Goldbach Conjecture Research Articles

A Goldbach Theorem for Laurent Polynomials with Positive Integer Coefficients

We establish an analogue of the Goldbach conjecture for Laurent polynomials with positive integer coefficients.

Implications of Ramsey Choice principles in ZF$\mathsf {ZF}$

AbstractThe Ramsey Choice principle for families of ‐element sets, denoted , states that every infinite set has an infinite subset with a choice function on . We investigate for which positive integers and the implication is provable in . It will turn out that beside the trivial implications , under the assumption that every odd integer is the sum of three primes (known as ternary Goldbach conjecture), the only non‐trivial implication which is provable in is .

A Detailed Proof of the Strong Goldbach Conjecture Based on Partitions of a New Formulation of a Set of Even Numbers

The Strong Goldbach's conjecture, a fundamental problem in Number Theory, asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers. Despite significant efforts over centuries, this conjecture remains unproven, challenging the core of mathematics. The known algorithms for attempting to prove or verify the conjecture on a given interval [a,b] consist of finding two sets of primes Pi and Pj such that Pi+Pj cover all the even numbers in the interval [a,b]. However, the traditional definition of an even number as 2n for n ∈ ℕ (where ℕ is the set of natural numbers), has not provided mathematicians with a straightforward method to obtain all Goldbach partitions for any even number of this form. This paper introduces a novel approach to the problem, utilizing all odd partitions of an even number of a new formulation of the form Eij = ni + nj + (nj - ni)n or alln ∈ ℕ. By demonstrating that there exist at least a pair of prime numbers in these odd partitions, the fact that the sum of any two prime numbers is even and there exists infinitely many prime numbers, this paper provides a compelling proof of the conjecture. This breakthrough not only solves a long-standing mathematical mystery but also sheds light on the structure of prime numbers.

A Necessary and Sufficient Condition for Proof of the Binary Goldbach Conjecture. Proofs of Binary Goldbach, Andrica and Legendre Conjectures. Notes on the Riemann Hypothesis

A Necessary and Sufficient Condition for Proof of the Binary Goldbach Conjecture. Proofs of Binary Goldbach, Andrica and Legendre Conjectures. Notes on the Riemann Hypothesis

Analyzing twin primes, Goldbach's strong conjecture and Polignac's conjecture

Here we analyze three well-known conjectures: (i) the existence of infinitely many twin primes, (ii) Goldbach's strong conjecture, and (iii) Polignac's conjecture. We show that the three conjectures are related to each other. In particular, we see that in analysing the validity of Goldbach's strong conjecture, one must consider also the existence of an infinite number of twin primes. As a consequence of how we approach this analysis, we also observe that if this conjecture is true, then so is Polignac's conjecture. Our first step is an analysis of the existence of infinitely many twin prime numbers. For this, using the formula 4((n−1)!+1) ≡−n (mod n(n + 2)) – satisfied if and only if (n, n + 2) are twin primes –, together with Wilson's theorem, we obtain conditions that must be met for two numbers to be twin primes. Our results, obtained from an analytic and functional study, lead us to conclude that there may exist infinitely many twin primes. Next, we consider the validity of Goldbach's strong conjecture. After showing that the conjecture is true for the first even numbers, we notice a pattern that we analyze for any even number, reducing it to three cases: (i) when the even number 2n is two times a prime number n; (ii) when the even number 2n is such that n=2m, with 2m-1 and 2m+1 twin primes; (iii) all other cases, i.e., for any 2n even number ∀n∈N with n > 1, n prime or not, with independence of n=2m being 2m-1 and 2m+1 twin primes or not. In this last case, we show that one can always find a certain r∈N such that 1 < r < n satisfying that n − r and n + r are primes, so that their sum is 2n. In this case, we use the reduction to absurd method, and our results lead us to conclude that Goldbach's strong conjecture is true to the best of our calculations, and Polignac's conjecture as well.

