# On ternary Egyptian fractions with prime denominator

## TL;DR

This paper investigates the distribution of ternary Egyptian fractions with prime denominators, establishing asymptotic bounds on the sum of the counts of such representations for primes up to x.

## Contribution

It provides new asymptotic bounds on the sum of counts of ternary Egyptian fractions with prime denominators, advancing understanding of their distribution.

## Key findings

- Sum of A_3(p) over primes p up to x grows between x(log x)^3 and x(log x)^5
- Establishes asymptotic bounds for the number of ternary Egyptian fractions with prime denominators
- Contributes to the analytic number theory of Egyptian fractions and prime distributions

## Abstract

Given a positive integer $n$ we let $A_k(n)$ be the number of positive integers $a$ such that $\frac{a}{n}=\frac{1}{m_1}+\frac{1}{m_2}+\cdots+\frac{1}{m_k}$ for some $m_1,m_2,\ldots,m_k\in {\mathbb N}$. We show that $x(\log x)^3\ll \sum_{p\le x} A_3(p)\ll x(\log x)^5$ as $x\rightarrow\infty$.