加载中…陈景润定理不是谎言的证据列表
(2019-02-05 14:50:05)陈景润定理不是谎言的证据列表
近年来，在反动文人王晓明的蛊惑下，国内出现一股反陈景润定理的“小高潮”，搅得陈景润在天之灵不得安息。
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附件：陈景润定理不是谎言的证据列表，文中“J. R.Chen”代表陈景润
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... The fact that the integer N − s is a product of two large integers, gives an idea of its factorization. In the existing literature, the decomposition of integers is an immense problem which has been posed in several ways and treated by different methods (for example [1,3,8]). ...
... If r = 2, then n = 2 α 1 t 2 which shows that t 2 is unlimited and consequently n is of the second form, otherwise (i.e. r > 2) the fact that n = 2 α 1 t 2 t α 3 3 ...t αr r where t 2 is odd and 2 α 1 t α 3 3 ...t αr r ≈ +∞ because 2 α 1 t α 3 3 ...t αr r > t 2 and the product t 2 .2 α 1 t α 3 3 ...t αr r = n ≈ +∞, shows that n is of the third form. ...
... r > 2) the fact that n = 2 α 1 t 2 t α 3 3 ...t αr r where t 2 is odd and 2 α 1 t α 3 3 ...t αr r ≈ +∞ because 2 α 1 t α 3 3 ...t αr r > t 2 and the product t 2 .2 α 1 t α 3 3 ...t αr r = n ≈ +∞, shows that n is of the third form. (a2) α 2 > 1. ...
Decomposition of terms in Lucas sequences
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Apr 2009
Abdelmadjid Boudaoud
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... Chen's theorem [26] is another weaker form of Goldbach, which states that every sufficiently large even integer is the sum of two primes, or a prime and a semiprime (product of two primes). By letting R(N ) denotes the number of ways N can be decomposed in this manner, Chen attempts to get a lower bound on the number of ways N can be decomposed as N = p + P 3 , where P 3 is some number with no more than three prime factors, using sieve theory methods. ...
... Minimising the upper bound while ensuring µ + (d) satisfies (2.7) and has sufficiently small support (to reduce the size of the summation) is, in general, very difficult. Nevertheless, this method is used by Chen [26] to construct the following upper bounds of sieves that we have related to the Goldbach conjecture and the twin primes conjecture § . ...
On a numerical upper bound for the extended Goldbach conjecture
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Dec 2017
David Quarel
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... (2) There are infinitely many primes p for which p + 2 is either prime or a product of two primes. (Chen [2], 1966.) (3) There is a number n 0 such that any even number n ≥ n 0 can be written as n = p+p with p prime and p either prime or a product of two primes. ...
... (3) There is a number n 0 such that any even number n ≥ n 0 can be written as n = p+p with p prime and p either prime or a product of two primes. (Chen [2], 1966.) (4) There are infinitely many integers n such that n 2 + 1 is either prime or a product of two primes. ...
Prime number theory and the Riemann zeta-function
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Jan 2005
Roger Heath-Brown
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... An approach based on the circle method demands that the number of variables is rather large compared to the degree. Thus when there are fewer variables, investigations have focused on solutions with few prime factors, a prototype result being that of Chen [6] on the binary Goldbach problem, a result using the weighted sieve. One can utilise again the circle method to cover cases where the number of variables is moderately large, see the recent work of Magyar-Titichetrakun [27], Schindler-Sofos [31] and Yamagishi [38]. ...
... An approach based on the circle method demands that the number of variables is rather large compared to the degree. Thus when there are fewer variables, investigations have focused on solutions with few prime factors, a prototype result being that of Chen [6] on the binary Goldbach problem, a result using the weighted sieve. One can utilise again the circle method to cover cases where the number of variables is moderately large, see the recent work of Magyar-Titichetrakun [28], Schindler-Sofos [31] and Yamagishi [40]. ...
Finite saturation for unirational varieties
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Yuchao Wang
Efthymios Sofos
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... Let P r be an integer with no more than r prime factors, counted with their multiplicities. In 1973 Chen [3] showed that there are infinitely many primes p with p + 2 = P 2 . Here are some examples of problems, concerning primes p with p + 2 = P r for some r ≥ 2. In 1937, Vinogradov [22] proved that every sufficiently large odd n can be represented as a sum ...
... see Mertens formula ([[12], ch.9, §9.1, Theorem 9.1.3]) and (13). ...
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Stoyan Ivanov Dimitrov
T. L. Todorova
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... Contrastly, the ordinary Goldbach's conjecture is still unsolved. A wellknown partial result is the theorem of Chen[2][3], who proved that every sufficiently large even number can be represented as the sum of a prime and the product of at most two primes. Ross[16] gave a simpler proof. ...
