On additive decompositions of the set of primes

We improve upon Hofmann and Wolkes bounds on the conceivable additive decompositions of the set of primes. We also prove that there cannot be a decomposition, where one set is a shifted copy of the other. This answers an open question of Hofmann and Wolke.We improve upon Hofmann and Wolkes bounds (see [2]) on the conceivable additive decompositions of the set of primes. Their result corresponds to the special case r 2 of our new bounds. We overcome a difficulty that arose in the application of the large sieve method in their work. We also prove that there cannot be a decomposition, where one set is a shifted copy of the other. This answers an open question of Hofmann and Wolke.

“…The proof is immediate when applying Montgomerys large sieve (see [1], Theorem 5.3.2) to the interval x p Y x, as was done in [2]. For the evaluation of L see [3].…”

mentioning

confidence: 99%

“…We improve upon Hofmann and Wolkes bounds (see [2]) on the conceivable additive decompositions of the set of primes. Their result corresponds to the special case r 2 of our new bounds.…”

mentioning

confidence: 99%

A remark on Hofmann and Wolke's additive decompositions of the set of primes

Elsholtz

2001

“…For an arbitrary prime pattern of this size, Hofmann and Wolke [9] implicitly obtained the following bound:…”

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confidence: 99%

Upper bounds for prime k -tuples of size log N and oscillations

Elsholtz

2004

We prove the estimatefor the number E k (N ) of k-tuples (n + a 1 , . . . , n + a k ) of primes not exceeding N , for k of size c 1 log N and N sufficiently large. A bound of this strength was previously known in the special case n − 2 i (1 i < log n log 2 ) only, (Vaughan, 1973). For general a i this is an improvement upon the work of Hofmann and Wolke (1996).The number of prime tuples of this size has considerable oscillations, when varying the prime pattern.

“…For any set A , we will use A (x) to denote the number of elements of A which are Ϲ x. With this notation A. Hofmann and D. Wolke [1] showed that if A is a set such that A + (A + 1) = P , then x log xx 1/2 . In this note we will show that no such A exists.…”

mentioning

confidence: 99%

“…For any set A , we will use A (x) to denote the number of elements of A which are Ϲ x. With this notation A. Hofmann and D. Wolke [1] …”

mentioning

confidence: 99%

On additive decompositions of the set of primes

Puchta¹

2002

We show that there is no set A of integers, such that (P − 1) ֤ A + A ֤ P ∪ (P − 1), where P denotes the set of primes.Let N be a set of integers. Following Wirsing[2], N is called additively decomposable, if there are sets A , B, such that A + B = {a + b|a ∈ A , b ∈ B} = N and both A and B have at least two elements. He showed that if N is probabilistically constructed with P(n ∈ N ) = 1 2 , we have with probability 1 that N is indecomposable, where N is any set which equals N up to finitely many elements. Let P be the set of primes. It is still unknown, whether P is decomposable. For any set A , we will use A (x) to denote the number of elements of A which are Ϲ x. With this notation A. Hofmann and D. Wolke [1] showed that if A is a set such that A + (A + 1) = P , then x log xx 1/2 . In this note we will show that no such A exists. Theorem 1. There is no setEspecially, if we had A + (A + 1) = P , the set A would contradict our theorem.P r o o f. Define A to be the set of residue classes (mod 30), such that A contains infinitely many elements from this class, B be the corresponding set for A + A and P for P. Then by the prime number theorem for arithmetic progressions and our assumption we get