The Goldbach Conjecture Is True

The Goldbach Conjecture Is True

On the binary Goldbach conjecture under primorial-based constraints

On the binary Goldbach conjecture under primorial-based constraints

Confirming Buya's and Bezaleel's Proof of the Binary Goldbach Conjecture using Bertrand's Postulate Paper Title

Confirming Buya's and Bezaleel's Proof of the Binary Goldbach Conjecture using Bertrand's Postulate Paper Title

Natural Numbers and the Strong Goldbach Conjecture

Natural Numbers and the Strong Goldbach Conjecture

A logical approach to validate the Goldbach conjecture: Part I

This paper is a first of the series of three papers which provide a general proof to validate the Goldbach conjecture. This conjecture states that every even number can be expressed as a summation of two prime numbers. At the onset, concept of successive-addition of‐digits‐of‐an‐integer‐number (SADN) and its properties in terms of basic algebraic functions like addition, multiplication and subtraction are discussed. SADN classifies odd numbers into 3 sequences—the S1, S3 and S5 sequences—which comprise of odd numbers having SADN 7 or 4 or 1; SADN 3 or 9 or 6 and SADN 5 or 2 or 8 respectively. The S1 and S5 sequences are of interest in the analysis. Furthermore, composites on the S1 sequence are derived as products of intra-sequence elements of the S1 and S5 sequences while composites on the S5 sequence are derived as products of inter-sequence elements of the S1 and S5 sequences. SADN also shows why such combinations for even numbers of SADN (1, 4, 7) will be found on the S5 sequence while those for even numbers of SADN (2, 5, 8) will lie on the S1 sequence and both the sequences have a role to play in identifying the prime number combinations for even numbers with SADN (3, 6, 9). Thereafter, analysis moves to calculating the total number of combinations for a given even number that would include combinations in the nature of two composites (c1 + c2); prime & composite (p + c) and two primes (p1 + p2). Identifying the total number of c1 + c2 and p + c combinations yield the number of p1 + p2 combinations. The logic employed in present discussion shows that at least one such p1 + p2 combination exists for the even numbers having SADN digit within 1 to 9. This encompasses all even numbers and hence generalizes this method for all even numbers.

Direct reference and the Goldbach puzzle

AbstractSo‐called Neo‐Russellians, such as Salmon, Braun, Crimmins, and Perry, hold that the semantic content of ‘ is ’ in a context is the singular proposition o, P, where is the referent of the name in , and is the property expressed by the predicate in . This is also known as the Neo‐Russellian theory. Using truth ascriptions with names designating propositions, such as ‘Goldbach's conjecture’, in this paper, I will argue that, together with highly plausible principles regarding a priori knowledge, the Neo‐Russellian theory leads to unacceptable consequences. I will call this ‘the Goldbach puzzle’. Since the solution to the Goldbach puzzle cannot be to reject the discussed principles regarding a priori knowledge, the puzzle will undermine the Neo‐Russellian theory.

Overview: How Does the Paper "The Continuity of Prime Numbers Can Lead to the Continuity of Even Numbers" Relate to the Goldbach Conjecture

The paper "The continuity of prime numbers can lead to the continuity of even numbers" proposes three clever methods to prove the Goldbach conjecture from another path that humans never anticipated and was also the correct path. "The continuity of prime numbers can lead to the continuity of even numbers" A preprint has previously been published (Xie ling. 2021).

Optimization applications of Goldbach's conjecture

Goldbach's conjecture is an unsolved legendary problem. In this paper, we investigate an optimization problem as an attempt to discuss potential applications of the conjecture. In an optimization perspective, the research question, modelled from design of base components, is to select a minimum number of primes to span a given set of target even integers. We formally define the problem, propose integer programming formulations, and develop solution procedures, which are then tested through a computational study.

A New Formulation of a Set of Even Numbers

We present a new definition of an Even number as an integer, E, of the form E = (n1 +n2 )+(n2.+n1)n, ˅ n1 , n2, n ϵ N. members of natural numbers. We have shown the new set representation of even numbers holds for all natural numbers. This new representation opens up new doors to the solution of the Strong Goldbach Conjecture. The proofs obtained here will have impressive application in partitioning a given even number into all pairs of odd numbers.