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Tomohiro Yamada
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... It is worth noting that the above mentioned results do not use the higher rank sieve and rely on the combinatorial version of the classical sieve. Note that for k = 2, the best result is due to Chen [1]. ...
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Ram Murty
Akshaa Vatwani
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... No counter-example has been found to date. Till now, the best theory result is J. R. Chen's 1966 theorem that every sufficiently large integer is the sum of a prime and the product of at most two primes[3]. A pair of primes (p, q) that sum to an even integer E = p + q are known as a Goldbach partition(Oliveira e Silva). ...
Divisor Goldbach Conjecture and its Partition Number
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Yan Kun
Hou-biao Li
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... Indeed, resolving the weak Goldbach conjecture will come through that every even number n ≥ 4 is the sum of at most 4 primes (see [21]). In 1973, using sieve theory methods Jing-run Chen (see [22]) showed that every sufficiently large even number can be written as a sum either of two primes or of one prime and one semiprime (i.e. a product of two primes), e.g. 100 = 23 + 7·11. In 1975, Montgomery and Vaughan (see [23]) showed that " most " even number is a sum of two primes. ...
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Emmanuel Markakis
N. Markakis
Christopher Provatidis
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... Although the conjecture has resisted our efforts, there has been spectacular partial progress. One well known result is Chen's theorem [3] that there are infinitely many primes such that p + 2 has at most two prime factors. In a different direction, building on the work of Goldston, Pintz, and Yldrm [5], it has recently been shown by Zhang [11] that there are bounded gaps between consecutive primes infinitely often. ...
Almost prime tripes and Chen's Theorem
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Xiannan Li
Roger Heath-Brown
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... [16,32,33, 34], so that any new clue for BGC formal proof may also help in TPC (formal) demonstration. Moreover, TPC may be weaker (and possibly easier to proof) than BGC (at least regarding the efforts toward the final formal proof) as the superior limit of the primes gap was recently " pushed " to be ≤246 [35][36,37,38]) has not been improved since a long time (at least by the set of proofs that are accepted in the present by the mainstream) except Cai's new proved theorem published in 2002 ( " There exists a natural number N such that every even integer n larger than N is a sum of a prime ≤ n 0.95 and a semi-prime " [39,40], a theorem which is a similar but a weaker statement than LC that hasn't a formal proof yet). ...
(VBGC - preprint - full version 1.5e - 23.02.2017 - 32 pages) The "Vertical" (generalization of) the Binary Goldbach's Conjecture (VBGC) as applied on “iterative” primes with (recursive) prime indexes (i-primeths)
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Jan 2017
Andrei-Lucian Dragoi
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... It should be noted that the best known asymptotic result in this area is that of Chen [1], who proved that every sufficiently large even number is the sum of a prime and another number which is the product of at most two primes. Chen's result is, at present, the closest one seems to be able to get towards a proof of the Goldbach conjecture, which famously asserts that every even integer greater than two can be written as the sum of two primes. ...
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Adrian Dudek
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... Using Vinogradov's method, Chudakov[2], Van der Corput[3], and Estermann[4]showed that " almost " all even numbers can be written as the sum of two primes. In 1973, Chen Jingrun using the methods of sieve theory showed that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime[6]. Considerable work has been done on Goldbach's weak conjecture that was finally proved in 2013 by Harald Helfgott[5]. His proof directly implies that every even number n ≥ 4 is the sum of at most four primes[7]. ...
Every even number of the form $2n=n^2+(p+1)^2-2np-d+1$ can be expressed as the sum of two primes
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Madieyna Diouf
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... Similarly, he showed that every sufficiently large even number is either the sum of two primes or a prime and a semi-prime, closely related to the Goldbach conjecture [2] [3]. ...
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Ali-John Hakan
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... of randomness is also fundamental in cognitive science and decision theory[Aerts. 2009, Kak. 1996. From a mathematical perspective, pseudo-randomness may be related to the complexity of certain number-theoretic properties[Cassaigne et al. 1999, Mauduit. 2002. Specifically, one may seek to define the unexpectedness of some properties of composition[Chen. 1978, Kak. 2014, Nicolas and Robin 1997that yield good randomness measure. In a recent paper, we showed that binary primes sequence can be used for computational hardening of pseudorandom sequences [Reddy and Kak 2016]. Here we go beyond that idea and generate pseudorandom sequences by folding the binary primes sequences using p columns. Thi ...
