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Exploring the Representation of Large Positive Integers as Sums of Prime Powers and Integer Powers: Analysis with Positive Density Subsets

Meng Gao

Abstract

In this paper, we use the transference principle to investigate the representation of sufficiently large positive integers as the sum of prime powers and integer powers, where the primes are drawn from a positive density subset of the set of all primes , and the integer powers are drawn from a positive density subset of k-th powers.

Exploring the Representation of Large Positive Integers as Sums of Prime Powers and Integer Powers: Analysis with Positive Density Subsets

Abstract

In this paper, we use the transference principle to investigate the representation of sufficiently large positive integers as the sum of prime powers and integer powers, where the primes are drawn from a positive density subset of the set of all primes , and the integer powers are drawn from a positive density subset of k-th powers.
Paper Structure (10 sections, 37 theorems, 6 equations)

This paper contains 10 sections, 37 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. $k\in \mathbb{N}\setminus\{1,2,4,8,9\}$
  4. Definitions
  5. Mean value estimate
  6. Conclusions
  7. $k\in \{4,8,9\}$
  8. Definitions
  9. Mean value estimate
  10. Conclusions

Key Result

Theorem 1.1

Let $s_{1}, s_{2} \in \mathbb{N},\ k\in \mathbb{N}\setminus\{1,2,4,8,9\},\ s_{1}\geq 16k\omega(k)+4k+4+s_{2}$ and $s_{1}+s_{2}>k^{2}+k$ . Let $\delta_{A}>1-1/2k$ and $k\delta_{A}+\mathcal{Z}_{k}\delta_{B}^{k}>\mathcal{Z}_{k}+k-1$. Then, for all sufficiently large integers $n \equiv s_{1}+s_{2} \ (\b where $p_{i} \in A$ for all $i \in \{1,\ldots,s_{1}\}$ and $n_{j}^{k} \in B$ for all $j \in \{s_{1}

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Proposition 1.9
  • Remark 1.10
  • ...and 46 more