The Strog Goldbach Cojecture: Proof for All Eve Itegers Greater tha 36 Persoal address: Dr. Redha M Bouras 5 Old Frakl Grove Drve Chael Hll, NC 754 PhD Electrcal Egeerg Systems Uversty of Mchga at A Arbor, Mchga Software Egeer ad Scetst at IBM Cororato rbours@c.rr.com Home hoe umber: 99-40-7459 February 0 th, 03
Abstract The Goldbach cojecture dates back to 74; we refer the reader to []-[] for a hstory of the cojecture. Chrsta Goldbach stated that every odd teger greater tha seve ca be wrtte as the sum of at most three rme umbers. Leohard Euler the made a stroger cojecture that every eve teger greater tha four ca be wrtte as the sum of two rmes. Sce the, o oe has bee able to rove the Strog Goldbach Cojecture. The oly best kow result so far s that of Che [3], rovg that every suffcetly large eve teger N ca be wrtte as the sum of a rme umber ad the roduct of at most two rme umbers. Addtoally, the cojecture has bee verfed to be true for all eve tegers u to 4.0^8, Jörg [4] ad Tomás [5]. I ths aer, we rove that the cojecture s true for all eve tegers greater tha 36. Ackowledgmets Frst I wsh to ackowledge Mr. Chrsta Goldbach ad Mr.Leohard Euler for ther tuto o ths cojecture whch has etertaed may mathematcas ad amateurs aroud the globe for 70 years. I ackowledge the followg eole who made a mact o my tellectual develomet: my Mathematcs structors Hgh School, Mr. Hamdouche ad Mr. Mchel Cheu Aaba, Algera, my Mathematcs Professor Maro Beedcty at the Uversty of Pttsburgh Pttsburgh Pesylvaa ad Electrcal Egeerg Professor Frederck Beutler at the Uversty of Mchga A Arbor Mchga. I also wsh to ackowledge my ste daughter, Kelly McGrath who took terest my work ad beleved I could solve ths roblem. I also ackowledge UNC Chael Hll Mathematcs Emertus Professor Mchael Schlessger for revewg the aer. Last but ot least, I would lke to ackowledge Jörg [4] ad Tomás [5] ad ther teams for verfyg the Goldbach cojecture for very large eve tegers. Ths verfcato was crucal to comlete ths roof for eve tegers less tha,370.
. Itroducto ad Problem Overvew Euler stated 74 what s so called the Strog Goldbach Cojecture that every eve teger N greater tha 4 ca be wrtte as the sum of two rmes. Some examles:. 0= 3+ 7= 5+ 5. 48 = 5 + 43 = 7 + 4 = + 37 = 7 + 3= 9 + 9 3. 5 = 5 + 47 = + 4= 3+ 9 4. 00= 3+ 97= + 89= 7+ 83= 9+ 7= 4+ 59= 47+ 53 However ths remas a usolved roblem for all eve tegers greater tha four; that s to the day o oe has bee able to rove that ths s the case. Please ote that 48 ca be wrtte as the sum of two rmes 5 ways; however, 5, a larger umber ca oly be wrtte as the sum of two rmes 3 ways. So t s ot ecessarly true that the larger the eve teger s, the larger the umber of ways t ca be wrtte as the sum of two rmes. Ths s true for suffcetly large eve tegers but s however out of the scoe of ths aer.. Overvew The best theoretcal results kow so far are those of Che [3] statg that every suffcetly large eve teger ca be wrtte as the sum of a rme ad the roduct of at most two rmes. Sce the, the most advacemet made was the emrcal verfcatos of the cojecture for eve tegers u to 4.0^8, Jörg [4] ad Tomás [5]. Some otable research aers o the Goldbach cojecture ca be foud [6]-[9]. The major dffculty solvg the cojecture s how to deal wth what s so called the error term, whch arses whe aroxmatg the floor fucto. May authors suled sold roofs that were ot correct because they omtted the error term. Our techque works for two reasos:.we aalyze the roblem asymtotcally the set of ratoal umbers whch reveals some ce asymtotc roertes of the umber of Goldbach ars ad the trasform t back to the set of tegers ad. We clude the error term the aalyss whch s crtcal solvg the Goldbach roblem. Ths techque s troduced Secto 4. 3