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Subhash Kak
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... A major achievement extending this result in which a more general system of linear equations is considered has been established by B. Green, T. Tao, and T. Ziegler (see[6],[7],[8]) and we refer the reader to[6, Theorem 1.8]for the precise statement. Another important achievement in this area is the well-known Chen's theorem[3]related to the twin prime conjecture. The theorem asserts that the equation x 1 − x 2 = 2 has infinitely many solutions ( 1 , p 2 ) where 1 has at most two prime factors and p 2 is prime. ...
Diophantine equations in semiprimes
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Shuntaro Yamagishi
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... Additionally, there are some arguments that Twin Prime Conjecture (TPC) [32] (which states that "there is an infinite number of twin prime (p) pairs of form ( ) , 2 p p + " ) may be also (indirectly) related to BGC as part of a more extended and profound conjecture [33,34,35], so that any new clue for BGC formal proof may also help in TPC (formal) demonstration. Moreover, TPC may be weaker (and possibly easier to proof) than BGC (at least regarding the efforts towards the final formal proof) as the superior limit of the primes gap was recently "pushed" to be ≤246 [36], but the Chen's Theorem I (that "every sufficiently large even number can be written as the sum of either 2 primes, OR a prime and a semiprime [the product of just 2 primes]" [37,38,39] ) has not been improved since a long time (at least by the set of proofs that are accepted in the present by the mainstream) except Cai's new proved theorem published in 2002 ("There exists a natural number N such that every even integer n larger than N is a sum of a prime ≤ n 0.95 and a semi-prime" [40,41] , a theorem which is a similar but a weaker statement than LC that hasn't a formal proof yet). ...
(VBGC - JAMCS - Original Research Article - 32 pages - 28.10.2017) The "Vertical" Generalization of the Binary Goldbach's Conjecture as Applied on "Iterative" Primes with (Recursive) Prime Indexes (i-primeths) (Journal of Advances in Mathematics and Computer Science [JAMCS] 25(2): 1-32, 2017; Article no.JAMCS.36895; ISSN: 2456-9968)
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Oct 2017
Andrei-Lucian Dragoi
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... Additionally, there are some arguments that Twin Prime Conjecture (TPC) [32] (which states that "there is an infinite number of twin prime (p) pairs of form ( ) , 2 p p + " ) may be also (indirectly) related to BGC as part of a more extended and profound conjecture [33,34,35], so that any new clue for BGC formal proof may also help in TPC (formal) demonstration. Moreover, TPC may be weaker (and possibly easier to proof) than BGC (at least regarding the efforts towards the final formal proof) as the superior limit of the primes gap was recently "pushed" to be ≤246 [36], but the Chen's Theorem I (that "every sufficiently large even number can be written as the sum of either 2 primes, OR a prime and a semiprime [the product of just 2 primes]" [37,38,39] ) has not been improved since a long time (at least by the set of proofs that are accepted in the present by the mainstream) except Cai's new proved theorem published in 2002 ("There exists a natural number N such that every even integer n larger than N is a sum of a prime ≤ n 0.95 and a semi-prime" [40,41] , a theorem which is a similar but a weaker statement than LC that hasn't a formal proof yet). ...
(VBGC - JAMCS - Original Research Article - 10.01.2017 - 32 pages) The “Vertical” Generalization of the Binary Goldbach’s Conjecture as Applied on “Iterative” Primes with (Recursive) Prime Indexes (i-primeths)
Article
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Jan 2017
Andrei-Lucian Dragoi
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... As to Goldbach conjecture, refer to Chen [1]. As to the twin prime number conjecture, refer to Zhang [3], Motohashi [2] and the references therein. ...
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Ze-Chun Hu
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陈景润工作的影响：
... There are many papers devoted to the study of problems involving primes and almost primes. For example, in 1973 J. R. Chen[4]established that there exist infinitely many primes p such that p+2 ∈ P 2. In 2000 Tolev[12]proved that for every sufficiently large integer N ≡ 3 (mod 6) the equation (1) has a solution in prime numbers p 1 , p 2 , p 3 such that p 1 + 2 ∈ P 2 , p 2 + 2 ∈ P 5 , p 3 + 2 ∈ P 7. Thereafter this result was improved by Matomäki and Shao[8], who showed that for every sufficiently large integer N ≡ 3 (mod 6) the equation (1) has a solution in prime numbers p 1 , p 2 , p 3 such that p 1 + 2, p 2 + 2, p 3 + 2 ∈ P 2. Recently Tolev[14]established that if N is sufficiently large, E > 0 is an arbitrarily large constant and 1 < c < 15 14 then the inequality[7]and Tolev[14]. ...
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