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A Diophantine inequality with five squares of Piatetski-Shapiro primes

S. I. Dimitrov

Abstract

Let $[\,\cdot\,]$ denote the floor function. Assume that $λ_1, λ_2, λ_3, λ_4, λ_5$ are nonzero real numbers, not all of the same sign, that $λ_1/λ_2$ is irrational, and that $η$ is a real number. Let $\frac{71}{72}<γ<1$ and $θ>0$. We prove that there exist infinitely many quintuples of primes $p_1,\, p_2,\, p_3,\, p_4,\, p_5$ satisfying the Diophantine inequality \begin{equation*} \big|λ_1p^2_1 + λ_2p^2_2 + λ_3p^2_3+ λ_4p^2_4 + λ_5p^2_5+η\big|<\big(\max p_j\big)^{\frac{71-72γ}{29}+θ}\,, \end{equation*} where $p_i=[n_i^{1/γ}]$, $i=1,\,2,\,3,\,4,\,5$. We also prove analogous theorems by raising the last variable in the inequality to the third and fourth powers.

A Diophantine inequality with five squares of Piatetski-Shapiro primes

Abstract

Let denote the floor function. Assume that are nonzero real numbers, not all of the same sign, that is irrational, and that is a real number. Let and . We prove that there exist infinitely many quintuples of primes satisfying the Diophantine inequality \begin{equation*} \big|λ_1p^2_1 + λ_2p^2_2 + λ_3p^2_3+ λ_4p^2_4 + λ_5p^2_5+η\big|<\big(\max p_j\big)^{\frac{71-72γ}{29}+θ}\,, \end{equation*} where , . We also prove analogous theorems by raising the last variable in the inequality to the third and fourth powers.
Paper Structure (14 sections, 18 theorems, 90 equations)

This paper contains 14 sections, 18 theorems, 90 equations.

Table of Contents

  1. Introduction and statement of the result
  2. Notations
  3. Auxiliary lemmas
  4. Initial steps
  5. Lower bound for $\mathbf{A_k(X)}$
  6. Upper bound for $\mathbf{J_1}$
  7. Upper bound for $\mathbf{J_5}$
  8. Lower bound for $\mathbf{J}$
  9. Estimation of $\mathbf{A_k(X)}$
  10. Upper bound for $\mathbf{B_k(X)}$
  11. Upper bound for $\mathbf{C_k(X)}$
  12. Proof of Theorem 1
  13. Proof of Theorem 2
  14. Proof of Theorem 3

Key Result

Theorem 1

Suppose that $\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is real. Let $\frac{71}{72}<\gamma<1$ and $\theta>0$. Then there exist infinitely many ordered quintuples of Piatetski-Sh

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 21 more