We ow formally troduce the Goldbach roblem. Let N be a eve teger greater tha 4 ad be the umber of odd rmes less tha N, say ascedg order = 3, = 5,...,. Sce < N < + ad our roof s based o varyg N throughout the rme umber le, we shall deote N by N ad defe: η < = PP, N {{ x, y}: x odd, x N, rme ad y N x s rme} η < = PC, N {{ x, y}: x odd, x N, rme ad y N x s comoste} η = CC, N {{ x, y}: x s a odd comoste ad y N x s comoste} Lettg π x be the umber of odd rme umbers less tha or equal to x, the:...3 π N = η N + η N P, P P, C.4 The left had sde of.4 s the umber of odd rme umbers less tha N ; t excludes the eve rme umber. The frst term o the rght had sde of.4 s the umber of rme umbers wth the sum of the rme ars equal to N term s the umber of rme ad comoste ars whose sum s equal to N.4, we have: η π N η N P, C PP, N, ad the secod also. From =.5 ad the Goldbach cojecture wll the hold f ad oly f η N > 0... Ma Results: PP, The aer s orgazed as follows. I Secto, we develo a exresso for η PC, N ad Secto 3 we develo a exresso for η, N. I Secto PP 4, we demostrate that the Strog Goldbach cojecture holds for suffcetly large eve tegers. I Secto 5, we determe a suffcetly large eve teger, N =,370 that the cojecture holds for every eve teger N N. We fally coclude the aer Secto 6. A aedx Secto 7 cotas Lemmas ad termedate results., so 4
. A Exresso for η N, PC We develo a exresso for η as a fucto of all odd rmes PC,,,..., less tha N. Our objectve s to show that for suffcetly large eve tegers 0 η PP, N >. As show.5, η PP, N deeds o two crtcal terms: η each =,,..., : N ad π N. We start by defg the followg sets for PC, C = { x odd, x< ad x 0 mod : x N mod, j =,,...,, ad j }. j We suose that N 0 mod, for each =,,...,. If for some j =,,...,, N 0 mod j, the η PC, N wll be smaller. For examle, f N 0 mod ad N 0 mod, for each j, =,,..., j the { N 3,3 } s a CC, ar; but f N 0 mod for each j =,,...,, the j N 3,3 s a PC, ar. As a result of.5, η, N wll the be j larger. It s the suffcet to cosder the case N 0 mod, for each =,,...,. We the wrte: η j j PC, N C C C C C C C 3 = = < = < < 3= = = +... + PP + C C..., C < <... < =. j C N N N = +... =, < 3=,, 3 3 + + + +, <... < < =,,...,, where δ s a error term ad for m > : N δ.3... 5
m m N + N C C...... C = m + + < <... <......... m= m < < < m + = m+ + N + + δ,,..., m,.4 < < < =...... where δ,,..., m s a error term. The error terms δ ad δ,,..., m are due to the fact that the umber of multles of... u to N s ether equal to m N /... or m N /... + ad caot be gored as the m sum of the s ca get large wth N. Omttg these error terms, wll make the roof of the Goldbach cojecture qute smle. Usg.3, we frst ote that: 3... N N N C = + = = < = < < 3= 3 + Combg.-.5, we obta: + N δ < <... <... = = η N + + + δ, <... < =... = +.5 N N N PC, N = + 3... = < =, < 3= 3 + N + N + N + N N + +... + + δ,, 3 < < 3= 3 < <... <... = +... δ, + < = < < 3= 3 < <... < =... + Evaluatg.6 s tatamout to: +... + +,,..., N δ.6 < <... <... = 6
