The value of additive forms at prime arguments

Roger J. Cook

Displaying similar documents to “The value of additive forms at prime arguments”

On the Piatetski-Shapiro-Vinogradov theorem

Angel Kumchev (1997)

Journal de théorie des nombres de Bordeaux

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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 ternary Diophantine inequality over primes

Roger Baker, Andreas Weingartner (2014)

Acta Arithmetica

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Let 1 < c < 10/9. For large real numbers R > 0, and a small constant η > 0, the inequality $| p ₁^{c} + p ₂^{c} + p ₃^{c} - R | < R^{- η}$ holds for many prime triples. This improves work of Kumchev [Acta Arith. 89 (1999)].

Shifted values of the largest prime factor function and its average value in short intervals

Jean-Marie De Koninck, Imre Kátai (2016)

Colloquium Mathematicae

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We obtain estimates for the average value of the largest prime factor P(n) in short intervals [x,x+y] and of h(P(n)+1), where h is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting $s_{q} (n)$ stand for the sum of the digits of n in base q ≥ 2, we show that if α is an irrational number, then the sequence ${(α s_{q} (P (n)))}_{n \in ℕ}$ is uniformly distributed modulo 1.

Sums of reciprocals of additive functions running over short intervals

J.-M. De Koninck, I. Kátai (2007)

Colloquium Mathematicae

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Letting f(n) = A log n + t(n), where t(n) is a small additive function and A a positive constant, we obtain estimates for the quantities $\sum_{x \leq n \leq x + H} 1 / f (Q (n))$ and $\sum_{x \leq p \leq x + H} 1 / f (Q (p))$ , where H = H(x) satisfies certain growth conditions, p runs over prime numbers and Q is a polynomial with integer coefficients, whose leading coefficient is positive, and with all its roots simple.

On a ternary Diophantine problem with mixed powers of primes

Alessandro Languasco, Alessandro Zaccagnini (2013)

Acta Arithmetica

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Let 1 < k < 33/29. We prove that if λ₁, λ₂ and λ₃ are non-zero real numbers, not all of the same sign and such that λ₁/λ₂ is irrational, and ϖ is any real number, then for any ε > 0 the inequality $| λ ₁ p ₁ + λ ₂ p ² ₂ + λ ₃ p ₃^{k} + ϖ | \leq {(m a x_{j} p_{j})}^{- (33 - 29 k) / (72 k) + ε}$ has infinitely many solutions in prime variables p₁, p₂, p₃.

On the sum of dilations of a set

Antal Balog, George Shakan (2014)

Acta Arithmetica

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We show that for any relatively prime integers 1 ≤ p < q and for any finite A ⊂ ℤ one has ${| p \cdot A + q \cdot A | \geq (p + q) | A | - (p q)}^{(p + q - 3) (p + q) + 1}$ .

The binary Goldbach conjecture with primes in arithmetic progressions with large modulus

Claus Bauer, Yonghui Wang (2013)

Acta Arithmetica

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It is proved that for almost all prime numbers $k \leq N^{1 / 4 - ϵ}$ , any fixed integer b₂, (b₂,k) = 1, and almost all integers b₁, 1 ≤ b₁ ≤ k, (b₁,k) = 1, almost all integers n satisfying n ≡ b₁ + b₂ (mod k) can be written as the sum of two primes p₁ and p₂ satisfying $p_{i} \equiv b_{i} (m o d k)$ , i = 1,2. For the proof of this result, new estimates for exponential sums over primes in arithmetic progressions are derived.

On the distribution of sparse sequences in prime fields and applications

Víctor Cuauhtemoc García (2013)

Journal de Théorie des Nombres de Bordeaux

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In the present paper we investigate distributional properties of sparse sequences modulo almost all prime numbers. We obtain new results for a wide class of sparse sequences which in particular find applications on additive problems and the discrete Littlewood problem related to lower bound estimates of the $L_{1}$ -norm of trigonometric sums.

On sum-sets and product-sets of complex numbers

József Solymosi (2005)

Journal de Théorie des Nombres de Bordeaux

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We give a simple argument that for any finite set of complex numbers $A$ , the size of the the sum-set, $A + A$ , or the product-set, $A \cdot A$ , is always large.

On Baker type lower bounds for linear forms

Tapani Matala-aho (2016)

Acta Arithmetica

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A criterion is given for studying (explicit) Baker type lower bounds of linear forms in numbers $1, Θ_{1}, . . ., Θ_{m} \in ℂ *$ over the ring $ℤ$ of an imaginary quadratic field . This work deals with the simultaneous auxiliary functions case.

The method of infinite ascent applied on $A^{4} \pm n B^{3} = C^{2}$

Susil Kumar Jena (2013)

Czechoslovak Mathematical Journal

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Each of the Diophantine equations $A^{4} \pm n B^{3} = C^{2}$ has an infinite number of integral solutions $(A, B, C)$ for any positive integer $n$ . In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when $A$ , $B$ and $C$ are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions $(A, B, C)$ of the Diophantine equation $a A^{3} + c B^{3} = C^{2}$ for any co-prime integer pair $(a, c)$ . ...