Goldbach Problem Research Articles

On the singular series in the Waring–Goldbach problem with shifted primes

On the singular series in the Waring–Goldbach problem with shifted primes

On the singular series in the Waring–Goldbach problem with shifted primes

On the singular series in the Waring–Goldbach problem with shifted primes

A result on Waring–Goldbach problem for cubes

In this paper, we prove that, with at most O(N5/6+e) exceptions, all positive integers n ⩽ N can be written as sums of a cube and four cubes of primes.

An improvement on Waring–Goldbach problem for unlike powers

Let pi be prime numbers. In this paper, it is proved that for any integer k≧5, with at most \(O\big(N^{1-\frac{1}{3k\times2^{k-2}}+\varepsilon}\big)\) exceptions, all positive even integers up to N can be expressed in the form \(p_{2}^{2}+p_{3}^{3}+p_{5}^{5}+p_{k}^{k}\). This improves the result \(O\big(\frac{N}{\log^{c}N}\big)\) for some c>0 due to Lu and Shan [12], and it is a generalization for a series of results of Ren and Tsang [15], [16] and Bauer [1–4] for the problem in the form \(p_{2}^{2}+p_{3}^{3}+p_{4}^{4}+p_{5}^{5}\). This method can also be used for some other similar forms.

The binary Goldbach problem with arithmetic weights attached to one of the variables

We study the binary Goldbach problem with arithmetic weights attached to one of the variables.

On a Diophantine problem with two primes and s powers of two

We refine a recent result of Parsell on the values of the form $\lambda_1p_1 + \lambda_2p_2 + \mu_1 2^{m_1} + ...m + \mu_s 2^{m_s}, $ where $p_1,p_2$ are prime numbers, $m_1,...c, m_s$ are positive integers, $\lambda_1 / \lambda_2$ is negative and irrational and $\lambda_1 / \mu_1$, $\lambda_2/\mu_2 \in \Q$.

The ternary Goldbach problem with arithmetic weights attached to two of the variables

We consider the ternary Goldbach problem with two prime variables of the form k 2 + m 2 + 1 and find an asymptotic formula for the number of its solutions.

A note on the Waring–Goldbach problem

The focus of this paper will be the extension of the Waring–Goldbach problem to all sufficiently large integers, without congruence restrictions. By reintroducing the effect of small primes, we are able to consider questions which more naturally resemble Waring's problem and the Goldbach conjecture. We extend the results of S.S. Pillai by considering the problem without the use of zero as an addend and we give a small improvement on the number of additional terms required.

THE TERNARY GOLDBACH PROBLEM WITH PRIMES FROM ARITHMETIC PROGRESSIONS

We establish Bombieri-Vinogradov's type result for the number of solutions of the ternary Goldbach problem with primes from arithmetic progressions.

Rank-one quadratic twists of an infinite family of elliptic curves

A conjecture of Goldfeld implies that a positive proportion of quadratic twists of an elliptic curve E /ℚ has (analytic) rank 1. This assertion has been confirmed by Vatsal [Math. Ann. 311: 791–794, 1998] and the first author [Acta Arith. 114: 391–396, 2004] for only two elliptic curves. Here we confirm this assertion for infinitely many elliptic curves E /ℚ using the Heegner divisors, the 3-part of the class groups of quadratic fields, and a variant of the binary Goldbach problem for polynomials.

On the ternary Goldbach problem with primes in arithmetic progressions having a common modulus

Pour ∈ > 0 et n impair suffisamment grand, nous montrons que, pour presque tout k < R := n 1/5-e :, il existe une representation n = p 1 + p 2 + p 3 avec des nombres premiers p i ≡ b i modulo k pour presque tout triplet admissible b 1 , b 2 , b 3 de residus modulo k.