Key Formula : where η + N N N PC, N = 3 + 5... = < = < < 3= 3 N + + ε,... < < < =... + N.7 ε N δ + δ, + δ,, 3 +... + δ,,..., s = < = < < 3= < <... < = a error term. 3. A Exresso for η N PP, We develo a exresso for η PP, N based o the odd rme umbers,,..., less tha N. Sce we already have a exresso forη PC, N.7, the ext ste s to develo a exresso for π N as a fucto of the odd rmes,,..., oly. Geerally a well kow exresso for π x exsts the lterature [0] but ts calculato volves also the eve rme ; t s gve by: x x x π x = π x + x +... < j j < j< k jk + 3. We avod 3. for cosstecy wth the exresso for η PC, N whch oly deeds o the odd rmes,,...,. Usg a Seve the set of odd umbers less tha or equal to N / whle excludg the umber, we have the followg: N N N N π N = + + +... + δ N, = < = < < 3= 3 3. for some error term δ N. Note that there s o error term 3. because the eve rme two s cluded 3. ad the umber of multles of... u to x s j k exactly equal to we get: x / jk.... Ths s ether s the case 3.. After smlfcato, 7
π N N N N N = + + +... = < = < < 3= 3 + + + N δ N < <... < = 3 3.3 Usg.7 ad 3.3 s tatamout to: Key Formula : N N N η PP, N = π N η PC, N = + + 4 + = < = N N + 6 +... + + ξ N, < < 3= 3 < <... < = 3.4 where ξ N = δ N ε N. 4. Every Suffcetly Large Eve Iteger Ca be Wrtte as the Sum of Two Prmes We demostrate that every suffcetly large eve teger ca be wrtte as the sum of two rmes. The ma dea behd the roof s the aalyss of η / PP, N N rather tha that of η PP, N. Ths ormalzato techque s crucal rovg the cojecture because t reveals some ce roertes of η / PP, N N ad also makes t ossble to deal wth the error termς N defed below. We ormalze 3.4 as we wrte: η N π N η N N N N PP, PC, = = + 4 + = < = + + + 6... ς N, < < 3= 3 < <... < =... 4. + 8
for some ew error term ς N. Let us ow troduce oe of the two crtcal terms of η /, N N : PP + + + 4 6... N = < = 3 < < = 3 H N + 4. +, < <... <... = whle the other beg ς N. After smlfcato, we obta: = + + + 3... N = < = 3 < < = 3 H N Usg 4. ad 4.3, we ca wrte: + + 4.3 < <... <... = η N π N η N PP, PC, = = H N + ς N N N 4.4 We also have by alyg4.3 where < N < : + + + + H N 3... If we defe: + H N = + N + N + = < = + + 4.5 < <... <... = BN 3 4..., + + + = < = < < 3= 3 < <... <... = 4.6 we the wrte: + BN + H N N + N + H N = 4.7 We show Lemma 7. Aedx 7 that: 9
4.8 = = = BN We ow rewrte H N gve by 4.3 as: H N = + E N, 4.9 N where: + + + 3 4... = < = 3 3 3 4 < < = < < < = 3 4 EN We show Lemma 7. Aedx 7 that: + + 4.0 < <... <... = = +, 3 = j= j j= j + EN ad Lemma 7.3 Aedx 7 that to 4.9, we obta: 4. EN as N. Alyg ths result lm H N N = 4. If we defe: the usg 4.4, we get: + ς, K N H N N N PP, N K η = 4.3 4.4 Sce η N < N /, we the have usg 4.4: PP, Key Iequalty 3: 0 K N < 4.5 We the deduce from 4.5, the followg three coclusos: lm ς N =, 4.6 N lm K N exsts ad L N 4.7 0
0 L 4.8 We cosder the two cases: 0< L L = 0 I case, L > 0, we fer the exstece of a teger m0 such for all tegers > m 0, K N > 0, whch mles usg 4.4 that η 0 PP, N > for all > m0. Ths the roves that every suffcetly large eve teger ca be wrtte as the sum of two rmes. Addtoally, we deduce from 4.4 that η N as N. Ths comletes the roof of Case. Q.E.D. PP, We ow cosder Case above where L = 0. There exsts a teger such that for all tegers m> > m, t must the be that 0 K N m K N. If for m some = m > m, K N = 0, t the must be that K N m = 0for all m m or equvaletly η 0 PP, N m = for all m m. However ths cotradcts the fact that there are ftely may eve tegers of the form, wth beg a rme umber, for whch η PP, N. Therefore t must be that K N > 0for all > m or equvaletly η, N > 0 for all > m. Ths fshes the roof of Case ad that PP every suffcetly large eve teger ca be wrtte as the sum of two rmes. Q.E.D. We wll show a future ublcato that η N as N Case PP, above usg a ew techque. We thus have arrved at the followg theorem: Theorem : Every suffcetly large eve teger ca be wrtte as the sum of two rmes. We ext aalyze the fucto K N as llustrated Fgure below.
