Large Odd Integer Research Articles

A density version of the Vinogradov three primes theorem

We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 5/8, then all sufficiently large odd positive integers can be written as the sum of three primes in A. The constant 5/8 in this statement is the best possible.

One Prime, Two Squares of Primes and Powers of 2

It is proved that every sufficiently large odd integer is the sum of one prime, two squares of primes and 35 powers of 2. This improves a previous result with 35 replaced by 83.

ON PAIRS OF ONE PRIME, TWO PRIME SQUARES AND POWERS OF 2

In this short paper, it is proved that every pair of large positive odd integers satisfying some necessary conditions can be represented in the form of a pair of one prime, two prime squares and k powers of 2. In particular, we have k = 332.

Cubes of primes and almost prime

It is proved that every sufficiently large odd integer n can be written as n = x + p 1 3 + p 2 3 + p 3 3 + p 4 3 where p 1 , p 2 , p 3 , p 4 are primes, and x has at most two prime factors.

On a Theorem of Prachar Involving Prime Powers

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Two results on powers of 2 in Waring–Goldbach problem

In this paper, it is proved that every sufficiently large odd integer is a sum of a prime, four cubes of primes and 106 powers of 2. What is more, every sufficiently large even integer is a sum of two squares of primes, four cubes of primes and 211 powers of 2.

The number of powers of 2 in a representation of large odd integers

The number of powers of 2 in a representation of large odd integers

A density version of Vinogradov's three primes theorem

Let 𝒫 denote the set of all primes. Suppose that P1, P2, P3 are three subsets of 𝒫 with d𝒫(P1) + d𝒫(P2) + d𝒫(P3) > 2, where d𝒫(Pi) is the lower density of Pi relative to 𝒫. We prove that for every sufficiently large odd integer n, there exist pi ∈ Pi such that n = p1 + p2 + p3.

On linear equations with prime variables of special type

Let P k denote any integer with no more than k prime factors, counted according to multiplicity. It is proved that for every sufficiently large odd integer n ≢ 1 mod 6 , the equation p 1 + p 2 + p 3 = n is solvable in prime variables p 1 , p 2 , p 3 such that p 1 + 2 = P 2 , p 2 + 2 = P 2 ′ , and for almost all sufficiently large even integer n ≢ 2 mod 6 , the equation p 1 + p 2 = n is solvable in prime variables p 1 , p 2 such that p 1 + 2 = P 2 .

Integers represented as the sum of one prime, two squares of primes and powers of 2

In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 83 83 powers of 2 2 .

Hua’s theorem with nine almost equal prime variables

We sharpen Hua’s result by proving that each sufficiently large odd integer N can be written as $$ N = p_1^3 + \cdots + p_9^3 with \left| {p_j - \sqrt[3]{{N/9}}} \right| \leqq U = N^{\tfrac{1} {3} - \tfrac{1} {{198}} + \varepsilon } $$ , where pj are primes. This result is as good as what was previously derived from the Generalized Riemann Hypothesis.

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.

Sets of permutations that generate the symmetric group pairwise

The paper contains proofs of the following results. For all sufficiently large odd integers n, there exists a set of 2 n − 1 permutations that pairwise generate the symmetric group S n . There is no set of 2 n − 1 + 1 permutations having this property. For all sufficiently large integers n with n ≡ 2 mod 4 , there exists a set of 2 n − 2 even permutations that pairwise generate the alternating group A n . There is no set of 2 n − 2 + 1 permutations having this property.

On sums of three integers with a fixed number of prime factors

Let N be sufficiently large odd integer. It is proved that the equation N=n1+n2+n3 has solutions, where ni has a fixed number of prime factors, and an asymptotic formula holds for the number of representations.

ON A CASE OF ALL PAIRS OF POLYNOMIAL ZEROS BAD

In this paper, we show, for a positive integer m and a large odd integer n, the polynomial equation (equation omitted) has a real zero on <TEX>$((n+1)^{1+\frac{1}{m}},{\;}(n+1)^{1+\frac{1}{m}}+{\frac{1}{2}})$</TEX> and <TEX>$((n+1)^{1+\frac{1}{m}}+{\frac{1}{2}},{\;}(n+2)^{1+\frac{1}{m}})$</TEX> respectively.

On the Waring-Goldbach Problem for Fourth and Fifth Powers

It is shown that every sufficiently large integer congruent to 14 modulo 240 may be written as the sum of 14 fourth powers of prime numbers, and that every sufficiently large odd integer may be written as the sum of 21 fifth powers of prime numbers. The respective implicit bounds 14 and 21 improve on the previous bounds 15 (following from work of Davenport) and 23 (due to Thanigasalam). These conclusions are established through the medium of the Hardy-Littlewood method, the proofs being somewhat novel in their use of estimates stemming directly from exponential sums over prime numbers in combination with the linear sieve, rather than the conventional methods which ‘waste’ a variable or two by throwing minor arc estimates down to an auxiliary mean value estimate based on variables not restricted to be prime numbers. In the work on fifth powers, a switching principle is applied to a cognate problem involving almost primes in order to obtain the desired conclusion involving prime numbers alone. 2000 Mathematics Subject Classification: 11P05, 11N36, 11L15, 11P55.

On the Ternary Goldbach Problem with Primes pi Such That pi + 2 are Almost-Primes

We prove that every sufficiently large odd integer can be represented as the sum of three primes p1, p2, p3, such that p1 + 2, p2 + 2 and p3 + 2 are almost-primes.

Checking the odd Goldbach conjecture up to 10²⁰

Vinogradov’s theorem states that any sufficiently large odd integer is the sum of three prime numbers. This theorem allows us to suppose the conjecture that this is true for all odd integers. In this paper, we describe the implementation of an algorithm which allowed us to check this conjecture up to 10 20 10^{20} .

On the Piatetski-Shapiro-Vinogradov theorem

In this paper we consider the asymptotic formula for the number of the solutions of the equation p 1 +p 2 +p 3 =N where N is an odd integer and the unknowns p i are prime numbers of the form p i =[n 1/γ i ]. We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case γ 1 =γ 2 =γ 3 =γ our theorem implies that every sufficiently large odd integer N may be written as the sum of three Piatetski-Shapiro primes of type γ for 50/53 < γ < 1.

A hybrid of theorems of Vinogradov and Piatetski-Shapiro

It was proved by Vinogradov that every sufficiently large odd integer can be written as the sum of three primes. We show that this remains the case when the primes so utilized are restricted to an explicit thin set. One may take, for example, the «Piatetski-Shapiro primes» p = [n 1/γ ] with any γ>20/21. By a similar argument it would follow that, for arbitrary θ, 0 0, one may take the set of primes for which {p θ }<P −λ