatng Prmoral: PrmeFact[]; PrmeFact[n_]:Prme[n]*PrmeFact[n-] Functon for calculatng : [n_,_]:(prmep[sqrt[n]];;; Whle[<,aMod[n,Prme[]];bMod[- n,prme[]];if[mod[,prme[]] a &&Mod[,Prme[]] b,u,if[mod[prmefact[],prme[]] a &&Mod[PrmeFact[],Prme[]] b,u,u]];(u*prmefact[]);];return[]) Functon for calculatng G (n) : Gn[n_]:(;;mPrmeP[n];Whle[ m,if[prmeq[n- Prme[]],];];Return[]) Functon for calculatng H (n) : cn[n_,_]:(c;if[mod[n,prme[]],c,c];return[c]) GnEt[n_]:(PrmeP[Sqrt[n]];umCelng[(n- )/];Do[;t;Do[tt*(Prme[]-cn[n,]),{,,- }];Celng[(t*cn[n,]*(n- ))/PrmeFact[]];umum,{,,}];etn--um;Return[et]) 9