The Order of Numbers and the Goldbach Conjecture

The Order of Numbers and the Goldbach Conjecture

Proofs of Beal’s Conjecture, Fermat’s Conjecture, Collatz Conjecture and Goldbach Conjecture

In this article the elementary mathematical methods are used to prove Beal’s Conjecture, Fermat’s Conjecture, Collatz Conjecture and Goldbach Conjecture.

Siegel Zeros, Twin Primes, Goldbach’s Conjecture, and Primes in Short Intervals

Abstract We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular, we prove for $$ \begin{align*} & \sum_{n \leq X} \Lambda(n) \Lambda(\pm n+h) \end{align*}$$an asymptotic formula that holds uniformly for $h = O(X)$. Such an asymptotic formula has been previously obtained only for fixed $h$ in which case our result quantitatively improves those of Heath-Brown (1983) and Tao and Teräväinen (2021). Since our main theorems work also for large $h$, we can derive new results concerning connections between Siegel zeros and the Goldbach conjecture and between Siegel zeros and primes in almost all very short intervals.

Algerbraic Proof Of Goldbach's Conjecture

Algerbraic Proof Of Goldbach's Conjecture

Advances in additive number theory

We obtain sufficient conditions to know if given a positive even integer number and a set of positive integer numbers being all even or all odd, such a number can be expressed as sum of two elements of this set. As consequence we obtain a result which, when applied to the prime numbers set, would prove Goldbach's Conjecture provided that certain conditions are satisfied. These hypothesis include Prime Consecutive Conjecture, which is a generalized form of Twin Prime Conjecture. In addition, we extend these results to sets of positive real numbers, even for two different sets. We also obtain a recurrent approximation of \pi(x) for enough large real x, being \pi the distribution function of the prime number set, which uses whichever expression of x as product of enough large factors. We also state this approximation in a more general context, give upper and lower bounds for the error, and show that this approximation is asymptotically equivalent to \pi(x).

The continuity of prime numbers can lead to even continuity (Relationship with Gold Bach’s conjecture)

N continuous prime numbers can combine a group of continuous even numbers. If an adjacent prime number is followed, the even number will continue. For example, if we take the prime number 3, we can get an even number 6. If we follow an adjacent prime number 5, we can get even numbers by using 3 and 5: 6, 8 and 10. If a group of continuous prime numbers 3, 5, 7, 11, ..., P, we can get a group of continuous even numbers 6, 8, 10, 12,..., 2n. Then if an adjacent prime number q is followed, the Original group of even numbers 6, 8, 10, 12,..., 2n will be finitely extended to 2(n + 1) or more adjacent even numbers. My purpose is to prove that the continuity of prime numbers will lead to even continuity as long as 2(n + 1) can be extended. If the continuity of even numbers is Discontinuous, it violates the Bertrand Chebyshev theorem of prime Numbers. Because there are infinitely many prime numbers: 3, 5, 7, 11,... We can get infinitely many continuous even numbers: 6, 8, 10, 12,... Get: Gold Bach conjecture holds. 2020 Mathématiques Subjectif Classification: 11P32, 11U05, 11N05, 11P70. Research ideas: If the prime number is continuous and any pairwise addition can obtain even number continuity, then Gold Bach’s conjecture is true. Human even number experiments all get (prime number + prime number). I propose a new topic: the continuity of prime numbers can lead to even continuity. I designed a continuous combination of prime numbers and got even continuity. If the prime numbers are combined continuously and the even numbers are forced to be discontinuous, a breakpoint occurs. It violates Bertrand Chebyshev's theorem. It is proved that prime numbers are continuous and even numbers are continuous. The logic is: if Gold Bach's conjecture holds, it must be that the continuity of prime numbers can lead to the continuity of even numbers. Image interpretation: turn Gold Bach’s conjecture into a ball, and I kick the ball into Gold Bach’s conjecture channel. There are several paths in this channel and the ball is not allowed to meet Gold Bach’s conjecture conclusion in each path. This makes the ball crazy, and the crazy ball must violate Bertrand Chebyshev's theorem.