Fgure : Grah of K N Theorem : The fucto K N has the followg roertes:. There exsts a teger such that 0 < K N < K N for all >.. There may exst a teger such that: 3 > K N K N < K N K N for all. + Proof: Alyg4.4, we have: 3 N K N K N N N + + = η P, P + η P, P N + N
η PP, N + η PP, N N + 4.9 We ote that as N +, the rght had sde of 4.9 aroaches zero. We the fer that for large values of, K N + K N 0. I the case K N s o-creasg for all tegers, the we sk above. Otherwse, we fer from 4. that H N creases ad from 4.6 that ς N decreases for large values of. There must the be a ot as llustrated Fgure above such that K N decreases for all. Sce K N L 0 as N, t must be that K N chages cocavty at some ot 3 >, that s, K N K N < K N K N for all. Ths fshes the roof. Q.E.D. + 3 5. The Goldbach Cojecture Holds for All Eve Itegers Greater tha or Equal to N = 36 The dea here s to fd the smallest teger such that the fucto B N / + creases for all > ; whch mles that the fucto H N creases wth. It also turs out that the ot s a flecto ot of H N as llustrated Fgure below. We rove our tuto that s also a flecto ot of the fucto K N. By usg ths last result, we the rove that the Goldbach cojecture holds for all eve tegers N > N = +. By a umercal calculato, we fd that =, = 37 ad N =,370. For comleteess, we frst fd the smallest ot such that 0 0 for all 0. B N < > From Lemma 7. Aedx 7, = = = BN 5. From5., we deduce that BN > 0 / <. By a smle calculato, = we fd that the smallest rme umber for whch 0 BN < s 3
0 0+ = 7, 0 0 = 3, 50 ad wth / 0.967 ad /.067. 0 = + = = = 0 N = = Fgure : Grahs of BN / ad HN + By alyg5., we fd that: BN BN + == + + 5. + + + + = = We deduce from 5. that: 4
BN BN 5.3 + > 0 > + + + = + ++ Sce + +, the + / + + + 4/3. A suffcet codto for 5.3 to hold s the: = / > 4/3 5.4 0 By a smle calculato, we fd that: / =.336 ad / =.3603. = = Thus the smallest rme for whch 5.4 holds s = 37, =. We the have: BN + BN > 0 for all = 5.5 + + Addtoally, the smallest rme less tha 37 for whch 5.3 holds s 9. By a smle calculato 7 / =.8whle + / + + + =.43. We coclude by = referrg to Fgure above that a eve teger for whch > 0 for all 7 H N s N = 7 + = 36 ; ad thus H N > 0 for all N N. We ext rove our tuto that s a flecto ot of K N. Theorem 3: Every eve teger greater tha or equal to 36 ca be wrtte as the sum of two rmes. Proof: Usg 4.7, we wrte: + BN + H N N + N + H N = 5.6 We show Lemma 7.4 Aedx 7 that: BN + 0 + + / N / N > 0 for all. Ths the comletes the roof that s creasg for all H N. Now we show that the ot defed above 0 s a flecto ot of H N or equvaletly for all : H N H N H N H N < 0 5.7 + + + Alyg 4.7 to 5.7, a suffcet codto for 5.7 to hold s the: 5
