关于哥德巴赫猜想的证明(美国康奈尔大学)英文原版论文 (抢先发布)

arXiv 1110 3465v1 math GM 16 Oct 2011 A PROOF OF THE GOLDBACH CONJECTURE SHAN GUANG TAN Abstract The Goldbach conjecture was proved in this paper The proof was by contradiction based on the fundamental theorem of arithmetic and the theory of Linear Algebra First by an assumption the Goldbach conjecture was converted into a group of linear equations Then by investigating solutions to the group of linear equations reductions to absurdity were derived to prove the assumption false Hence the Goldbach conjecture was proved that even numbers greater than 2 can be expressed as the sum of two primes Introduction Goldbach s conjecture is one of the oldest unsolved problems in number the ory and in all of mathematics It states Every even number greater than 2 is a Goldbach number a number that can be expressed as the sum of two primes 1 7 For small values of even numbers N the Goldbach conjecture has been verifi ed for N 1 609 1018and some higher small ranges up to 4 1018by T Oliveira e Silva 1 Considerable work has been done on the weak Goldbach conjecture 1 The strong Goldbach conjecture is much more diffi cult Chen Jingrun showed in 1973 using the methods of sieve theory that every suffi ciently large even number can be written as the sum of either two primes or a prime and a semiprime the product of two primes 1 4 In 1975 Hugh Montgomery and Robert Charles Vaughan showed that most even numbers were expressible as the sum of two primes More precisely they showed that there existed positive constants c and C such that for all suffi ciently large numbers N every even number less than N is the sum of two primes with at most CN1 cexceptions In particular the set of even integers which are not the sum of two primes has density zero 1 5 Linnik proved in 1951 the existence of a constant K such that every suffi ciently large even number is the sum of two primes and at most K powers of 2 Roger Heath Brown and Jan Christoph Schlage Puchta in 2002 found that K 13 works 1 6 This was improved to K 8 by Pintz and Ruzsa 1 7 The Goldbach conjecture was proved in this paper The proof was by contra diction based on the fundamental theorem of arithmetic and the theory of Linear Algebra First by an assumption the Goldbach conjecture was converted into a group of linear equations Then by investigating solutions to the group of lin ear equations reductions to absurdity were derived to prove the assumption false Hence the Goldbach conjecture was proved that even numbers greater than 2 can be expressed as the sum of two primes Date September 22 2011 and in revised form January 22 2012 2010 Mathematics Subject Classifi cation Primary 11A41 11A99 Key words and phrases number theory Goldbach conjecture 1 2SHAN GUANG TAN 1 Lemmas of linear equations Defi nition 1 1 Let an even number N 2n and Qr q1 q2 qr denote a set of odd primes where r 2 and 3 q1 q2 qr 1 2n q r 2n 1 4 for i 1 2 r since qr 1 2n 1 4 Then we obtain 1 6 det A 1 1 r 1a2 1a3 2 ar r 1a1 r 1 1 r 1a1a2 ar 1ar and 1 7 A B 1b1 2 b1 r b2 11 b2 r br 1br 2 1 where 1 8 bi i 1 i 1 2 r bi j 1 i j Qi 1 k jak j i 2 3 r bi j 1 i j r Qi 1 k 1ak Qr k jak i i bi j bi i 1 1 i j 1 1 Qi 1 k j 1ak Qr k i 1ak i 1 2 r 1 j i br j br r 1 br j br 1 1 j 1 1 Qj 1 k 1ak j 2 3 r Thus solutions to Equation 1 1 should be 1 9 xi 2n Pr j 1bi j 1 1 r 1a1a2 ar i 1 2 r The ratio of xito xkin Solution 1 9 is 1 10 xi xk Pr j 1bi j Pr j 1bk j i k 1 2 r i 6 k Let defi ne 1 11 ci i ai i 1 2 r ci j aibi j 1 i j Qi k jak j i 2 3 r ci j aibi j 1 i j r Qi k 1ak Qr k jak i i ci j ci i 1 1 i j 1 1 Qi 1 k j 1ak Qr k i 1ak i 1 2 r 1 j i cr j cr r 1 cr j cr 1 1 j 1 1 Qj 1 k 1ak j 2 3 r Thus solutions to Equation 1 1 can be written 1 14 xi 2n ai Pr j 1ci j 1 1 r 1a1a2 ar i 1 2 r The ratio of xito xkin Solution 1 14 can be written 1 15 xi xk ak ai Pr j 1ci j Pr j 1ck j i k 1 2 r i 6 k By using Expression 1 15 and substituting xr a1 ar Pr j 1cr j Pr j 1c1 j x1 into the fi rst equation of the group of linear equations 1 1 in the form 1 3 we obtain 1 a1 Pr j 1cr j Pr j 1c1 j x1 2n When n since ai 2n 1 4 for i 1 2 r there are lim n 2n 1 4 andlim n 1 ai 0 i 1 2 r A PROOF OF THE GOLDBACH CONJECTURE5 According to Expressions 1 12 1 13 we obtain lim n cr j cr 1 0 j 2 3 r lim n ci j ci i 1 0 i 1 2 r 1 j 6 i 1 and 1 16 lim n Pr j 1ci j Pr j 1ck j ci i 1 ck k 1 lim n Pr j 1 ci j ci i 1 Pr j 1 ck j ck k 1 1 where i k 1 2 r Thus when n we have lim n x1 2n 1 a1 and similarly for i 1 2 r 1 17 lim n xi 2n 1 ai where xican not be an integer since aiis a positive integer and xishould not be a prime factor of n Hence when n no group of primes qifor i 1 2 r can satisfy Equation 1 1 and at least one of xifor i 1 2 r in solutions to Equation 1 1 is not an integer The proof