Additive Representation in Thin Sequences, VII: Restricted Moments of the Number of Representations

Let $R_{s,k}(n)$ denote the number of representations of $n$ as the sum of $s$ positive integral $k$th powers. We develop methods to establish asymptotic formulae for moments of $R_{s,k}(n)$ in which $n$ is restricted to a thin sequence consisting of values of a given polynomial. As a first example, we discuss the case of cubes, and present conclusions for sums of 6 and 7 cubes in which the polynomial is quadratic, and for sums of 7 cubes in which the polynomial is cubic. We also consider the case wherein the exponent $k$ is large, and briefly describe corresponding results for the binary Goldbach problem.

A mean value theorem on the binary Goldbach problem and its application

We study the binary Goldbach problem with one prime number in a given residue class, and obtain a mean value theorem. As an application, we prove that for almost all sufficiently large even integers n satisfying n ≢ 2(mod 6), the equation p1 + p2 = n is solvable in prime variables p1, p2 such that p1 + 2 = P3, and for every sufficiently large odd integer \({\bar n}\) satisfying \({\bar n}\) ≢ 1(mod 6), the equation p1 + p2 + p3 = \({\bar n}\) is solvable in prime variables p1, p2, p3 such that p1 + 2 = P2, p2 + 2 = P3. Here Pk denotes any integer with no more than k prime factors, counted according to multiplicity.

The binary Goldbach problem with one prime of the form [formula omitted

We prove that almost all integers n ≡ 0 or 4 (mod 6) can be written in the form n = p 1 + p 2 , where p 1 = k 2 + l 2 + 1 with ( k , l ) = 1 . The proof is an application of the half-dimensional and linear sieves with arithmetic information coming from the circle method and the Bombieri–Vinogradov prime number theorem.

Refinements of Goldbach's conjecture,and the generalized Riemann hypothesis

We present three remarks on Goldbach's problem. First we suggest a refinement of Hardy and Littlewood's conjecture for the number of representations of $2n$ as the sum of two primes positing an estimate with a very small error term. Next we show that if a strong form of Goldbach's conjecture is true then every even integer is the sum of two primes from a rather sparse set of primes. Finally we show that an averaged strong form of Goldbach's conjecture is equivalent to the Generalized Riemann Hypothesis; as well as a similar equivalence to estimates for the number of ways of writing integers as the sum of $k$ primes.

Waring–Goldbach Problem for Unlike Powers

In this paper, it is proved that with at most $$ O{\left( {N^{{\frac{{65}} {{66}}}} } \right)} $$ exceptions, all even positive integers up to N are expressible in the form $$ p^{2}_{2} + p^{3}_{3} + p^{4}_{4} + p^{5}_{5} $$ . This improves a recent result $$ O{\left( {N^{{\frac{{19193}} {{19200}} + \varepsilon }} } \right)} $$ due to C. Bauer.

On the number of representations in the ternary Goldbach problem with one prime number in a given residue class

For J 3 ( n ) = ∑ p 1 + p 2 + p 3 = n p 1 ≡ a 1 ( mod q 1 ) log p 1 log p 2 log p 3 , it is shown that ∑ q 1 ⩽ n θ max ( a 1 , q 1 ) = 1 J 3 ( n ) - n 2 2 ϕ ( q 1 ) S 3 ( n ) ≪ A n 2 ( log n ) - A for any A and any θ < 1 / 2 , what improves a work of Tolev; S 3 ( n ) is the corresponding singular series. A special form of a sieve of Montgomery is used.

On the Waring–Goldbach problem for seventh powers

We use sieve theory and recent estimates for Weyl sums over almost primes to prove that every sufficiently large even integer is the sum of 46 46 seventh powers of prime numbers.

On the Waring–Goldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers

AbstractWe investigate exceptional sets in theWaring–Goldbach problem. For example, in the cubic case, we show that all but O(N79/84+∊) integers subject to the necessary local conditions can be represented as the sum of five cubes of primes. Furthermore, we develop a new device that leads easily to similar estimates for exceptional sets for sums of fourth and higher powers of primes.

The quadratic Waring–Goldbach problem

It is conjectured that Lagrange's theorem of four squares is true for prime variables, i.e. all positive integers n with n≡4 ( mod 24) are the sum of four squares of primes. In this paper, the size for the exceptional set in the above conjecture is reduced to O(N 3 8 +ε ) .