+ BN BN < 05.8 N N N N + + + + + + We show Lemma 7.5 Aedx 7 that 5.8 holds for all =. We shall ext rove that K N > 0 for all =. Usg the result of Theorem, the f K N s o-creasg for all, the K N > 0ad thus η 0 PP, N > for all ad the roof s comlete. Otherwse there exsts a teger ad a flecto ot > such that: 3 0 < K N m < K N for all m> ad K N K N < K N K N for all + 3 5.9 5.0 If 3, there s othg to rove as from5.9, 3 We ow cosder the case < <. Usg5.0, we have: 3 K N > 0 for all. : + k : K N K N < K N K N 3 + k + k + k + k 5. We deduce from 5. that f for some k, K N + = 0, ad k K N + > 0, k t must the be that: K N < K N < 0, + k + k 5. whch mles that k 0 K N + <. Ths however cotradcts 5.9 so that K N > 0 for all whch mles that η 0 PP, N > for all = 7. We thus have show that the Goldbach cojecture s true for all eve tegers greater tha N = + = 36. Q.E.D. Geeralzato: For j > 0, let us cosder the decomosto of the eve teger jn as the sum of two odd umbers, the frst oe say x, j N < x< jn, ad the secod oe jn x,0< jn x < N. By reeatg the exact same roof of Theorem above usg the same deftos for η N, η N, η N as gve by.-.3 but PP, PC, CC, cosderg the rmes the terval N, N oly, we obta the followg: 6
Theorem 4: For ay teger j, every suffcetly large eve teger jn ca be wrtte as the sum of two rmes, oe the terval j N, jn ad the other the terval 0, N. 6. Cocluso I ths aer, we have show the followg:. Every suffcetly large eve teger ca be wrtte as the sum of two odd rmes.. Ay eve teger greater tha 36 ca be wrtte as the sum of two rmes. We used a uque techque to aroach the Goldbach cojecture. The method cossts of aalyzg the roblem the ratoal set of tegers ad also takg to accout the error fucto. The techque revealed some ce roertes of η / PP, N N that eabled us to rove the above results. We also aalyzed the Goldbach roblem by varyg the eve teger N. Sce Jörg [4] ad Tomás [5] verfed the Goldbach cojecture for eve tegers u to 4.0^8, t the follows from 3 above that the Goldbach cojecture holds for all eve tegers greater tha 4 ad thus s ow a theorem: Theorem 5- The Strog Goldbach Cojecture: Every eve teger greater tha 4 ca be wrtte as the sum of two rmes. We easly coclude that the weak Goldbach cojecture Every odd teger greater tha 7 ca be wrtte as the sum of three odd rmes s ow true. It follows from Theorem 3 by addg three to each eve umber greater tha four. 7. Aedx 7.. Lemma 5.3 = = = BN Proof: Recall from 4.6 that: = + 3 4..., = < = < < 3= 3 BN 5.4 7
Let us defe: + +... = < = < < 3= 3 DN 5.5 3 + 4 5 +..., = < = < < 3= 3 CN 5.6 By usg5.4, we fd that: BN + = BN 3 + 4 5 +... 5.7 From 5.4 ad 5.6, we get: + = < = < < = 3 3 CN BN DN = + +... = = < = < < 3= 3 5.8 We ow show by ducto o that: below that: DN 5.9 = = Alyg to 5.9 =, we fd that D 9 = / 3. We obta the same value whe alyg 5.5 to =. Assumg 5.9 holds for, we shall show t hold for +. Usg 5.5, we fd that 5.9 for +. Equatos 5.8 ad 5.9 lead to: DN + = / + DN, so that holds = CN = + N + + BN By usg5.7, 5.8 ad5.9, we obta wth < < : 5.0 CN BN = BN = BN BN + = + + + BN + + = We suose 5.3 holds for we obta: = 5. ad show t also holds for +. Usg 5. ad5.3, BN + = = = + + = = = = = + = = + = 8
