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Sums of four generalized polygonal numbers of almost prime length

Bosco Ng

Abstract

In this paper, we consider sums of four generalized polygonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restriction on m modulo 30, we show that for n sufficiently large, it can be represented as such a sum, where the parameters are restricted to have at most 988 prime factors.

Sums of four generalized polygonal numbers of almost prime length

Abstract

In this paper, we consider sums of four generalized polygonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restriction on m modulo 30, we show that for n sufficiently large, it can be represented as such a sum, where the parameters are restricted to have at most 988 prime factors.
Paper Structure (8 sections, 15 theorems, 154 equations)

This paper contains 8 sections, 15 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eisenstein series components
  4. Computation of local densities of $X^\textbf{d}$.
  5. Ratio of Eisenstein series.
  6. Error from Cusp form
  7. Main term and error term from sieving
  8. Sieving and proof of theorem 1.1

Key Result

Theorem 1.1

Suppose that $m$ is odd, $m-4\not\equiv 0 \pmod3$ and $m-4\not\equiv 0 \pmod5.$ Further suppose $\text{ord}_p(\prod_{j=1}^4\alpha_j)\leq 1$ for any odd prime $p.$ Then for $n$ sufficiently large, the equation (1.1) is solvable with each $x_j$ containing at most $988$ prime factors.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • Lemma 3.5
  • Lemma 3.6
  • proof
  • ...and 16 more