Waring–Goldbach problem in short intervals

Let k ≥ 2 and s be positive integers. Let θ ∈ (0, 1) be a real number. In this paper, we establish that if s > k ( k + 1) and θ > 0.55, then every sufficiently large natural number n , subject to certain congruence conditions, can be written as n = p 1 k + ⋯ + p s k , , where p i (1 ≤ i ≤ s )...

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Bibliographic Details
Published inIsrael journal of mathematics Vol. 261; no. 2; pp. 637 - 669
Main Author Wang, Mengdi
Format Journal Article
LanguageEnglish
Published Jerusalem The Hebrew University Magnes Press 01.06.2024
Springer Nature B.V
Subjects
Algebra
Analysis
Applications of Mathematics
Congruences
Group Theory and Generalizations
Integers
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Real numbers
Theoretical
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Summary:Let k ≥ 2 and s be positive integers. Let θ ∈ (0, 1) be a real number. In this paper, we establish that if s > k ( k + 1) and θ > 0.55, then every sufficiently large natural number n , subject to certain congruence conditions, can be written as n = p 1 k + ⋯ + p s k , , where p i (1 ≤ i ≤ s ) are primes in the interval ( ( n s ) 1 k − n θ k , ( n s ) 1 k + n θ k ] . The second result of this paper is to show that if s > k ( k + 1 ) 2 and θ > 0.55, then almost all integers n , subject to certain congruence conditions, have the above representation.
Bibliography:ObjectType-Article-1
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content type line 14
ISSN:0021-2172
1565-8511
1565-8511
DOI:10.1007/s11856-023-2590-9