+ = = = + = + = = + + + = = + + = 5. = = Usg4.6, we fd that B 9 = / 3 ad usg4.8, we also fd that B 9 = / 3. Therefore 4.8 holds for all tegers. Q.E.D. = 7.. Lemma = + 3 = j= j j= j + EN Proof: Recall from 4.0 that: 5.3 + + + 3 4... = < = 3 3 3 4 < < = < < < = 3 4 EN + + 5.4 < <... <... = A smle calculato leads to: BN EN + = EN + 5.5 + Usg ether 5.3 or 5.4lead to E 9 = / 3so 5.3 holds for =. Assumg that 5.5 holds for we easly show that 5.3 holds for +. Q.E.D. 7.3. Lemma 3 lm EN N = 5.6 Proof: We ca wrte the egatve of the secod term o the rght had sde of 5.3 as: 5.7 = = j= j j= j + = + j= j = + j= j Sce for all, / j <, t follows that: j= 9
> > = j= j j= j + = + j= j > 5.8 = + j= Usg the well kow lower boud, lease see []: + l l j + l π / 6, 5.9 we get: There exsts a teger j= j π 5 l l + l = + j= j = 6 m 0 such that l l l π / 6 5 / for all m0 j 5.30 + > >. Ths the mles that the rght had sde of the above coverges to as so that the exresso 5.7 coverges to as ; hece EN as N ad the roof s comlete. Q.E.D. 7.4. Lemma 4 Proof: Sce BN + + N + N + ad + + > for all 7 < N < < N < + + + + +, the: + + + < < N N 5.3 5.3 For 7, B N / + > 0 lease see Secto 5 after5. so that for large, BN / + > 0. As, both sdes of 5.3 aroach zero. Ths comletes the roof of 5.3. Q.E.D. 7.5. Lemma 5 + + + BN + BN < for all = 7 N + N + N + N + + 5.33 Proof: Sce < N < ad < N < + + + +, the: + + < < N N + + + 5.34 0
Smlarly, + + + + < < N N + 3 + + + + 5.35 Combg 5.34 ad 5.35 leads to: + + + + < < N N N N 5.36 + 3 + + + + + As, both the left ad rght had sdes of 5.36 aroach zero. Hece the desred result follows usg the fact that the rght had sde of 5.33s greater tha zero usg 5.5. Ths comletes the roof. Q.E.D. 8. Refereces. htt://e.wkeda.org/wk/goldbach's_cojecture. htt://mathworld.wolfram.com/goldbachcojecture.html 3. J. R. Che, O the reresetato of a larger eve teger as the sum of a rme ad the roduct of at most two rmes. Sc. Sca 6 973, 57--76. 4. Jörg Rchste, Verfyg the Goldbach cojecture u to 4.0^4, Mathematcs of Comutato, vol. 70, o. 36,. 745-749, July 000. 5. Tomás Olvera e Slva, Goldbach Cojecture Verfcato u to 4.0^8, Deartameto de Eletróca, Telecomucações e Iformátca Uversdade de Avero, Arl 4 th, 0. 6. Kev O Bryat, Goldbach Talks, htt://www.math.ucsd.edu/, October 3 st, 00 7. Peter Schorer, A Possble Aroach to Provg Goldbach s Cojecture, Hewlett- Packard Laboratores, Palo Alto, CA, March 7 th, 00. 8. George Greaves, Seves Number Theory, Srger-Verlag Berl Hedelberg New York. 9. Yua Wag, The Goldbach Cojecture, Seres Pure Mathematcs, Vol. 4, Secod Edto, Academa Sca, Cha. 0. htt://e.wkeda.org/wk/prme-coutg_fucto. htt://e.wkeda.org/wk/dvergece_of_the_sum_of_the_recrocals_of_the_ rmes