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Generalization of Romanoff's theorem

Artyom Radomskii

Abstract

We generalize Romanoff's theorem. Also, we obtain a result on sums related to Euler's totient function.

Generalization of Romanoff's theorem

Abstract

We generalize Romanoff's theorem. Also, we obtain a result on sums related to Euler's totient function.
Paper Structure (5 sections, 11 theorems, 102 equations)

This paper contains 5 sections, 11 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preparatory Lemmas
  4. Proof of theorems \ref{['T1']} and \ref{['T2']}
  5. Acknowledgements

Key Result

Theorem 1.1

Let $a$, $d$, and $s$ be integers with $a>1$, $d\geq 1$, and $s\geq 1$. Let be a polynomial with integer coefficients such that $b_d >0$ and $R: \mathbb{N}\to \mathbb{N}$. Then there is a constant $c=c(R, a, s)>0$, depending only on $R$, $a$, and $s$, such that for any positive integer $N$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 12 more