of the lemma is completed Lemma 1 4 By Defi nition 1 1 and with det A 6 1 when xiis the solution to Equation 1 1 and corresponding to the prime qjifor i 1 2 r for any positive number n at least an inequality of xiand qjican be obtained as following 1 18 xi qji 2 1 ai 6 2n 3 1 i r The inequality shows at least one of xifor i 1 2 r in solutions to Equation 1 1 is not an integer On the other hand for any given small positive value a positive number n can be found to obtain at least an inequality xi qji when n n Proof Let consider the value i xi qji for i 1 2 r When n by Lemma 1 3 and Expression 1 17 we have lim n xi qji 2n 1 ai qji 2n aiqj i qji 1 qji 1 qji 1 ai qji 1 qji 1 ai 2 1 ai 6 2n 3 where qji 1 qji 2 ai 6 2n 3 6SHAN GUANG TAN Then for any positive number n 6 2n 3 where qji 1 qji aiqji n 1 2 ai 2n 3 and in the assumption of aiqji qji 1 2n for the i 1 th equation of the group of linear equations 1 1 in the form 1 3 when xiis considered Therefore for any positive number n 6 2n 3 On the other hand for any given small positive value we always can choose a positive number n to make 6 2n 3 Thus for n n we obtain at least an inequality of xiand qjifor 1 i r as following xi qji 2 1 ai ai n 1 6 2n 3 6 2n 3 Hence for any positive number n at least an inequality 1 18 of xiand qjican be obtained by inequalities 1 19 1 20 Also for any given small positive value a positive number n can be found to obtain at least an inequality xi qji when n n The proof of the lemma is completed 2 Proof of the Goldbach conjecture Theorem 2 1 Every even number greater than 2 is a Goldbach number a number that can be expressed as the sum of two primes A PROOF OF THE GOLDBACH CONJECTURE7 Proof Let consider any even number N 2n greater than 2 When n is a prime p we have N p p that is N can be expressed as the sum of two primes Otherwise n should be a composite number Let a set of primes P p1 p2 pl denote all odd primes smaller than or equal to n where 3 p1 p2 pl n We can form a group of equations 2 1 p1 c1 2n p2 c2 2n pl cl 2n where for i 1 2 l ciis an odd number By the fundamental theorem of arithmetic n can be written n 2 0p 1 i1 p 2 i2 p s is 1 i1 i2 is l where 0is a non negative integer and 1 2 sare positive integers Thus for i i1 i2 is cimust be a composite number and the corresponding equation can be removed from the group of equations 2 1 Hence let Psand Q denote two subsets of P where Ps pi1 pi2 pis P Q P Ps q1 q2 qm P 3 q1 q2 qm n Also we can form a group of equations 2 2 q1 d1 2n q2 d2 2n qm dm 2n where diis an odd number for i 1 2 m It is obvious that diand n have no common prime factor and qiis not a prime factor of dior n Assume that all of odd numbers difor i 1 2 m are composite numbers By the fundamental theorem of arithmetic for i 1 2 m dican be written 2 3 di q i 1 i 1 q i 2 i 2 q i mi i mi where mi is the sum of diff erent prime factors of diand i 1 i 2 i miare positive integers It is obvious that qi 1 qi 2 qi mi Q Let Qr q1 q2 qr Q denote a set of primes which satisfy 3 q1 q2 qr 1 2n q r n and qrbe smaller than or equal to the maximum prime of qi mifor i 1 2 m When qj Qris equal to qi kwhich is a prime factor of di let denote ai j di qj q i 1 i 1 q i 2 i 2 q i k 1 i k q i mi i mi 8SHAN GUANG TAN Thus we can form a group of linear equations 2 4 x1 a1 j1xj1 2n x2 a2 j2xj2 2n xr ar jrxjr 2n or in the matrix form as Equation 1 1 When all of odd numbers difor i 1 2 m are composite numbers Equation 1 1 can be solved and solutions to Equation 1 1 should satisfy 2 5 xi qi i 1 2 r Now let investigate solutions to Equation 1 1 to verify whether the assumption that all of odd numbers difor i 1 2 m are composite numbers is true or false When rank A r 1 since q1is not a prime factor of d1 d1must be a prime p and satisfi es N q1 p that is N can be expressed as the sum of two primes When det A 1 by Lemma 1 2 all of xifor i 1 2 r are not primes Hence solutions xican not satisfy Equation 2 5 and at least one of difor i 1 2 r is a prime p and satisfi es N qi p that is N can be expressed as the sum of two primes Then in the case of det A 6 1 for any positive number n or for n n where n is a positive number it can be proved by lemmas 1 3 1 4 or verifi ed 1 by Lemma 1 4 that at least one of xifor i 1 2 r in solutions to Equation 1 1 is not an integer Hence solutions xican not satisfy Equation 2 5 and at least one of di for i 1 2 r is a prime p and satisfi es N qi p that is N can be expressed as the sum of two primes In the above investigation of the solutions to a group of linear equations reduc tions to absurdity are derived and the assumption that all of odd numbers difor i 1 2 m are composite numbers is proved false Hence it is proved that even numbers greater than 2 can be expressed as the sum